Spectral triangles of Zakharov-Shabat operators in ... · O—n k–for all kÆ0; and Trubowitz...

Spectral triangles of Zakharov-Shabatoperators in weighted Sobolev spaces

Jan-Cornelius MolnarDecember 19, 2011

b

b

b

b

µn

τn

λ+n

λ−n

Supervised by Prof. Dr. Jürgen PöschelInstitut für Analysis, Dynamik und Modellierung, Universität Stuttgart

Contents

1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 The forward problem 112 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Gap estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 The inverse problem 275 Adapted Fourier coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 Abstract regularity results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 Lower gap length bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 Regularity: Potentials of real or imaginary type . . . . . . . . . . . . . . . . . . 389 Regularity: Potentials of general type . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Extensions 4210 A note on non-monotone weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211 Second order gap estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Weighted Sobolev spaces 51A Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51B An inequality for products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54C Construction of unbounded weights . . . . . . . . . . . . . . . . . . . . . . . . . 55D The Cut-Off Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57E Equivalent weights and monotonicity . . . . . . . . . . . . . . . . . . . . . . . . 58

5 Background Information 60F Analytic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60G Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63H Auxiliary gap lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66I Three auxiliary estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

The End 74References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Eidesstattliche Erklärung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2

1 Overview

1 Overview

The nonlinear Schrödinger equation on the real line

i@tu � �@2xu� 2�juj2u; x 2 R; � � �1;

is a cubic perturbation of the wave function of a free one-dimensional particle. The case

� � 1 is called defocusing and � � �1 focusing. This evolution equation provides a suit-

able setting to describe slow modulations of a localized wave packet in dispersive media,

and arises in many applications such as in hydrodynamics, nonlinear optics, nonlinear

acoustics, plasma waves and biomolecular dynamics.

The linear dispersion, which tends to spread the wave packet, is balanced by the cubic

non-linearity describing the self interaction of the wave with itself. This allows one of

the most successful applications of the Nls equation, the description of soliton waves in

optical fibers.

After the Kdv equation, the Nls equation was the second evolution equation known

to be integrable by the inverse scattering method [27]. When considered with periodic

boundary conditions

u�x � 1� � u�x�;

it can be written in a hamiltonian form and admits a Lax-pair formalism. This enables us

to view solutions of the Nls equation as the potentials of Zakharov-Shabat operators. The

spectrum of the latter encodes many properties of the Nls solutions. Instead of analyzing

the Nls equation directly, we thus focus on these much simpler linear operators.

In this treatise we pursue the assertion that the regularity of such a solution is closely

related to the asymptotic behaviour of certain spectral data of the associated operator.

More precisely, the Fourier coefficients of the solution and the associated spectral gap

lengths belong to the same weighted Sobolev space, hence they share the same asymptotic

behaviour. Since the spectrum of the associated Zakharov-Shabat operator is invariant

under the Nls flow, a solution of the Nls equation stays in the same weighted Sobolev

space for all time. One can use this relationship to show that the Birkhoff coordinates,

which linearize the Nls flow [12, 22], preserve regularity. As integrating the Nls equation

in these coordinates is trivial, well-posedness of the Nls equation follows in all weighted

Sobolev spaces with subexponential weight. This encompasses high regularity spaces

such as standard Sobolev spaces and Gevrey functions.

December 19, 2011 3

1 Overview

Hamiltonian formalism & Lax pair representation

To set the stage for the Nls hamiltonian system, let L2 ˝ L2�T;C� denote the space of

1-periodic, L2-integrable, complex functions on the circle T � R=Z. As phase space we

introduce L2 ˝ L2 � L2, with elements ’ � �’�;’��, and associated Poisson structure

given by

fF;Gg ˝ �iZT�@’�F @’�G � @’�F @’�G�dx;

where @’�F and @’�F denote the components of the L2-gradient @’F of a C1-functional F .

The hamiltonian system with Hamiltonian

HNls �ZT�@x’�@x’� �’2�’2��dx

is then given by

i@t’� � @’�HNls � �@2x’� � 2’�’2�;i@t’� � �@’�HNls � @2x’� � 2’�’2�;

(?)

which we also call the general Nls equation.

The defocusing Nls equation is obtained, when the general Nls equation is restricted

to the invariant subspace of states of real type

L2r ˝n’ 2 L2 : ’� �’

o; ’� � �’�;’��:

On the other hand, we obtain the focusing Nls equation by restriction to the invariant

subspace of states of imaginary type

L2i ˝n’ 2 L2 : ’� � �’

o:

Note that the Nls Hamiltonian requires some regularity to be well defined, for example

the standard Sobolev space Hm with m Æ 1. However, the initial value problem for the

Nls equation on the circle is well posed on L2 as well – see [2].

Zakharov & Shabat [27] discovered a Lax-pair for the Nls equation. This enables us to

view a solution of the general Nls equation as the potential of a linear operator. More

precisely, we consider ’ � �’�;’�� as the potential of the Zakharov-Shabat operator

L�’� �

i�i

!d

dx�

’�’�

!;

December 19, 2011 4

1 Overview

where blank matrix entries represent 0. A tedious computation yields that ’ is a solution

of the Nls equation (?) if and only if

@tL�’� � �A�’�; L�’��;

where the second operator of the Lax-pair is given by

A�’� � 2i@2x � i’�’� @x’� � 2’�@x@x’� � 2’�@x �2i@2x � i’�’�

!:

Many properties of Nls solutions ’ can be recovered from spectral data of L�’�. Of

particular interest is that the Nls flow leaves the periodic spectrum of this operator invari-

ant, so many important features of the Nls dynamics are revealed by the static spectrum

of L�’�. We thus focus on these linear Zakharov-Shabat operators, instead of analyzing

the Nls equation directly.

Spectrum

The periodic spectrum of L�’� is well known to be pure point and more precisely [22]

to consist of a sequence of pairs of complex eigenvalues ��n�’� and ��n�’�, listed with

multiplicities, such that

��n�’� � n� � ‘2n; n 2 Z;

locally uniformly in ’. Here, ‘2n denotes a generic ‘2-sequence.

If we employ a lexicographical ordering on the complex numbers, ordering them first

by their real and second by their imaginary part, the periodic spectrum is given by an

unbounded bi-infinite sequence of eigenvalues, such that

: : : ä ��n�1 ä ��n ä ��n ä �

�n�1 ä : : : ;

where equality or inequality may occur in every place.

Suppose the potential ’ is of real type as in the case of the defocusing Nls equation.

Then L�’� is self-adjoint and so the periodic spectrum is real. By Floquet theory, the

periodic eigenvalues characterise the spectrum of L when considered on the real line

without boundary conditions. In this case the spectrum is absolutely continuous and

given by

specR�L� � R n[n2Z��n; ��n�:

December 19, 2011 5

1 Overview

The complementary and possibly empty intervals ��n; ��n� precisely describe the gaps in

the continuous spectrum, hence one speaks of spectral gaps. The length n ˝ ��n � ��nis called the nth spectral gap length and we denote by �’� ˝ � n�n2Z the two sided

sequence of spectral gap lengths of the potential ’. When some gap length vanishes, the

corresponding gap is called collapsed, and otherwise open.

In contrast, for a general potential ’, the operator L�’� is not necessarily formally

self-adjoint. The gap lengths are defined as before, however, they are no longer real, but

complex numbers. They also cannot be interpreted as lengths of gaps in the continuous

spectrum, when L�’� is considered on R. Nevertheless we will call them gap lengths, and

speak of open and collapsed gaps.

Statement of results

Our main concern is the relationship between the regularity of the potential ’ and the

asymptotic behaviour of n. This question has a long history in the case of Hills’ operator

@2x � q with q being a 1-periodic real L2-potential. Its spectrum is real, pure point, and

consists of an unbounded sequence of periodic eigenvalues

�0 < �1 à �2 < �3 à �4 < : : : ;

with gap lengths defined by n � �2n � �2n�1. Marcenko & Ostrovskiı [18] observed that

q 2 Hk�S1;R�aXnÆ1n2k 2n <1;

for nonnegative integers k. Even earlier, Hochstadt [11] showed that

q 2 C1�S1;R�a n � O�n�k� for all k Æ 0;

and Trubowitz [26] then proved that

q 2 C!�S1;R�a n � O�e�an� for some a > 0:

When the Kdv flow was realized as an isospectral deformation of Hill’s operator, other

regularity classes, such as Gevrey-functions, have been considered, as well as complex

potentials q. We point the reader to Pöschel [20, 21] for a self-contained presentation of

recent progress.

Analogous results for the Zakharov-Shabat system are due to Grébert & Kappeler [9, 10],

Djakov & Mityagin [5–7], and more recently Kappeler, Serier & Topalov [17]. The technical

December 19, 2011 6

1 Overview

details are a bit more complicated than in Hill’s case, mainly due to the fact that the

distance of two distinct eigenvalues is bounded. The bottom line, however, is the same as

for Hill’s operator, the gap lengths characterise the regularity of the potential just like the

Fourier coefficients do.

To allow a more precise statement, we introduce weighted Sobolev spaces [13, 14]. A

submultiplicative, and normalized weight is a symmetric function w : Z! R with

wn Æ 1; wn�m à wnwm;

for all n andm. The class of all such weights is denoted by M and Hw is the Hilbert space

of complex 1-periodic functions u �Pk2Zuke2�ikx with finite w-norm

kuk2w ˝Xk2Zw22kjukj2:

We write Hw ˝ Hw � Hw for the 2-vector version, and denote by hw the analog for

complex sequences u � �uk�, where

k�uk�k2Zk2w ˝Xk2Zw22kjukj2:

The forward problem, that is controlling the asymptotic behaviour of the gap lengths

by the regularity of the potential, is analyzed in chapter 1. Using the language of weighted

Sobolev spaces our result may be stated as follows.

Theorem 1 If ’ 2Hw and w 2M, then �’� 2 hw . ˇ

Chapter 2 is then dedicated to the inverse problem of recovering the regularity of the

potential from the asymptotic behaviour of its gap lengths. To obtain a true converse

to Theorem 1, we first have to exclude weights of exponential growth, as there exist po-

tentials with finitely many open gaps, so called finite gap potentials, that have poles and

therefore are not entire functions. Second, we have to restrict ourselves to the subclass

of potentials of real or imaginary type, to get a first inverse result.

Theorem 2 Let w 2M and ’ be of real or imaginary type with �’� 2 hw .

a) If w is subexponential, then ’ 2Hw .

b) If w is exponential, then ’ 2Hw" , for some " > 0, where w" � e"j�j. ˇ

December 19, 2011 7

1 Overview

Theorem 1 is due to Grébert & Kappeler [9, 10] and can be found in the generality stated

here in [6]. The inverse result was obtained by Djakov & Mityagin [6] for the case of real

type potentials and was later extended by Kappeler et al. [17] to potentials of imaginary

type. We present a self-contained proof of both theorems, which applies simultaneously

to the case of real and imaginary type potentials. A clever use of shifted w-norms re-

duces the amount of technical estimates needed to a minimum and completely avoids

the consideration of »slowly growing« weights. This significantly simplifies the original

approaches, and allows a straightforward generalization to potentials not necessarily of

real or imaginary type.

As a byproduct we obtain the following result due to Tkachenko [25] about finite gap

potentials.

Theorem 3 For each w 2 M the class of general, real type and imaginary type finite gap

potentials is dense inHw ,Hw \L2r andHw \L2i , respectively. ˇ

When the potential ’ is neither assumed to be of real nor of imaginary type, the gap

lengths may not encode any information about the regularity of the potential. Indeed,

for Hill’s operator with q allowed to be complex, Gasymov [8] provided an explicit con-

struction of potentials of any regularity, whose gaps are all collapsed. Additional spectral

data are needed to infer the regularity in this case. To this end, Sansuc & Tkachenko

[23] considered the quantities �n � �n � �n, where �n ˝ ��n � ��n�=2 is the mid point of

the nth spectral gap and �n is the nth Dirichlet eigenvalue. Later, Djakov & Mityagin [6]

adapted this approach to the Zakharov-Shabat system. To further extend their results we

introduce auxiliary gap lengths ��’� ˝ ��n�n2Z, which need to satisfy

(a) �n is continuously differentiable,

(b) �n vanishes whenever ��n � ��n with a geometric multiplicity of two, and

(c) there are real numbers ��n and ��n such that

@�n � tn � o�1�; tn � 12�

e2�in��n�x�; e�2�in��

�n�x�

�;

locally uniformly in’, where the factor 1=2 is arbitrary but simplifies computations

involving normalized quantities.

The auxiliary gap lengths ��’� complement the gap lengths �’� in the case of a general

potential, and allow us to obtain the regularity results for the general case by the same line

December 19, 2011 8

1 Overview

of arguments as used in the real or imaginary case. One easily verifies that �n � �n � �nand even �n � �n � �n, with �n the nth Dirichlet and �n the nth Neumann eigenvalue,

satisfies the conditions (a)-(c), which justifies the introduction of auxiliary gap lengths.

Since the latter include �n � �n � �n as a special case, we can further extend the results

of [6, 17].

Theorem 4 Suppose w 2M and ’ 2 L2 is a general potential with gap lengths �’� and

auxiliary gap lengths ��’�.

(a) If ’ 2Hw , then �’� 2 hw and ��’� 2 hw .

(b) Conversely, if �’� 2 hw and ��’� 2 hw , then ’ 2 Hw when w is subexponential,

and ’ 2Hw" for some " > 0 when w is exponential. ˇ

Djakov & Mityagin [6] considered the quantities ��n, �n � �n and ��n as the vertices of a

spectral triangle �n and

�n�’� ˝ j n�’�j � j�n�’�j

as a measure of its size. With this notation, we can summarize our results as follows.

Theorem 5 If w is subexponential, then

’ 2Hw a ��’� 2 hw : ˇ

Well-posedness

Now suppose the potential ’ also depends differentiably on time t, and denote it by

’t . In the defocusing and focusing case the gap lengths �’t� completely suffice to

characterize the regularity of ’t . Moreover, the Nls flow leaves the periodic spectrum

and in particular the gap lengths invariant, that is �’t� � �’0�. Thus, if we restrict

ourselves to subexponential weights, we obtain the following implications clockwise from

top left to bottom left:

’0 2Hw =) �’0� 2 hw

=)

’t 2Hw (= �’t� 2 hw :

Consequently, any solution of the defocusing or focusing Nls equation, whose initial con-

dition ’0 belongs to Hw , stays in the weighted Sobolev space Hw for all time, and is

even uniformly bounded.

December 19, 2011 9

1 Overview

We sketch an approach to further extend this result – see also [16]. Consider the global

Birkhoff coordinates �xn; yn�n2Z on L2r constructed for the defocusing Nls equation in

[22]. They are the cartesian counterpart to action angle coordinates �In; �n�n2Z and lin-

earize the Nls equation to

xn � !nyn;

yn � �!nxn;

where !n only depends on the actions In � 1=2�x2n �y2n�. Moreover,

jxnj2 � jynj2 � O�j�nj2�

on some complex neighbourhood of L2r in L2. Using Theorem 5 one can show that for

every subexponential weight w the Birkhoff map : Hwr ! hwr � hwr , which essentially is

a nonlinear Fourier transformation that introduces Birkhoff coordinates, is a real analytic

diffeomorphism. Thus the regularity of any initial condition is preserved in Birkhoff co-

ordinates as ’o 2 Hwr is mapped to the sequence space hwr � hwr , where integrating is

trivial. Consequently, the defocusing Nls equation is globally well posed in the sense of

Hadamard, i.e. solutions exist for all time, are unique, and depend continuously on their

initial data. This encompasses all high regularity classes generated by subexponential

weights, such as Sobolev and Gevrey spaces.

The general Nls equation, including the focusing case, also admits Birkhoff coordinates,

albeit, only locally around the zero solution. These are sufficient to infer the local well-

posedness of the general Nls equation in weighted Sobolev spaces.

Acknowledgment. I wish to thank my supervisor Jürgen Pöschel for careful leading and

valuable help with writing this thesis.

December 19, 2011 10

Chapter 1

The forward problem

Let L2 ˝ L2�T;C� denote the space of 1-periodic, L2-integrable, complex functions on the

circle T � R=Z. We consider Zakharov-Shabat operators

L�’� �

i�i

!d

dx�

’�’�

!;

with ’ � �’�;’�� being a vector potential taken from L2 ˝ L2 � L2.The periodic spectrum of L is defined with respect to the L2-dense domain of 1-periodic

or 1-antiperiodic functions that are together with their derivative L2-integrable,

Dper ˝nf 2 H1��0;1�;C��H1��0;1�;C� : f�0� � �f�1�

o:

To avoid distinguishing between the periodic and the anti-periodic case, we use the fact

that the periodic spectrum coincides with the spectrum of L considered on �0;2� endowed

solely with periodic boundary conditions. Consequently, a complex number � is a periodic

eigenvalue if and only if the equation

Lf � �f

admits a nontrivial solution in L2�, which denotes the 2-periodic version of L2. As the

periodic spectrum is known to be pure point, it is given by the set of all such �. Moreover,

the following result is well known [22] and will be eventually recovered in the following:

Proposition 1.1 The periodic spectrum of L�’� consists of a sequence of pairs of complex

eigenvalues ��n�’� and ��n�’�, listed with multiplicities, such that

��n�’� � n� � ‘2n; n 2 Z;

locally uniformly in ’. Here, ‘2n denotes a generic ‘2-sequence. ˇ

11

Chapter 1 The forward problem

bb

b

b b

b

b

bb

b

b

b

b

b

b b

oDN DN+1

Figure 1.1: Distribution of the periodic eigenvalues.

Consequently, when N Æ 1 is chosen sufficiently large, each disc

Dn ˝ f� : j��n�j à �=6g ; jnj Æ N;

contains exactly two eigenvalues ��n. The remaining part of the spectrum is given by an

even number of periodic eigenvalues located in a bounded disc centered at the origin. If

we employ a lexicographical ordering on the complex numbers by

a ä b :a

8>>>>>><>>>>>>:Rea < Reb

or

Rea � Reb and Ima à Imb;

then the periodic spectrum may be represented as an unbounded bi-infinite sequence of

eigenvalues with

: : : ä ��n�1 ä ��n ä ��n ä �

�n�1 ä : : : ;

where equality or inequality may occur in every place. While ��n is well defined through

the lexicographical ordering for every n 2 Z, only for n sufficiently large ��n and ��n may

be considered as an isolated pair of eigenvalues close to n� .

Definition The nth spectral gap length of ’ 2 L2 is given by

n ˝ ��n � ��n;

and �’� � � n�n2Z denotes the entire sequences of spectral gap lengths of ’. ˇ

This chapter is dedicated to the investigation of the forward problem. That is, control-

ling the asymptotic behaviour of the spectral gap lengths in terms of the regularity of the


2 Preparation

potential ’. The proposition above gives a first rough answer, as �’� is an ‘2-sequence

for any ’ 2 L2. Using the language of weighted Sobolev spaces – see appendix A for an

introduction – we can give a more precise answer.

Theorem 1 If ’ 2Hw and w 2M, then �’� 2 hw . ˇ

The proof of Theorem 1 is based on a Lyapunov-Schmidt reduction called Fourier block

decomposition, which reduces the infinite-dimensional eigenvalue equation Lf � �f for

� close to n� to a two-dimensional system described by a 2 � 2 matrix Sn��. More to

the point, � is a 2-periodic eigenvalue of L if and only if Sn�� is singular. In view of

this correlation, the nth gap length n is just the spacing of the two roots of the analytic

function detSn��, and thus may be estimated in terms of the coefficients of Sn. In turn,

the asymptotic behavior of these coefficients is closely related to the regularity of’, such

that we can infer the claim. This approach was first used by Kappeler & Mityagin [14] to

obtain a similar result for Hill’s operator, and was later extended by Grébert & Kappeler

[9, 10] to the Zakharov-Shabat system.

Our proof makes use of shifted w-norms, which completely avoids the delicate esti-

mates of Neumann expansions as well as the need to introduce »slowly growing« weights

and the use of iterative arguments for convolution operators implemented in [14, 17]. As

a side effect, this allows a straightforward extension of the inverse problem to general

potentials, as well as an improved two term gap length estimate which is presented in

section 11.

2 Preparation

We start with the free equation ’ � 0. The 2-periodic eigenvalues of L are then exactly

the double eigenvalues k� for k 2 Z. A basis of corresponding eigenfunctions in L2�, the

2-periodic version of L2, is given byne�k ; e

�k : k 2 Z

o;

where

e�k ˝ �e�k;0�; e�k ˝ �0; ek�; ek�x� ˝ e� ikx:

This basis is orthonormal with respect to the L2� inner product

f ; g

� ˝ �f�; f��; �g�; g�� 12Z 20

�f�g� � f�g�

�dx;


2 Preparation

and any 2-periodic vector-function f admits the following Fourier series expansion in L2�,

f � �f�; f�� Xk2Z

�f�k e�k; f�k ek

� � Xk2Z

�f�k e

�k � f�k e�k

�:

In the following we tag functions whose Fourier expansion are given in e�l with the sub-

script � and functions whose Fourier expansion are given in el with the subscript �. More-

over, we introduce the 2-periodic versions Hw� andHw� of Hw andHw , respectively. The

w-norm of a 2-periodic function u �Pkukek is given by

kuk2w ˝Xk2Zw2k jukj2;

and the w-norm of a 2-periodic vector function f � �f�; f�� by

kfk2w ˝ kf�k2w � kf�k2w :

We now turn to the case of a nonzero potential ’ taken fromH o, where o � 1 denotes

the trivial weight, that is H o � L2. For j�j sufficiently large, the eigenvalue equation

Lf � �f can then be viewed as a perturbation of the free equation, hence we expect

solutions � to be close to some n� . For this reason, we cover the complex plane with the

closed strips

Un ˝ f� : jRe��n�j à �=2g ;

and consider the equation on each of these strips separately.

nπ (n + 1)π(n − 1)π

UnUn−1 Un+1

b

b

b b bb

λ−n λ+

n

Figure 1.2: Periodic eigenvalues for sufficiently large jnj.

For � 2 Un and jnj sufficiently large, the dominant modes of a solution f are e�n and e�n .

Therefore, it makes sense to separate these modes from the others by a Lyapunov-Schmidt

reduction. To this end, consider the splitting

H o� � Pn �Qn� sp

�e�n; e�n

� sp�e�k ; e

�k : k � n

:


2 Preparation

The projections onto Pn and Qn are denoted by Pn and Qn, respectively.

Then write the eigenvalue equation Lf � �f as

A�f � �f ;

with the unbounded operators

A� � ��

i�i

!d

dx; � �

’�

’�

!:

Recall that ’� �Pl2Z’�l e�l by the �-convention, thus we get for f � �f�; f�� in the

domain of �,

�f � �’�f�;’�f��

Xk;l2Z

�’�l�kf

�k e�l; ’�l�kf

�k el

��

Xk;l2Z

�’�l�kf

�k e�l �’�l�kf�k e�l

�:

Note that A� leaves the spaces Pn and Qn invariant, hence by writing

f � u� v � Pnf �Qnf ;

we can decompose the equation A�f � �f into the two equations

A�u � Pn��u� v�;A�v � Qn��u� v�;

called the P - and the Q-equation, respectively.

Our first goal is to solve the Q-equation on the strips Un by writing �v as a func-

tion of u, thus reducing the infinite dimensional eigenvalue equation Lf � �f to a two-

dimensional system on Pn.

Since A�e�m � �� m��e�m, the operator A� is on one hand clearly unbounded, while

one checks that for m � n

min�2Un

j��m�j Æ jn�mj Æ 1:

So, on the other hand, the restriction of A� to Qn is boundedly invertible for all � 2 Un.

We may therefore multiply the Q-equation from the left by �A�1� , giving

�v � �A�1� Qn��u� v�� Tn��u� v�;


2 Preparation

with Tn ˝ �A�1� Qn. The latter equation may be written as

�Id� Tn��v � Tn�u:

Hence writing �v as a function of u amounts to inverting �Id� Tn�.With foresight to section 4 we consider operator norms induced by shifted w-norms

[21]. For any u in Ho� the i-shifted w-norm is given by

kuk2w;i ˝ kueik2w �Xk2Zw2k�ijukj2; i 2 Z:

An i-shift ofu shifts the Fourier coefficients to the left by i. To extend this notion to vector

functions f � �f�; f�� 2H o�, consider the Fourier series of the i-shifted components

f�e�i �Xk2Zf�k e�k�i; f�e�i �

Xk2Zf�k ek�i:

We observe that the Fourier coefficients of f� and f� are shifted into opposite directions.

To compensate this, we define the i-shifted w-norm of a vector function f by

kfk2w;i ˝ kf�k2w;�i � kf�k2w;i � kf�e�ik2w � kf�e�ik2w�Xk2Z

�w2�k�ijf�k j2 �w2k�ijf�k j2

��Xk2Z

�w2k�ijf�k j2 �w2k�ijf�k j2

�:

Lemma 1.2 If ’ 2Hw with w 2M, then for any n 2 Z and any � 2 Un,

Tn � �A�1� Qn

is a bounded linear operator onHw� . Moreover, for any i 2 Z, and any f 2Hw�

kTnfkw;i à 2k’kwkfkw;�i: ˇ

Proof. Given a function f 2Hw� , we first write Tnf � �g with g � A�1� Qnf . An estimate

of the weighted L1-norm of g by Young’s inequality will then give the desired estimate of

kTnfkw;i .

Recall that the restriction of A� to Qn is boundedly invertible. Thus the function

g � A�1� Qnf �Xm�n

� f�m��m� e

�m �

f�m��m� e

�m

�� g�; g��


2 Preparation

is well defined. With the use of Hölder’s inequality we obtain for the weighted L1-norm

kg�e�ik1;w �Xm�n

wm�ijf�mjj��m�j à

� Xm�n

1jn�mj2

�1=2kf�kw;�i à 2kf�kw;�i;

uniformly for � 2 Un. And, in a similar fashion, kg�eik1;w à 2kf�kw;i. Note that

k�gkw;i � k�’�g��e�i; �’�g��eikw � k’��g�e�i�;’��g�ei�kw :

The claim now follows with the estimate above and the weighted Young inequality B.1,

kTnfk2w;i � k�gk2w;i à k’k2w�kg�eik21;w � kg�e�ik21;w

�à 4k’k2w

�kf�k2w;i � kf�k2w;�i

�� 4k’k2wkfk2w;�i: v

Provided ’ 2 Hw the operator Tn is in particular bounded on Hw with respect to

operator norm induced by the 0-shifted w-norm,

kTnkw à 2k’kw :

In order to achieve the bounded invertibility of �Id � Tn� we show that kT 2nko ! 0 as

jnj ! 1. Consequently, �Id � T 2n� is boundedly invertible on H o�, for all jnj sufficiently

large, using a Neumann series. This gives, in view of the identity

�Id� Tn��1 � �Id� Tn��Id� T 2n��1;

the desired result for �Id� Tn�.To avoid technical complications we further assume that all weights taken from M are

monotone, that is

wn à wn�1; n Æ 0:

The case of general, eventually non-monotone weights is discussed in section 10.

Lemma 1.3 Suppose ’ 2Hw with w 2M. Then for any n 2 Z and any � 2 Un,

kT 2nkw;n à 4k’kw k’kwpjnj � kRn’kwwn

!;

where Rn’ denotes the nth remainder of a function ’,

Rn’ �XjkjÆn

�’�k e�k �’�k e�k �: ˇ


2 Preparation

Proof. As in the preceding lemma, given f � �f�; f�� 2Hw� , we write T 2nf � �g with

g � A�1� Qn�A�1� Qnf :

Then a straightforward computation yields

g ˝Xk;l�n

� ’�k�l�� k�

f�l�� l� e

�k �

’�k�l�� k�

f�l�� l� e

�k

�� g�; g��:

To proceed, we estimate the weighted L1-Norms kg�e�nk1;w and kg�e�nk1;w . An applica-

tion of Lemma I.2 gives

kg�e�nk1;w àXk;l�n

wk�nj’�k�ljjn� kj

jf�l jjn� lj

à 4 k’�kwpjnj � kRn’�kwwn

! Xlw2�l�njf�l j2

!1=2

� 4 k’�kwpjnj � kRn’�kwwn

!kf�e�nkw ;

as well as,

kg�enk1;w àXk;l�n

w�k�nj’�k�ljjn� kj

jf�l jjn� lj

à 4 k’�kwpjnj � kRn’�kwwn

! Xlw2l�njf�l j2

!1=2

� 4 k’�kwpjnj � kRn’�kwwn

!kf�enkw :

Since kT 2nfkw;n � k�gkw;n � k’��g�e�n�;’��g�en�kw , we can use the above estimates

together with the weighted Young inequality B.1 to obtain

kT 2nfk2w;n à k’k2w�kg�e�nk21;w � kg�enk21;w

�à 16k’k2w

k’kwpjnj � kRn’kwwn

!2 �kf�k2w;�n � kf�k2w;n

�

� 16k’k2w k’kwpjnj � kRn’kwwn

!2kfk2w;n: v

It follows from Lemma 1.3 that T 2n is a 1=2-contraction on H o� for all jnj sufficiently

large, where the threshold of jnj can be chosen locally uniformly in ’ on H o due to the

Rn’ term. The situation becomes even more transparent, when the weight w is assumed

to be unbounded, as Hw is then compactly embedded into H o by Rellich’s Theorem


3 Reduction

C.3. Hence the contraction property holds uniformly in ’ on bounded subsets of Hw .

Moreover, in view of Lemma C.1 we may always assume the unboundedness of w when

’ 2Hw is given. In this case Lemma 1.3 says

kT 2nko; kT 2nkw;n à8wnk’k2w ; wn �min

�qjnj;wn

�;

and the threshold for jnj can explicitly be chosen uniformly on bounded subsets ofHw .

To simplify notation, we denote the subclass of unbounded, monotone weights by M?.

Moreover, we introduce the weighted balls

Bwm ˝ f’ 2Hw : k’kw àmg ; m Æ 1;

which are centered at the origin. The following proposition summarizes our observations.

Proposition 1.4 For each w 2M?, and each m Æ 1 there exists an Nm;w Æ 1 such that

kT 2nko; kT 2nkw;n à 1=2; jnj Æ Nm;w ;

uniformly for � 2 Un and ’ 2 Bwm. ˇ

3 Reduction

With the preparation of the preceding section we can reduce the infinite dimensional

eigenvalue equation Lf � �f to a two-dimensional system.

For the remaining part of this chapter m Æ 1 and w 2 M? are considered to be fixed,

and is Nm;w is chosen according to Proposition 1.4 above. In view of the identity

Tn ˝ �Id� Tn��1 � �Id� Tn��Id� T 2n��1;

one then finds a unique solution

�v � TnTn�u

of the Q-equation for each jnj Æ Nm;w . Substituting this solution into the P -equation

yields

A�u � Pn�Id� TnTn��u � PnTn�u:

Writing the latter as

Snu � 0; Sn ˝ A� � PnTn�;


3 Reduction

we immediately conclude that there exists a one-to-one relationship between a nontrivial

solution u of Snu � 0 and a nontrivial 2-periodic solution f of Lf � �f . An immediate

consequence is the following lemma.

Reduction Lemma 1.5 Suppose jnj Æ Nm;w , then a complex number � 2 Un is a periodic

eigenvalue of L if and only if it is a root of the determinant of Sn. ˇ

Recall that Pn is the orthogonal projection onto the two-dimensional space Pn. The

matrix representation of an operator B on Pn is given by

�Be�n; e�n

��;� :

Therefore, we find for Sn the representation

A� � ��n�

��n�

!; PnTn� �

a�n b�nb�n a�n

!;

with the coefficients of the latter matrix given by

a�n ˝DTn�e�n; e�n

E; b�n ˝

DTn�e�n; e�n

E;

b�n ˝DTn�e�n; e�n

E; a�n ˝

DTn�e�n; e�n

E:

We point out that the coefficient functions of Sn depend on � and ’. They share some

remarkable symmetries which we will discuss in the following.

Lemma 1.6 Let ’ 2 Bwm with w 2M?. For any � 2 Un and any jnj Æ Nm;w ,

a) a�n � a�n,

b) a�n�� a�n�� and b�n�� b�n��, when ’� � �’�. ˇ

Therefore, we drop the superscript of a�n in the sequel.

Proof. a): Recall that Tn � �A�1� Qn. From evaluating the diagonal operators A� and Qn

at e�k we conclude

A�� A� � PA�P; Q�n � PQnP � Qn;

where � denotes complex conjugation and

P �

11

!:


3 Reduction

Since A� leavesQn invariant, we get �A�1� Qn�� PA�1� QnP . Moreover, we have �� P�P

such that

�Tn�� A�1� Qn�� PTn�P:

An inspection of the Neumann expansion of Tn� yields �Tn�� PTn�P . Together with

e�n � Pe�n we then obtain

a�n �DTn�e�n; e�n

E�De�n; �Tn��e�n

E�DPe�n; Tn�Pe�n

E�DTn�e�n; e�n

E� a�n:

b): Suppose ’� � �’�, then we have �� such that

Tn�� A�1� Qn � PP�PPA�1� PPQnPP � �PTn��P:

Using the identity Tn � �Id� Tn��Id� T 2n��1, we get

Tn�� P�Id� Tn��Id� T 2n��1P:

So it follows that

a�n�� DTn��e�n; e�n

E�DPe�n; Tn��Pe�n

E� �

De�n; P Tn��P�e�n

E� �

De�n; �Id� Tn��Id� T 2n��1�e�n

E:

Since Tn is an anti-diagonal operator, T 2n is a diagonal. Hence we obtain

a�n�� De�n; �Id� Tn��Id� T 2n��1�e�n

E�

De�n; Tn��Id� T 2n��1�e�n

E�


E� a�n��:

For the case of b�n we get on the other hand,

b�n�� DTn��e�n; e�n

E�DPe�n; Tn��Pe�n

E� �

De�n; P Tn��P�e�n

E� �


E� �

De�n; �Id� T 2n��1�e�n

E� b�n��: v


4 Gap estimates

The coefficients an and b�n are given as scalar products on the two dimensional space

Qn. In fact, their representation can be reduced from a two dimensional to a one dimen-

sional. To this end, we write the anti-diagonal operator � and the diagonal operators A�

and Qn as

� �

’�’�

!; A� �

A��

A��

!; Qn �

Q�n

Qn

!;

where A��e�k � �� k��e�k and Q�ne�k � �nke�k. Then we may write

Tn �

T�nT�n

!; T�n ˝’� �A�� 1Q�n:

Evaluating the identity Tn � �Id� Tn��Id� T 2n��1 using this representation of Tn, gives

Tn �

Id T�nT�n Id

! �Id� T�n T�n ��1

�Id� T�n T�n ��1!:

So it follows that

an �DTn�e�n; e�n

E�DTn�Id� T 2n��1�e�n; e�n

E�DT�n �Id� T�n T�n ��1�’�en�; en

E;

as well as

b�n �DTn�e�n; e�n

E�D�Id� T 2n��1�e�n; e�n

E�D�Id� T�n T�n ��1�’�e�n�; en

E;

b�n �DTn�e�n; e�n

E�D�Id� T 2n��1�e�n; e�n

E�D�Id� T�n T�n ��1�’�en�; e�n

E:

We will make repeated use of these representations in the sequel.

4 Gap estimates

In the previous section we observed that for jnj Æ Nm;w a complex number � 2 Un is a

periodic eigenvalue of L if and only if the 2� 2-matrix

Sn�� n� � an �b�n

�b�n ��n� � an

!

is singular. The coefficients of Sn turn out to be analytic functions in �, thus we can use

Rouche’s theorem to deduce that detSn has exactly two roots in Un, namely the periodic


4 Gap estimates

eigenvalues ��n and ��n. Our goal is then to find a suitable estimate of the nth gap length

n � ��n � ��n.

As a starting point we use the representations

an � hT�n �Id� T�n T�n ��1�’�e�n�; e�ni;b�n �’�2n � hT�n T�n �Id� T�n T�n ��1�’�e�n�; e�ni;

which we obtain from the identity �Id�T�n T�n ��1 � Id�T�n T�n �Id�T�n T�n ��1 together with

the results of Lemma 1.6. Moreover, we introduce the following notion for the sup-norm

of a complex valued function

jf jU ˝ sup�2U

jf��j:

Lemma 1.7 Suppose ’ 2 Bwm with w 2 M?. Then for any jnj Æ Nm;w the coefficients an

and b�n are analytic functions on Un with bounds

janjUn à4w2nk’k2w ; jb�n �’�2njUn à

8w2nk’k3w : ˇ

Proof. Since kT 2nko à 1=2, the Neumann expansion of �Id� T�n T�n ��1 converges uniformly

for � 2 Un. Thus an and b�n are the uniform limit of analytic functions and hence analytic

on Un.

To estimate janj put u � �Id � T�n T�n ��1�’�e�n�. As kT 2nkw;n à 1=2, we immediately

get

kukw;n à k�Id� T�n T�n ��1kw;nk’�e�nkw;n à 2k’�kw :

Writing u �Pm2Zumem a straightforward computation yields

an �T�nu; e�n

� � Xm�n

’�n�m��m�um:

Thus we can apply Hölder’s inequality to obtain

janj àX

jn�mj>jnj

j’�n�mjjn�mj jumj �

1w2n

Xjn�mjàjnj

wnj’�n�mj �wnjumj

à1jnjk’�kokuko � 1

w2n

Xjn�mjàjnj

w2n�mj’�n�mj2!1=2 X

mw2n�mjumj2

!1=2

à 1jnj �

1w2n

!k’kwkukw;n

à4w2nk’k2w ;


4 Gap estimates

where we used the monotonicity of the weight together with the fact that jn �mj à jnjimplies jn�mj Æ 2jnj � jn�mj Æ jnj.

To proceed with b�n we put v � T�n �Id� T�n T�n ��1�’�e�n�. Then Lemma 1.2 gives

kvkw;�n à 2k’�kwk�Id� T�n T�n ��1kw;nk’�e�nkw;n à 4k’k2w :

Writing v �Pm2Z vne�n we compute

b�n �’�2n �T�n v; en

� � Xm�n

’�n�m��m�vm;

so by the same line of arguments

jb�n �’�2nj à 1jnj �

1w2n

!k’kwkvkw;�n

à8w2nk’k3w :

The claimed bound of b�n �’�2n follows in a similar fashion. v

The preceding lemma implies that the determinant of Sn

detSn � ��n� � an�2 � b�nb�n

is an analytic function in �, which is close to �� n��2 for jnj sufficiently large. We will

use this fact to infer that the determinant has two roots � close to n� .

Lemma 1.8 Let ’ 2 Bwm with w 2 M?. Then there exists an N?m;w Æ 1, such that, for any

jnj Æ N?m;w , the determinant of Sn has exactly two complex roots ��, �� in Un, which are

contained in the disc

Dn � f� : j��nj à �=6g ;

and satisfy

j�� j2 à 6jb�njjb�nj: ˇ

Proof. Let h � ��n� � an, and choose N?m;w Æ 1 such that

janjUn ; jb�njUn à�24; jnj Æ N?m;w :

Since

janjUn � jb�njUn < inf�2UnnDn

j��n�j;


4 Gap estimates

bnπ

λ−n

λ+n

γn

Un

γn+1

γn−1

Un+1Un−1

Dn

Figure 1.3: Localization of the periodic eigenvalues and their spectral gaps.

it follows from Rouche’s Theorem that h has a single root in Dn, just as �� n��. In

a similar fashion, we obtain that h2 and detSn have the same number of roots in Dn,

namely two, while detSn clearly has no root in Un nDn.

To put it differently, we write detSn � g�g� with

g� � ��n� an � �n; �n �qb�nb�n ;

where the branch of the root is immaterial. Then each root � of detSn is either a root

of g� or g�, respectively, and thus satisfies � � �n � an�� n��. To estimate the

distance of the two roots, we note that an is analytic on Un, so Cauchy’s estimate F.1 gives

j@�anjDn àjanjUn

dist�Dn; @Un�à�=24�=3

� 18:

The claim now follows with

j�� j à jan�� an��j � j�n�� n��j

à18j�� j � 2j�njUn : v

For jnj sufficiently large, detSn has exactly two roots on each strip Un, counted with

multiplicities. Since the strips cover the complex plane, those roots have to be the 2-

periodic eigenvalues ��n according to the Reduction Lemma 1.5. So we obtain the following

preliminary individual gap estimate

j n�’�j2 � j��n � ��n j2 à 6jb�njjb�nj à 3�jb�nj2 � jb�nj2�:

Since jb�nj à j’�2nj�jb�n�’�2nj, it remains to show that b�n is hw -close to’�2n, to conclude

that �’� 2 hw when ’ 2Hw .


4 Gap estimates

Proposition 1.9 If ’ 2 Bwm with w 2M?, then for any N Æ Nm;w

XjnjÆN

w22njb�n �’�2nj2 à210

w2Nk’k6w : ˇ

Proof. Making use of the representation of the coefficient b�n we obtain

b�n �’�2n �DT�n T�n �Id� T�n T�n ��1�’�en�; e�n

E:

To allow an invocation of Lemma I.3, put un � �Id � T�n T�n ��1�’�en�. Then �un� is a

sequence of Hw� functions, that is uniformly bounded by

kunkw;�n à k�Id� T�n T�n ��1kw;�nk’�enkw;�n à 2k’�kw ; jnj Æ Nm;w :

Expanding un into its Fourier series, un �Pkuk;ne�k, we obtain by a straightforward

computation

jhT�n T�nun; e�nij �� Xk;l�n

’�n�l�� l�

’�l�k�� k� uk;n

�� à Xk;l�n

j’�n�ljjn� lj

j’�l�kjjn� kj juk;nj:

Notice that

kunk2w;�n �Xk2Zw2k�njuk;nj2;

i.e. the negative shift of un by �n translates into a positive shift of its Fourier coefficients.

With that we can invoke Lemma I.3 to obtain

XjnjÆN

w22njhT�n T�nun; e�nij2 à210

w2Nk’k6v :

The case of b�n can be treated in a similar fashion. v

We are now in a position to prove Theorem 1. Suppose ’ 2 Bwm for some m Æ 1. Then

we can choose jnj Æ max�Nm;w ; N?m;w�, such that Lemma 1.8 applies giving us two roots

�� and �� of detSn contained in Dn � Un and satisfying

j n�’�j2 � j�� j2 à 6jb�nb�njUn à 3�jb�nj2Un � jb�nj2Un�:

Moreover, one easily checks that

12w22njb�nj2 à w22nj’�2nj2 �w22njb�n �’�2nj2:

Now, it follows from the preceding proposition that �’� 2 hw .


Chapter 2

The inverse problem

So far we have studied the forward problem of controlling the asymptotic behaviour of

the gap lengths �’� in terms of the regularity of the potential ’. Using the language of

weighted Sobolev spaces, our result may be summarized as

’ 2Hw ) �’� 2 hw :

We now turn to the inverse problem of recovering the regularity of the potential ’

from the spectral gap lengths �’�. Gasymov [8] observed that the gap lengths of Hill’s

operator @2x � q may not contain any information about the regularity of the potential q,

when q is allowed to be complex. A similar observation can be made in our situation.

Consider the potential

� ˝XmÆ1

�qme2m; pme2m�:

Then �Id � T�n T�n ��1��en� is given as a power series in e2� ix with lowest term en�2. It

follows that an � b�n � 0 and

Sn � ��n� b�n0 ��n�

!:

So the periodic spectrum of � consists entirely of double periodic eigenvalues n� , all

gaps are collapsed, and hence �� 2 hw for every weight w. However, the coefficients

qm and pm can be chosen freely, such that � may have any regularity, thus � 2Hw does

not hold for every w.

To avoid this situation, we first restrict ourselves to certain subclasses ofH o-potentials.

The space of potentials of real type is defined by

H or ˝

�’ 2H o : ’� �’ ; ’� � �’�;’��:

27

Chapter 2 The inverse problem

When ’ is of real type, the operator L is formally self-adjoint. Hence the periodic spec-

trum of L is real, and so are the gap lengths.

The imaginary counterpart ofH or is the space of potentials of imaginary type,

H oi ˝

�’ 2H o : ’� � �’ � iH o

r :

For ’ of imaginary type, the operator L is in general not formally self-adjoint. However,

each pair of eigenvalues is adjoint, that is ��n � ��n, such that all gap lengths are purely

imaginary.

Note that H or and H o

i are real subspaces of H o, not complex ones. Moreover, the

potential � given above is obviously neither of real nor of imaginary type. Indeed, the

requirement for the potential to be of real or imaginary type imposes a certain symmetry

on the Fourier coefficients linking them to the periodic spectrum. As a consequence the

gap lengths completely encode the regularity of ’ in the case of subexponential weights.

Still, for exponential weights the situation is not as clear cut as in the forward case,

since there exist certain finite gap potentials, potentials with finitely many open gaps, that

are real analytic but whose extension to the complex plane has poles. For example, one

can use Weierstrass }-functions to construct a meromorphic potential }? with a single

gap [24]. On one hand, �}?� 2 hw for any weight w, since except for one all gaps are

collapsed. On the other hand, }? is meromorphic with poles in the complex plane. Let

d � minz2P�}?� jImzj denote the shortest possible distance of a pole to the real axis.

Then }? 2 Hwa with wa � eaj�j if and only if �2��a à d. Thus we can not expect a true

converse to the forward problem for exponential weights.

Theorem 2 Suppose ’ 2H o is of real or imaginary type and �’� 2 hw .

(a) If w is subexponential, then ’ 2Hw .

(b) If w is exponential, then ’ 2Hw" with w" � e"j�j for all sufficiently small " > 0. ˇ

The proof of Theorem 2 is based on two observations. First, the Fourier coefficients’�2nof the potential are hw -close to the off-diagonal coefficients b�n of the matrix Sn. Second,

for potentials of real or imaginary type, the latter coefficients can be bounded in terms

of the nth gap length. Pöschel [20] used these facts in the similar Hill case to define a

perturbed Fourier series map

� F�’�;



which, according to the inverse function theorem, is a near identity diffeomorphism that

respects regularity. The Fourier coefficients of are particularly suited to be compared

to the gap lengths, allowing the following conclusion

�’� 2 hw ) 2Hw )’ 2Hw :

This approach was extended by Kappeler et al. [17] to Zakharov-Shabat systems with

potentials of real or imaginary type. Due to the special symmetries in this case, the Fourier

coefficients of can be bounded by the spectral gap lengths �’�, which establishes

Theorem 2.

Incidentally, we may recover an observation due to Tkachenko [25] about finite gap

potentials.

Theorem 3 For each w 2 M the class of general, real type and imaginary type finite gap

potentials is dense inHw ,Hwr andHw

i , respectively. ˇ

We now turn to the inverse problem for general potentials ’ 2 H o. By Gasymov’s

observation, additional spectral data are needed to recover the regularity of the potential.

For this reason, Sansuc & Tkachenko [23] in Hill’s case, and later Djavok & Mityagin [6] for

Zakharov-Shabat operators, considered the quantities �n � �n��n, where �n denotes the

nth Dirichlet eigenvalue and �n the mid-point of the nth spectral gap. To further simplify

and extend their results, we follow Pöschel’s treatment of Hill’s operator [21] and define

an abstract set of auxiliary spectral quantities.

Definition Suppose ’ 2 Hw with w 2 M. A family of auxiliary gap lengths ��’� is a

bi-infinite sequence with

(a) �n is continuously differentiable on some neighbourhood U �Hw of ’,

(b) �n vanishes whenever ��n � ��n has also a geometric multiplicity of two, and

(c) there exist real numbers ��n ; ��n and an integer NU Æ 1, such that

k@’�n � tnko à 1=16; tn � 12�e2�in��

�n��; e�2�in��

�n��

�;

uniformly on U for jnj Æ NU . ˇ

For example, one can choose �n � �n��n, with �n being the nth Dirichlet or Neumann

eigenvalue of ’, and �n � ��n��n�=2 the mid point of the nth gap – see also appendix G



for the definition of Dirichlet and Neumann spectra. The factor 1=2 in the definition of

tn is due to the usual normalization of the Dirichlet and Neumann eigenfunctions used

to express the gradient @�n in this case. We show in appendix H that this choice of �n

actually satisfies the conditions (a) - (c) whenever w is unbounded. This justifies the

introduction of auxiliary gap lengths, which cover �n � �n � �n as a special case.

Similar to the case of the spectral gap lengths �’�, the asymptotic behaviour of ��’�

is determined by the regularity of the potential ’. On the other hand, no additional

theory is required to recover the regularity of a general potential ’ from the gap lengths

complemented by a family of auxiliary gap lengths. Essentially, it is property (a) - (c) that

we need to bound the Fourier coefficients of � F�’� by the auxiliary gap lengths, so we

obtain the following inverse result without any additional cost.

Theorem 4 Suppose w 2M, ’ 2H o and ��’� is a family of auxiliary gap lengths.

(a) If ’ 2Hw , then �’� 2 hw and ��’� 2 hw .

(b) Conversely, if �’� 2 hw and ��’� 2 hw , then ’ 2 Hw when w is subexponential,

and ’ 2Hw" for some " > 0 if w is exponential. ˇ


5 Adapted Fourier coefficients


In this section we define an analytic perturbation of a given potential ’, which adapts

the Fourier coefficients of ’ to the spectral gap lengths while preserving regularity. The

key idea is to consider the coefficients of the 2�2-matrix Sn as analytic functions of their

potential and to use them as the new Fourier coefficients for the perturbed potential.

For simplicity, we assume that ’ 2 Hw with some unbounded and monotone weight

w 2 M? and not only w 2 M. In this case, Hw is compactly embedded into L2 by

Rellich’s Theorem, and consequently the coefficients of Sn are well defined uniformly on

each ball Bwm when jnj is sufficiently large. In the context of regularity preservation, we

need some notation to say when a weight v is “stronger” than a weightw. For this reason,

we introduce the ordering

v Æ w :a vn Æ wn for all n:

Clearly, any weight w 2 M satisfies w Æ o where o � 1 denotes the trivial weight, and

Bvm � Bwm whenever w à v .

Lemma 2.1 Let w 2M?. One can choose Nm;w Æ 1, such that for any weight v Æ w

kT 2nkv ;n

��Bv2m à 12 ; jnj Æ Nm;w :

Moreover, the coefficient functions

an : Un �Bw2m ! C; b�n : Un �Bw2m ! C;

are analytic in � and ’, and uniformly bounded,

janjUn�Bw2m ; jb�njUn�Bw2m à �=48: ˇ

Proof. To estimate the norm of T 2n we apply Lemma 1.3 on the ball Bvm, which gives

kT 2nkv ;n��Bvm à 8m2

vnà8m2

wn:

Since w is assumed to be unbounded, we can choose Nm;w Æ 1 such that

1wn

à1

212m2 ;

where the choice 212 turns out to be useful later. Then we have kT 2nkv ;n��Bv2m à 1=2 even

on the ball of radius 2m and for any weight v Æ w. In particular, T 2n is a 1=2-contraction



for each ’ 2 Bw2m, hence the coefficients of Sn are well defined on Bw2m. The analytic

dependence on � and the estimates

janjUn�Bwm à4m2

w2n; jb�njUn�Bwm à j’�2n � b�n �’�2njUn�Bwm à

mw2n

� 8m3

w2n:

follow from Lemma 1.7. Moreover, the functions an and b�n are each given as a Neumann

series in T 2n, which converges absolutely and uniformly on Bw2m by the contraction prop-

erty of T 2n. Hence, Lemma F.2 gives the analytic dependence on the potential ’. v

The idea is to perturb’ using the analytic coefficients b�n of Sn as they are on one hand

close to the Fourier coefficients, while they are on the other hand comparable to the gap

lengths. To remove the dependency of Sn on �, we fix a � 2 Un for each given ’ and

n. More to the point, we choose � � �n�’� such that the diagonal of Sn vanishes, thus

greatly simplifying all terms involving Sn.

Throughout this chapter Nm;w is supposed to be chosen according to the lemma above.

Lemma 2.2 Let w 2 M?. For any jnj Æ Nm;w there exists a unique analytic function

�n : Bw2m ! C such that,

a) j�n �n�jBw2m à �=48,

b) �n�’� is real for any ’ of real or imaginary type, and

c) �n �n� � an��n; ��Bw2m � 0. ˇ

Proof. Let E denote the space of analytic functions � : Bw2m ! C equipped with the usual

metric induced by the topology of uniform convergence. For jnj Æ Nm;w , the claim can

then be written as a fixed point problem of the map

T� ˝ n� � an��; ��

on the closed subspace E? of functions in E with properties a) and b). Note that � � n�satisfies a) and b), hence E? is not empty.

Each � in E? maps the ball Bw2m into the disc D0n � f� : j��n�j à �=48g � Un by

assumption. Consequently,

jT��n�jBw2m � jan��; ��jBw2m à janjUn�Bw2m à �=48;

hence T� satisfies a). Moreover, for ’ of real or imaginary type, an��;’� is real for real

� by Lemma 1.6. Therefore, T� also satisfies b) and thus T maps E? into E?.



The contraction property now follows from

d�T�1; T�2� � jT�1 � T�2jBw2m à j@�anjD0n�Bw2mj�1 ��2jBw2m ;

and Cauchy’s estimate

j@�anjD0n�Bw2m àjanjUn�Bw2m

dist�D0n; @Un�à�=48�=6

� 18:

Hence we find a unique fixed point �n with the properties as claimed. v

We now define the perturbation map Fm on Bwm by

Fm�’� ˝X

jnj<Nm;w

�’�2ne

�2n �’�2ne�2n

�� XjnjÆNm;w

�b�n��n; ��e�2n � b�n��n; ��e�2n

��’:

For each jnj Æ Nm;w the Fourier coefficients of the 1-periodic function � Fm�’� are

given by �2n � b�n��n;’�, and

Sn��n;’� � 0 �2n �2n 0

!;

since the diagonal of Sn��n; �� vanishes. These new Fourier coefficients are adapted to

the gap lengths, thus we call Fm adapted Fourier coefficient map. Indeed, for ’ of real or

imaginary type we make the following conclusion in section 8

�’� 2 hw ) � Fm�’� 2Hw :

With the help of some auxiliary gap lengths an analogous result for general potentials is

obtained in section 9.

Next we obtain the diffeomorphism property of Fm, which finally enables us to conclude

that ’ has the same regularity as – with possibly a slight loss of regularity in the

exponential case.

Proposition 2.3 If w 2M?, then Fm maps Bwm intoHw . For any weight v Æ w its restric-

tion to Bvm is an analytic diffeomorphism

Fm��Bvm : Bvm ! Fm�Bvm� �H v ;

such that

12k’kv à kFm�’�kv à 2k’kv ;

and Bvm=2 � Fm�Bvm�. Moreover, kdFm � IdkBvm à 1=8 and Fm preserves type, i.e.

Fm�Bvm \H or � � Fm�Bvm�\H o

r ; Fm�Bvm \H oi � � Fm�Bvm�\H o

i : ˇ


6 Abstract regularity results

Proof. Since �n maps the ball Bw2m into Un for every jnj Æ Nm;w , the coefficient functions

b�n��n; �� are well defined on Bw2m. Moreover, for each v Æ w we have kT 2nkv ;n à 1=2 on

Bv2m � Bw2m. Thus Proposition 1.9 applies giving the estimate

kFm � idk2v;Bv2m � sup’2Bv2m

XjnjÆNm;w

v22n�jb�n��n; ��’�2nj2 � jb�n��n; ��’�2nj2

�

à218m6

w2Nm;wà�2m�2

162;

by our choice of Nm;w from Lemma 2.1. Therefore, Fm is locally bounded while each

coordinate function is analytic in ’. Hence Fm is analytic on Bv2m by Lemma F.3. In

particular, Cauchy’s estimate applies, giving

kdFm � Idkv;Bvm àkFm � idkv;B2mdist�Bm; @B2m� à

18:

The preservation of H or and H o

i follows from the symmetry of the coefficients b�nobserved in Lemma 1.6. Finally, the inverse function theorem F.4 together with the fact

that Fm�0� � 0 gives the remaining properties. v

We now prove Theorem 3. Suppose ’ 2 Bwm with w 2 M?. When �2n � �2n � 0 for

some jnj Æ Nm;w , then Sn vanishes identically at �n and hence the nth gap is collapsed.

Consequently, if

Fm�’� 2 GrN � span f�e�k; ek� : jkj à Ng ;

with N sufficiently large, then at most 2N � 1 gaps of ’ are open. As the union of the

spaces GrN is dense in Bwm \ H or and Fm is a diffeomorphism, it follows that the finite

gap potentials are dense in Bwm \H or . The remaining cases are treated similarly, so the

theorem is proved for unbounded weights. Moreover, Bwm is dense in Bom for any weightw,

so the claim immediately follows for bounded weights. The remaining cases of imaginary

and general type potentials are proved analogously.


What can we say about the regularity of ’, when � Fm�’� 2Hw is known?

Proposition 2.4 If ’ 2 Bwm with w 2M? and

Fm�’� 2 Bvm=2;

for a weight v Æ w from M, then ’ 2 Bvm �H v . ˇ



Proof. Since the weightw is assumed to be unbounded, Proposition 2.3 applies and hence

Fm : Bwm ! Hw is an analytic diffeomorphism onto its image. Moreover, v Æ w by as-

sumption, thus the restriction of Fm to Bvm is also an analytic diffeomorphism onto its

image with Bvm=2 � Fm�Bvm�. Therefore, the unique preimage ’ of Fm�’� 2 Bvm=2 has to

be in Bvm. v

In general, we have no a priori bound on kFm�’�kv . However, we can modify the

weight v to obtain such a bound while not affecting the asymptotic behaviour in the

subexponential case. More to the point, for v" ˝ e"j�j and any " > 0 sufficiently small,

w" �min�v; v"� 2M

by the Cut-Off lemma D.1. If v is subexponential, then w" and v share the same asymp-

totics, while w" � v" for all sufficiently small " > 0, when v is exponential.

It makes sense to modify only that part of the weight v Æ w, which is actually stronger

than w, while leaving w untouched, as ’ 2 Hw is granted anyway. To this end, we

introduce the notion of product weights

�wv�n ˝ wn � vn:

Obviously wv is submultiplicative if w and v are and w à wv for any weight v 2M.

Proposition 2.5 Suppose ’ 2 Bwm with w 2M?, and

Fm�’� 2H vw

for some v 2M.

(a) If v is subexponential, then also ’ 2H vw .

(b) If v is exponential, then ’ 2H v"w for all sufficiently small " > 0. ˇ

Proof. We may assume that m Æ 8k’kw , as increasing m does not change our assump-

tions. Put � Fm�’�, then k kvw <1 by assumption, and hence kRN kvw à k kw for

some N Æ 1. With w" �min�v; v"� we then obtain

k k2w"w � k � RN k2w"w � kRN k2w"wà k � RN k2v"w � kRN k2vwà e"Nk � RN k2w � k k2wà �1� e"N�k k2wà 4k k2w ;


7 Lower gap length bound

for " > 0 sufficiently small. Thus, the preceding lines together with Proposition 2.3 yield

k kw"w à 2k kw à 4k’kw àm=2;

such that 2 Bw"wm=2 . Since w"w Æ w, it follows from Proposition 2.4 that ’ 2 Bw"wm .

Moreover, Hw"w � H vw , provided v is subexponential, while we have w" � v" for any

exponential v when " > 0 is sufficiently small. v


Since Fm respects regularity, recovering the regularity of the potential’ from the asymp-

totic behaviour of its gap lengths �’� amounts to dominating the Fourier coefficients of

� Fm�’� by n, such that

�’� 2 hv ) 2H v :

By Gasymov’s observation this cannot hold for arbitrary potentials, thus we impose addi-

tional conditions on .

Lemma 2.6 Let ’ 2 Bwm with w 2M?. If � Fm�’� and

19à

�� 2n �2n

�� à 9;for jnj Æ Nm;w , then

j �2n �2nj à j nj2 à 9j �2n �2nj: ˇ

Note that the assumption on is satisfied for ’ of real or imaginary type by the sym-

metry of the adapted Fourier coefficients.

Proof. We begin by writing detSn � g�g� with

g� ˝ ��n� � an � �n; �n �qb�nb�n :

In contrast to the situation of Lemma 1.8, the assumption on effects that g� is continu-

ous and even analytic. Indeed, recall �2n � b�n��n�, thus

�n ˝ �n��n� �qb�n��n�b�n��n� � 0; rn ˝ j�nj > 0;



rn/2

2rn

bαnb

bαn + ξn

αn − ξn

Don

D+n

D−n

Figure 2.1: The discs D�n contained in Don.

so we may choose �n as a fixed branch of the square root locally around �n. Even more,

when the disc Don ˝ f� : j��nj à 2rng is considered, we have

j@�b�njDon à

jb�njUndist�Do

n; @Un�à

�=48�=2� j�n �n�j � 2rn à

121;

where we used j�n �n�j à �=48 and rn à �=48. Note that the same estimate holds for

@�an. Combined with Cauchy’s estimate this yields

jb�n � b�n��n�jDon à j@�b�njDo

n2rn à221rn;

so b�n does not vanish on Don and the function �n is analytic there. Hence we can use

Rouche’s theorem to refine our localization of the roots ��n of g� in the following.

By fixing the last two terms of g� at �n, we obtain h� ˝ ��n� �an��n��n��n�. We

want to compare h� and g� on the disc

D�n ˝�� : j�� n � �n�j < rn=2

� Don:

Since h��n � �n� � �n � �n��n� � 0, we have

jh�j@D�n � jh� � h��n � �n�j@D�n �rn2:

In the following lines we will show that on Don � D�n

j@��njDon à

421:


8 Regularity: Potentials of real or imaginary type

As a consequence,

jh� � g�jD�n à jan��n�� anjDon � j�n��n�� njDo

n

à�j@�anjDo

n � j@��njDon

�2rn <

rn2� jh�j@D�n :

Thus, it follows from Rouche’s theorem that g� has a unique root contained in D�n , while

there is no root in Un n D�n . In a similar fashion, we find the unique root of g� to be

contained in D�n ˝�� : j�� n � �n�j < rn=2

. Since the roots of g� are roots of detSn,

they have to coincide with ��n and hence

rn à j��n � ��nj à 3rn;

which is the claim.

It remains to show the estimate for @��n on Don. To this end, we write

jb�nj � jb�nb�nj1=2��b�nb�n

��1=2 ;and recall that

13rn à jb�n��n�j à 3rn; jb�n � b�n��n�jDo

n à221rn:

Thus, by the triangle inequality we obtain

521rn �

�13� 221

�rn à jb�njDo

n à�3� 2

21

�rn � 6521rn:

Treating b�n completely analogous we arrive at��b�nb�n��Don

;��b�nb�n

��Don

à 13;

which finally yields the desired estimate

j@��njDon à

j@�b�njDon

2

��b�nb�n��1=2Don

� j@�b�njDo

n

2

��b�nb�n��1=2Don

à421: v

8 Regularity: Potentials of real or imaginary type

We are now in a position to prove Theorem 2. Suppose ’ is of real or imaginary type and

�’� 2 hv with v 2M. The sequence

rn ˝ j’�2nj � j’�2nj � v2nj nj; n 2 Z;


9 Regularity: Potentials of general type

is clearly square summable, hence by Lemma C.1 there exists an unbounded and monotone

weight w 2 M? such that �rn� 2 hw . Consequently, �’� 2 hwv and ’ 2 Bwm for some

m Æ 1, thus

� Fm�’�

is well defined. Since ’ is of real or imaginary type, we have by Lemma 1.6

b�n�� b�n��:

Moreover, � � �n�’� is real by Lemma 2.2, and therefore

j �2nj � jb�n��n�j � jb�n��n�j � j �2nj:

If �2n; �2n � 0, then Lemma 2.6 applies giving

j �2nj2 � j �2nj2 à j nj2;

which is obviously also true in the case �2n � �2n � 0. It follows that 2 Hwv and

Proposition 2.5 yields the claimed regularity of ’.


In the preceding section, we solved the inverse problem for potentials of real or imaginary

type. The particular symmetry of the adapted Fourier coefficients of � Fm�’� allows

us to compare them with the gap lengths �’�. By Gasymov’s observation, this fails for

general potentials. To this end, we consider additional spectral data given by a family

of auxiliary gap lengths ��’� � ��n�n2Z, which is characterized by the following three

properties:

(a) �n is continuously differentiable on some neighbourhood U �Hw of ’,

(b) �n vanishes whenever ��n � ��n also has a geometric multiplicity of two, and

(c) there exist real numbers ��n , ��n and an integer NU Æ 1 such that

k@�n � tnko à 1=16; tn � 12�e2�in��

�n��; e�2�in��

�n��

�;

uniformly on U for jnj Æ NU .



The properties (a) - (c) are easily verified for �n � �n � �n, with �n the nth Dirichlet

or Neumann eigenvalue and �n the mid-point of the nth spectral gap – see appendix H.

These properties enable us to bound the adapted Fourier coefficients of , so we can

proceed as in the case of potentials of real or imaginary type, and obtain the general case

with no additional costs.

Lemma 2.7 If ’ 2 Bwm with w 2 M? has auxiliary gap lengths ��’� on some neighbour-

hood U � Bwm, then there exists an N Æmax�Nm;w ; NU�, such that

j�n�’�� &�n �2n � &�n �2n�j à14�j �2nj � j �2nj�; jnj Æ N;

with &�n � e2�in��n=2 and &�n � e�2�in��n=2. ˇ

Proof. We may assume that U is star convex with respect to ’. Put � Fm�’� and let

V � Fm�U�. Then we can choose N Æmax�NU ; Nm;w� such that for each jnj Æ N

o ˝Xk�n� �2ke

�2k � �2ke�2k� 2 V:

Fix such an n, then ’o � F�1m � o� 2 U . Per definitionem, the 2nth Fourier coefficients of

o vanish, hence Sn�’o� � 0. Consequently, �n�’o� is a double periodic eigenvalue of

’o with a geometric multiplicity of two, so by assumption (b)

�n�’o� � 0:

Let ’s ˝ s’ � �1� s�’o, then ’s 2 U for 0 à s à 1 by star convexity, and hence

�n�’� � �n�’�� n�’o�

�Z 10h@�n�’s�;’ �’oir ds

� htn;’ �’oir � h�n;’ �’oir ;

with �n �R 10 �@�n � tn��’s�ds. In addition,

’ �’o � F�1m � �� F�1m � o�

�Z 10

dF�1m � s�� o�ds

� � o ��m� � o�;



with �m �R 10 �dF

�1m � Id�� s�ds. A combination of both representations yields

�n�’� � htn; � oir � htn;�m� � oir � h�n;’ �’oir :

Clearly, htn; � oir � �2nt�n ; e2n

� � �2n t�n ; e�2n� � &�n �2n � &�n �2n and ktnko à 1.

Moreover, k�nko à 1=16 by assumption (c), and

k�mko à kdF�1m kokdFm � Idko à16;

by Proposition 2.3. Altogether we obtain

j�n�’�� htn; � oi j à�ktnkok�mko � k�nko�1� k�mko�

�k � ok

à14k � ok: v

With this preparation we can prove Theorem 4. Suppose ’ 2 Hw with w 2 M. Then

for each jnj Æ N we have according to the preceding Lemma,

j�n�’�j2 à 54�j �2nj � j �2nj

�2à104

�jb�nj2Un � jb�nj2Un

�:

By the same line of arguments as in the proof of Theorem 1, we get ��’� 2 hw .

Conversely suppose ’ 2 L2 and �’� 2 hv as well as ��’� 2 hv with v 2M. So

rn ˝ j’�2nj � j’�2nj � v2nj n�’�j � v2nj�n�’�j

is an ‘2-sequence, hence �rn� 2 hw for some weightw 2M?. Consequently, �’� 2 hvw ,

��’� 2 hvw and ’ 2 Bwm for some m Æ 1, thus � Fm�’� is well defined.

For any jnj Æ N , with the hypothesis of Lemma 2.6 satisfied, we have

j �2nj2; j �2nj2 à j n�’�j2:

Otherwise, we may assume that j �2nj Æ 9j �2nj, such that by the preceding Lemma

j�n�’�j Æ j&�n �2n � &�n �2nj �14

�j �2nj � j �2nj

�Æ1289j �2nj �

14109j �2nj

Æ16j �2nj:

We conclude that in this case

136j �2nj2; j �2nj2 à j�n�’�j2;

so 2H vw . The claim now follows with Proposition 2.5.


Chapter 3

Extensions

10 A note on non-monotone weights

For the sake of brevity and clarity we restricted ourselves to monotone weights. However,

this has been a purely technical restriction and Theorem 1-5 remains valid when we drop

this requirement and consider weights that are just submultiplicative. This is due to the

fact that for a given potential ’ 2 H w , with w 2 M an arbitrary weight, we can always

find an unbounded and monotone weightw 2M?, such that k’kww <1 – see Lemma C.1.

This enables us to reuse our initial approach by considering weights v � ww 2 M given

as the product of an unbounded monotone weight w and an arbitrary other weight w.

Since we do not use the »other« weight w explicitly, we introduce the following notation

to suppress the product structure:

v Ý w :a v � ww for some w 2M:

Note that v Ý w implies v Æ w. Now, given a potential ’ 2 H v we may always assume

that v Ý w with an unbounded and monotone weight w.

In our treatment of the forward problem in chapter 1 we first used the monotonicity of

the weight to obtain an estimate for kT 2nkw;n (Lemma 1.3) and second to show that b�n is

hw -close to ’�2n (Lemma 1.9). We can easily adapt the proofs to the situation of a weight

v Ý w with w monotone and unbounded.

Lemma 3.1 Suppose w 2M?. Then for any weight v Ý w and any n 2 Z,

kT 2nkv ;n à 8k’kv k’kvpjnj � kRn’kvwn

!à16wnk’k2v : ˇ

Proof. Using the addendum to Lemma I.3 instead of Lemma I.3 itself, the proof is identical

to that of Lemma 1.3. v

42


When we fix m Æ 1 and w 2 M?, we can choose an Nm;w Æ 1 such that �Id � Tn� is

boundedly invertible onH v� for any jnj Æ Nm;w , any ’ 2 Bvm and any v Ý w. It is crucial

to note that the threshold Nm;w only depends on w and works for every weight v Ý w

not necessarily monotone. The subsequent arguments of chapter 1 can be repeated word

by word, just the final Lemma needs a slight modification.

Lemma 3.2 Suppose ’ 2 Bvm with v 2M. If v Ý w with w 2M?, then for any N Æ Nm;w

XjnjÆN

v22njb�n �’�2nj2 à210

wNk’k6v : ˇ

Proof. Using the addendum to Lemma I.3, we can copy the proof of Lemma 1.9. v

To adapt our treatment of the inverse problem in chapter 2 it suffices to fix a weight

w 2M? and to replace our general assumption v Æ w with the assumption v Ý w. Then

v Ý w can be chosen arbitrarily and does not need to be monotone, since we solely used

the monotonicity of w.

These somewhat roundabout arguments are indeed necessary, as a given weightw 2M

may not be equivalent to a monotone weight, that is

Hw � Hv

for any monotone weight v . We want to give an example of such a weight in the following.

For w 2M consider the smallest monotone upper bound

wn � max0àkàn

wk;

and the greatest monotone lower bound

wn � infkÆnwk:

Both are submultiplicative weights, when w is submultiplicative, and clearly w à w à w.

If �wn=wn� is bounded, then the induced norms of those three weights are equivalent,

and thus the generated Sobolev spaces are the same

Hw � Hw � Hw :

On the other hand, if �wn=wn� is unbounded, we have by Proposition E.1

Hw Ì Hw ;



and consequently, Hw � Hv for any monotone weight v .

Such weights with �wn=wn� being unbounded are called strongly oscillating, and one

may ask whether they exist at all. Thus we construct an example in the sequel. Since

addition is easier visualized than multiplication, we consider the equivalent case of sub-

additive weights

w � logw; w 2M:

Clearly, w � v is subadditive, when w and v are. Moreover, the 2-periodic function

��x� �

8>><>>:x; 0 à x à 1;

2� x; 1 < x à 2;gives rise to the subadditive triangular weights

�kn � k ��n=2k2�1�; k Æ 1;

whose height grows linear with k, while its width is 2k2 .

1

2

2 4 8 16 32 64

Figure 3.1: The iterations of �kn for k � 1; : : : ;3.

It follows that

wso�n� ˝XkÆ1�kn

is subadditive and clearly unbounded. More to the point,

wso�2n2�1� �

XkÆnk ��2n2�1=2k2�1� Æ n;

while

wso�2n2� �

Xk>nk ��2n2=2k2�1� �

XlÆ1

n� l2l2�2nl�1

àXlÆ1

n� l4nl

àn� 14n�1 � 1 ! 0:

So wso � exp�wso� is strongly oscillating.


11 Second order gap estimates


In this section, we obtain the first two terms in the asymptotics of the gap lengths n�’�.

They will be revealed by further analyzing the gap length representation known from

Lemma 1.8

n � an��n�� an��n�� n�n��n�� n�n��n�; �n ˝qb�nb�n ; (?)

where any configuration of the signs ��n ; ��n 2 f�1;�1g has to be considered due to pos-

sible discontinuities of �n.

It turns out that the coefficients an��n� do not account to the low order asymptotics of

n�’�, thus we focus on the terms involving �n. In particular, we consider the first two

terms in the Neumann series expansion of

b�n � h�Id� T�n T�n ��1�’�e�n�; e�ni�’�2n � ’�2n � h�T�n T�n �2�Id� T�n T�n ��1�’�e�n�; e�ni;

where the second term ’�2n depends on � and is given by

’�2n ˝ hT�n T�n �’�e�n�; e�ni:

The first term in the asymptotics of n�’� is just’�2n’�2n, the product of the 2nth Fourier

coefficients of the potential ’, while the second term involves ’�2n and ’�2n.

Furthermore, it turns out that for’ 2Hw any improvement of the estimates of n�’�

is limited by the regularity of ’ and thus related to the weight w. To express the asymp-

totics in terms of the growth of w, it is handy to introduce the subclass

Ms ˝�w 2M : w monotone, unbounded and

Xn�0w2n=n2 <1

�:

Having fixed notations, we can state the main result of this section.

Theorem 6 Let ’ 2 Bvm with v Æ w and w 2Ms . Then

XjnjÆNm;w

v22nw2nmin� j n � 2�nj2 à Cm;w ; wn �min

�qjnj;wn

�;

with a real constant Cm;w depending only on m and w, and

�n ˝q�’�2n � ’�2n��’�2n � ’�2n�

��n�

: ˇ



When’ is a potential of real type, then the periodic eigenvalues ��n and the gap lengths

n are real. Together with the symmetries of b�n observed in Lemma 1.6, we obtain an

even better estimate in this case.

Supplement to Theorem 6 Suppose ’ is also of real type, thenXjnjÆNm;w

v22nw4nj n � 2�nj2 à Cm;w : ˇ

Remark. When the weight is assumed to be of the formwn � hni� for some � < 1=2, then

the estimate of Theorem 6 can be improved. More to the point, using Hölder’s inequality

together with the fact thatXnÆ1

1hniq� <1;

XnÆ1

1hnip�1�� <1;

for some p;q Æ 1, one can show that the estimate of j n � 2�nj2 even holds for w3ninstead of w2n – see [9]. However, the class Ms is more general, since it encompasses,

among others, weights of the form hnih�n� with h�n� " 1=2 in a slow manner as n ! 1.

Therefore, we will not rework the special treatment for weights of the form hni� here.

Moreover, when potentials of real type are considered, the approach taken in [9] does

does not yield a stronger result than the one presented here. ˙

We begin by expanding n using the representation (?). This gives

min� j n � 2�nj à jan��n�� an��n�j �min� j��n�n��n�� n�n��n�� 2�nj:

The first summand, called the an-term, is easily estimated and does not contribute to the

low order asymptotics of n. However, the second summand, called the �n-term, requires

more attention. We start with the an-term.

Lemma 3.3 For each ’ 2 Bvm with v Æ w, we haveXjnjÆNm;w

v22nw4njan��n�� an��n�j2 à 16m4k �’�k2v : ˇ

Proof. For jnj sufficiently large, the periodic eigenvalues ��n are located in the disc Dn �Un with dist�Dn; @Un� � �=3 Æ 1. Hence, Cauchy’s estimate F.1 gives

jan��n�� an��n�j à j@anjDnj nj à janjUnj nj:

The claim now follows with the bound of an we found in Lemma 1.7,

janjUn à4m2

w2n; jnj Æ Nm;w : v



We now turn to the �n-term. First of all, we improve our localization of the periodic

eigenvalues ��n. Given ’ 2 Bwm, we have by Lemma 1.7

janjUn � jbnjUn à13m3

wn<16m3

wn; jnj Æ Nm;w :

Similar to Lemma 1.8, it follows with Rouche’s Theorem that for such n the eigenvalues

��n and ��n are contained in

D�n ˝ f� : j��n�j à rng � Dn; rn ˝ 16m3=wn:

On this even smaller disc, we obtain the following preliminary estimate of the �n-term.

Lemma 3.4 Let ’ 2 Bvm with v Æ w. Then for each jnj Æ Nm;w

min� j��n�n��n�� n�n��n�� 2�nj2 à 36j� 2n � �2njD�n : ˇ

Proof. Put ��n � ��n�n��n� and note that we have for each � 2 D�n

min� j�n�� nj à j� 2n�� 2nj1=2 à M ˝ j� 2n � �2nj1=2D�n :

Consequently, �n�� pb�n��b�n�� is for such � contained in either of the two possibly

intersecting discs

C�n � fz : jz � �nj à Mg ; C�n � fz : jz � ��n�j à Mg ;

but may jump between them when � varies due to possible discontinuities of the root.

Suppose j�nj à 2M . When ��n and ��n are contained in the same disc, there is nothing

to do, so we may assume the contrary. For ��n 2 C�n , we then find

j��n � ��n�j à j��n � ��n�j � 2j�nj à 5M:

Hence the the claim follows under this assertion on �n.

Conversely suppose j�nj > 2M , so the two discs C�n are disjoint. Moreover,

jb�nb�n � �2njD�n à M2 <j�nj24

by our choice of M . For each � 2 D�n , thus b�n��b�n�� is contained in the disc of radius

j�2nj=4 around �2n and never vanishes. So we can choose a fixed branch of the square

root to effect that �n �pb�nb�n is analytic on all of D�n . It follows from Lemma 1.8 that

��n � ��n � 1, hence ��n � �n��n�. Let ��n; ��n� � D�n denote the straight line segment

connecting ��n and ��n in the complex plane. By continuity ��n; ��n�� is either contained

in C�n or in C�n and cannot jump. Thus ��n and ��n are contained in the same disc, which

gives the claim. v



b

ρn

−ρnb

b

MC+

n

C−n

o

b

bσn(λ+

n)

σn(λ−n)

Figure 3.2: �n��n; ��n�� in the case j�nj > 2M .

With this preparation estimating the �n-term reduces to estimating j� 2n � �2njD�n , which

is fairly straightforward. Recall that �2n � �’�2n � ’�2n��’�2n � ’�2n��n� . To obtain a

comparable representation of � 2n and �2n we define

b�n ˝ b�n �’�2n; ��n ˝ ’�2n��n� :

Then we may write

� 2n � �’�2n � b�n��’�2n � b�n�; �2n � �’�2n � ��n��’�2n � ��n�;

from what we obtain the following expansion

� 2n � �2n �’�2n�b�n � ��n��’�2n�b�n � ��n�� b�nb�n � ��n��n:

The estimates needed to attack this expansion are collected in the following lemma.

Lemma 3.5 For each ’ 2 Bvm with v Æ w and w 2Ms we have

XjnjÆNm;w

v22nw2njb�nj2Un à 210Cwm6;

with a real constant Cw depending only on the weight w. Moreover,

XjnjÆNm;w

v22nw4njb�n � ’�2nj2Un à 218Cwm10: ˇ

Proof. We proceed as in Proposition 1.9 and let un � �Id� T�n T�n ��1�’�en�. Then �un� is

a sequence of Hv -functions which is uniformly bounded by

kunkv ;�n à k�Id� T�n T�n ��1kv ;�nk’�enkv ;�n à 2m:



Moreover, b�n �T�n T�nun; e�n

�, so we consider

v22nw2njb�nj2 � v22nw2njhT�n T�nun; e�nij2;

and follow the argumentation of Lemma I.3 by splitting up the latter term into �k;n, �l;n

and �\;n. Since w 2Ms it isPn�0w2n=jnj2 à Cw <1. Thus we obtain by exactly the same

arguments as in the proof of the lemmaXjnjÆNm;w

��2k;n � �2l;n

�à 27Cwm6;

XjnjÆNm;w

�2\;n à 26m6:

Summing up yields the first claimXjnjÆNm;w

v22nw2njb�nj2Un à 210Cwm6:

To obtain the second claim, let u?n � T�n T�n �Id � T�n T�n ��1�’�en�. Clearly �u?n� is a

sequence of Hv -functions, with individual bound

ku?nkv ;�n à16wnk’k2vk�Id� T�n T�n ��1kv ;�nk�’�en�kv ;�n à

32m3

wn

for each n. Moreover, b�n � ’�2n � hT�n T�nu?n; e�ni. Repeating the arguments above word

by word we arrive atXjnjÆNm;w

v22nw4njb�n � ’�2nj2Un à 218Cwm10: v

We are now in a position to proof Theorem 6. First of all, we apply Lemma 3.4 to obtain

12v22nw

2nmin� j n � 2�nj2 à v22nw2njanj2Unj nj2 � 36v22nw2nj� 2n � �2njD�n :

The former term is summable by Lemma 3.3, and for the latter we found the expansion

� 2n � �2n �’�2n�b�n � ��n��’�2n�b�n � ��n�� b�nb�n � ��n��n:

Since j��nj à jb�njUn � jb�n � ’�2njUn , the square terms may be estimated by

12v22nw

2n�jb�nb�nj � j��n��nj�

à v22nw2n�jb�nj2Un � jb�nj2Un � jb�n � ’�2nj2Un � jb�n � ’�2nj2Un�:

Thus they are summable by Lemma 3.5. It remains to consider the mixed terms,

v22nw2nj’�2n�b�n � ��n��’�2n�b�n � ��n�jUnà v22n

�j’�2nj2 � j’�2nj2�� v22nw4n�jb�n � ��nj2Un � jb�n � ��nj2Un�:



The former term is clearly summable, while an application of Hadamard’s lemma gives

for the latter

b�n � ��n � b�n � ’�2n � ’�2n � ’�2n��n�

� b�n � ’�2n �Z 10@’�2n��1� s�n� � s��n��ds:

Since j��n�j à rn � 16m3=wn and j@’�2njD�n à j’�2njUn by Cauchy’s estimate, we get

132m3v

22nw

4njb�n � ��nj2 à v22nw4njb�n � ’�2nj2 � v22nw2nj’�2nj2;

so the mixed terms are summable by Lemma 3.5.

Altogether we obtain

XjnjÆNm;w

v22nw2nj� 2n � �2njD�n à Cw;m;

with a real constant Cm;w only depending on m and the weight w. This completes the

proof of the theorem.

When ’ is of real type, the eigenvalues ��n and ��n are real. Moreover, we infer from the

symmetry b�n�� b�n�� that ��n � ��n � 1 as well as

�n��n� �qb�n��n�b�n��n� � jb�n��n�j;

and similarly

�n �q�’�2n � ��n��’�2n � ��n� � j’�2n � ��nj:

Thus j�n��n��n��n��2�nj à 2jb�n ��njD�n , which gives the supplement to Theorem 6.


Chapter 4

Weighted Sobolev spaces

The main concern of this thesis is to compare the regularity of a 1-periodic L2-function ’

to the asymptotic behaviour of some spectral data given as a bi-infinite sequence �’� �� n�. In our case, the regularity of ’ is sufficiently described by its Fourier coefficients,

so we seek a tool to compare their asymptotic behaviour with that of the data �’�. This

is where weighted Sobolev spaces come into play, giving a measure for the decay rate of

sequences. In the sequel we collect some basic facts about weighted Sobolev spaces and

present a mechanism to construct unbounded monotone weights for a given L2-function.

Together with the compact embedding theorems this allows us to transfer local results

on L2 to uniform results on bounded subsets of some weighted Sobolev space.

A Definitions and examples

Let L2 ˝ L2�T;C� denote the standard Hilbert space of L2-integrable, complex-valued

functions on the circle T ˝ R=Z. A function u 2 L2 admits a Fourier series expansion

u �Xk2Zuke2k; e2k�x� ˝ e2�ikx:

Definition A normalized, submultiplicative weight is a function w : Z! R, with

w�n � wn; wn Æ 1; wn�m à wnwm;

for all n and m. The space of all such weights is denoted by M. Moreover, the w-norm of

a function u 2 L2 is defined through

kuk2w ˝Xk2Zw22kjukj2;

and Hw is the Hilbert space of all such functions with finite w-norm. ˇ

51

Appendix A: Definitions and examples

The trivial weight w � 1 is denoted by o and

Ho � L2 �[w2M

Hw ;

as every weight is assumed to be at least 1.

Definition A weight w 2M is called monotone when

wn à wm; 0 à n àm: ˇ

More precisely, such a weight is monotonically increasing. However, monotonically de-

creasing weights are bounded and thus generate the space Ho � L2. They do not provide

any information about asymptotic behaviour, hence we excluded them from the definition

of monotone weights.

Moreover, by submultiplicity,

wn à wn1 :

For this reason a submultiplicative weight cannot grow faster than exponentially. A more

precise measure of growth is given by the following

Definition and Lemma A.1 The characteristic of a weight w 2M,

��w� ˝ limn!1

log�wn�n

;

exists, is nonnegative and satisfies ��w� � infn log�wn�=n [19, No. 98]. ˇ

Weights with different characteristics clearly give rise to different weighted Sobolev

spaces, while weights with the same characteristic not necessarily give rise to the same

Sobolev space. We discuss the question when two given weights generate the same

weighted Sobolev space in appendix E, for now we content ourselves with the characteris-

tic as a first order quantity to distinguish weights. Two important subclasses of weights

are the following:

Definition Let w 2M.

(a) If ��w� > 0, then w is called exponential.

(b) If w is a monotone weight, and log�wn�=n converges eventually monotonically to

zero as n!1, then w is called subexponential. ˇ


Appendix A: Definitions and examples

Note that there exist weights that are neither exponential nor subexponential.

Examples a. Let hni ˝ 1� jnj. Then for any s Æ 0, the polynomial Sobolev weight

wn � hnis

gives rise to the usual Sobolev space Hs .

b. For s Æ 0 and a > 0, the exponential Abel weights

wn � hnis eajnj;

give rise to the spaces Hs;a of L2-functions, which can be analytically extended to the

open strip fz : jImzj < a=2�g of the complex plane with traces inHs on the boundary

lines. Note that the characteristic of such a weight is ��w� � a and therefore does not

provide any information about the polynomial factor hnis for s Æ 0.

For example Weierstrass }-functions are real analytic, 1-periodic functions and thus

belong to some Hs;a for a > 0 sufficiently small. However, they have poles in the

complex plane and thus are not entire functions.

c. In between are, among others, the subexponential Gevrey weights

wn � hnis eajnj�; 0 < � < 1;

which give rise to the Gevrey-spaces Hr ;s;� , as well as weights of the form

wn � hnir exp�

ajnj1� �log hni��

�;

that are lighter than Abel weights but heavier than Gevrey weights.

d. The space of 1-periodic entire functions is given by

H! �\a>0Ho;a

This one as well as other classes of entire functions may be described by weights with

��w� � 1, growing faster than exponentially and thus called superexponential. Such

weights are not submultiplicative, hence our methods do not apply directly. Still one

can use results for exponential weights, to infer results for the superexponential case

– see [4, 21]. �


Appendix B: An inequality for products

The examples show that certain weights give rise to certain regularity classes of func-

tions. To compare the regularity of functions in Hw with the asymptotics of spectral data,

it is convenient to introduce a counterpart of Hw for sequences.

Definition The w-norm of a complex-valued sequence �uk�k2Z is given by

k�uk�k2w ˝Xk2Zw22kjukj2;

and hw denotes the space of all such sequences with finite w-norm. ˇ

In particular for any 1-periodic function u �Pk2Zuke2k, we have

u 2 Hw a �uk� 2 hw :

B An inequality for products

The product u �v of two Hw functions u and v does in general not belong to Hw . There-

fore, we extend our definition of Hw to Lp-based w-norms, which enables us to estimate

such products. The Lp-w-norm of a function f is given by

kfkp;w ˝�Xk2Zwp2kjfkjp

�1=p:

Note that for p � 2 this definition is different from the usual Lp-norm defined using the

integral. In particular Hölder’s inequality does not hold in this context, as u � v involves

the convolution of the Fourier coefficients of u and v . Instead, we use Young’s inequality

to estimate the w-norm of products.

Weighted Young inequality B.1 Suppose u 2 Lp, v 2 Lq and

1� 1r� 1p� 1q:

Then for any w 2M,

kuvkr ;w à kukp;wkvkq;w : ˇ

Proof. The Fourier coefficients of the product of two functions are given as the convolu-

tion of the original Fourier coefficients

uv �Xk;l2Z

uk�l vl e2k:


Appendix C: Construction of unbounded weights

Therefore, we have by submultiplicity

kuvkr ;w à Xk2Z

��Xl2Zw2k�2luk�l �w2l vl

��r!1=r

� k�w2kuk�� w2lvl�krà k�w2kuk�kp � k�w2kvk�kq� kukp;wkvkq;w ;

where the ordinary Young inequality for sequences was applied to obtain the last esti-

mate. v

Since we use this inequality repeatedly in the case r � p � 2 and q � 1, we explicitly

note that

ku � vkw à kukwkvk1;w :

C Construction of unbounded weights

The Fourier coefficients of a given L2-function clearly decay to zero. However, this decay

may happen arbitrary slow, and there is no a priori estimate on the convergence rate.

Therefore, the following result may come to some surprise.

Lemma C.1 For any L2-function u, there exists an unbounded and monotone weight w,

such that kukw is finite. ˇ

We denote the subclass of weights in M that are monotone and unbounded with

M? ˝ fw 2M : wn % 1 as jnj ! 1g :

A consequence of the lemma above is the sufficiency of this subclass in the sense that

Ho �[

w2M?

Hw ;

while obviously Hw Ì Ho for any w 2M?. Before we attack the proof, we want to elabo-

rate a criterion for submultiplicity. Since addition is in general easier than multiplication,

we consider the equivalent case of subadditivity. More to the point, the submultiplicity of

a weight v 2M implies the subadditivity of w � log�v�,

wn�m � log�vn�m� à log�vn�� log�vm� � wn �wm;

for all n and m. Conversely, v � exp�w� is submultiplicative, when w is subadditive.


Appendix C: Construction of unbounded weights

Lemma C.2 Suppose w : N ! R� is a positive function with wn nondecreasing and wn=n

nonincreasing. Then w is subadditive. ˇ

Proof. Let m;n Æ 0, then we have by positivity and monotonicity of wn=n,

wn�m � n�mn�mwn�m � nwn�mn�m �mwn�m

n�mà n

wnn�mwm

m� wn �wm:

It remains to consider wn�m for 0 à m à n. However this to the former case by mono-

tonicity,

wn�m à wn�m à wn �wm: v

Proof of Lemma C.1 We may assume that �uk� has infinitely many nonzero terms, other-

wise the w-norm of u is finite for any weight w. Let

�n ˝ r�1n ; rn ˝ XjkjÆn

jukj2!1=4

:

Then �n is clearly positive, increasing and unbounded. Moreover,

�2n�junj2 � ju�nj2� � 2junj2 � ju�nj2

2r 2nà 2

r 4n � r 4n�1r 2n � r 2n�1

� 2�r 2n � r 2n�1�:

From this calculation we get

NXn�1�2n�junj2 � ju�nj2� à 2r 21 à 2kuko;

for any N Æ 1. The function w : N! R defined by

w1 ˝ log�1� �1� > 0; wn ˝min�

log�1� �n�; nn� 1wn�1

�for n > 1;

is positive, unbounded and satisfies the conditions of the previous lemma, as

wn à wn�1; andwn�1n� 1 à

wnn:

Thus the symmetric extension of w to Z given by w0 ˝ 0 and w�n ˝ wn for n Æ 1, is

subadditive. It follows that w � ew is submultiplicative, nondecreasing, unbounded, and

kukw <1. v

Unbounded weights are handy in many situations. In particular we have the following

result.


Appendix D: The Cut-Off Lemma

Rellich’s Theorem C.3 If w 2M?, then Hw is compactly embedded into L2. ˇ

Proof. Suppose that un * u in Hw . Clearly, the Fourier coefficients of un converge point-

wise to those of u, since each h�; eki is continuous on Hw .

Moreover, by weak convergence, kun �ukw is bounded by some m Æ 1. Let " > 0, and

choose K Æ 1, such that m2=w22K < ". Then

kun �uk2o �Xjkj<K

junk �ukj2 �XjkjÆK

junk �ukj2 à " �kun �uk2ww22K

à 2";

for all n sufficiently large. v

When w is unbounded and monotone, Rellich’s Theorem allows us to extend results

which hold locally uniformly on L2 to uniform results on bounded subsets of Hw .

D The Cut-Off Lemma

In section 6 we employ an a priori bound on a given function u 2 Hw . Pöschel’s Cut-

Off Lemma [21, Lemma 9] allows us to modify the weight w by cutting off a potential

large chunk of kukw arising from finitely many modes while not affecting the asymptotic

behaviour when w is subexponential.

Cut-Off Lemma D.1 If w is either subexponential or exponential, then

w" �min�w;v"� 2M; v" � e"j�j;

for all sufficiently small positive ". ˇ

Proof. If w is exponential, then w" � v" for all sufficiently small positive ", and thus

w" 2M.

Now suppose that w is subexponential and let

w � logw; w" � logw":

All we have to show is subadditivity of w" which implies submultiplicity ofw". Clearly w"

is monotone, as w and v" are. Moreover, wn=n converges eventually to zero, hence for

any " > 0 sufficiently small there exists an integer N" Æ 1, such that

wiiÆ " >

wnn>wmm

for 1 à i à N" < n <m:


Appendix E: Equivalent weights and monotonicity

Consequently,

�w"�n �

8>><>>:�v"�n; n à N";

wn; N" < n;

thus �w"�n=n is nonincreasing. So Lemma C.2 applies giving the claim. v

E Equivalent weights and monotonicity

Two weightsw;v 2M giving rise to the same weighted Sobolev space Hw � Hv are called

equivalent. It is the purpose of this section to give a criterion for equivalence.

When both quotients �wn=vn� and �vn=wn� are bounded, the weights w and v clearly

give rise to equivalent norms and thus generate the same spaces. It turns out that the

converse is also true.

Proposition E.1 Two weights w;v 2 M are equivalent if and only if both �wn=vn� and

�vn=wn� are bounded. ˇ

Before we attack the proof we provide a basic fact about sequences.

Lemma E.2 Suppose �an� is a real, positive, and unbounded sequence, then there exists a

real, positive sequence �bn�, such that

XnÆ1bn <1; and

XnÆ1an � bn � 1: ˇ

Proof. Choose a subsequence ank , such that ank Æ k2 for every k Æ 1. Put ck � ank and

dk � k�2. Clearly,PnÆ1 dk < 1, while ck � dk Æ 1 and hence is not summable. We now

define

bn ˝

8>><>>:dk; nk � n;0; otherwise:

ThenPnÆ1 bn �

PkÆ1 dk <1, while on the other hand

XnÆ1an � bn �

XkÆ1ck � dk � 1: v


Appendix E: Equivalent weights and monotonicity

Proof of Proposition E.1. We may assume that w and v are unbounded, as otherwise the

claim follows immediately. Suppose �vn=wn� is unbounded, then, by the preceding Lemma,

we find a 1-periodic function h �Pn2Z hnen 2 Hv with

Xn2Z

v2nw2nv2njhnj2 � 1:

Let g �Pn2Z gnen with gn � vnhn=wn, then kgkw � khkv and

kgk2v �Xn2Z

v2nw2nv2njhnj2 � 1:

Consequently, Hw � Hv . v


Chapter 5

Background Information

F Analytic maps

We present a very short collection of facts about analytic maps in this appendix, mainly to

fix notions and notations. A more complete survey including proofs can be found in [15].

For a nice exposition of infinite dimensional holomorphy see also Dineen [3].

In the following let E and F denote complex Banach spaces, and U � E an open neigh-

bourhood.

Definition A map f : U ! F is said to be differentiable at q 2 U , if there exists a bounded

linear map

dqf : E ! F;

called the derivative of f at q, such that

kf�q � h�� f�q�� dqf�h�k � o�khk�:

When f is differentiable at any q 2 U , it is said to be analytic on U . ˇ

The derivative dqf is uniquely determined, and in the analytic case defines a map

df : U ! L�E; F�; q , dqf ;

which is again analytic on U .

When E is a Hilbert space and f complex valued, there is yet another interpretation of

its derivative. We restrict ourselves to the case E � L2��0;1�;C� with f : U ! C an analytic

function on some neighbourhood U � L2. For each q 2 U the derivative dqf is a bounded

60

Appendix F: Analytic maps

linear functional on L2. Thus, by the Riesz representation theorem, there exists a uniquely

determined function @qf 2 L2, such that

dqf�h� � hh; @qf ir ˝Z 10h � @qf dx

for every h 2 L2. We call @qf the gradient of f at q, and denote the gradient map of f by

@f : U ! E; q , @qf :

The derivative of an analytic function f can be expressed as an integral involving only

f itself by Cauchy’s formula. An important application is the following estimate, which

enables us to bound the norm of the derivative of f in terms of the norm of f , where

jf jU ˝ supq2U

jf�q�jF ; jdf jU ˝ supq2U

jdfqjL�E;F�:

Cauchy’s estimate F.1 Let f : U ! F be analytic and V � U an open subset. Then

jdf jV à jf jUdist�V ; @U�

: ˇ

Another notable fact is that the uniform limit of analytic maps is again analytic. We

only need the following weaker version for power series.

Proposition F.2 Suppose f : U ! C is given as a power series of homogeneous polynomials

f �XkÆ0Pk:

If the power series converges absolutely and uniformly on any compact subset of U , then f

is analytic on U . ˇ

Furthermore, when f : U ! H maps from an open subset of a complex Banach space

into a Hilbert space, we have the following characterization of analyticity.

Proposition F.3 Let f : U ! H map into a Hilbert space H with orthonormal basis �en�.

Then f is analytic on U if and only if f is locally bounded, and each coordinate function

fn �f ; en

�: U ! C; n 2 Z;

is analytic on U . ˇ


Appendix F: Analytic maps

Finally we provide a version of the inverse function theorem for near-identity analytic

maps. To simplify notation we define

Bm ˝ fx 2 E : jxj àmg ; m Æ 0;

to be the ball of radius m centered at the origin, and let j � jm ˝ j � jBm .

Inverse function theorem F.4 Suppose f : Bm ! E is analytic, and

jf � idjm àm=k; k > 4:

Then f is an analytic diffeomorphism onto its image, and this image covers Bm=2. ˇ

Proof. Let F denote the space of analytic functions g : Bm=2 ! Bm with jg� idjm=2 àm=kendowed with the standard metric. Clearly, the operator

Tg � id� �f � id� � g

maps F into F , since

jTg � idjm=2 � j�f � id� � gjm=2 à jf � idjm àm=k:

Moreover, T is a contraction on F as

jTg � Thjm=2 àZ 10

��df � Id��1� s�g � sh��m=2 ds jf � gjm=2

à jdf � Idjm=k�m=2jg � hjm=2;

and by Cauchy’s estimate

jdf � Idjm=k�m=2 à jf � idjmm=2�m=k à

1=k1=2� 1=k < 1:

The unique fixed point g : Bm=2 ! Bm is the analytic inverse to f . v


Appendix G: Spectra

G Spectra

In this appendix we give all necessary definitions and derive some facts about the spectra

of Zakharov-Shabat operators

L�’� �

i�i

!d

dx�

’�’�

!;

acting on various dense subspaces of L2��0;1�;C� � L2��0;1�;C�, where ’ � �’�;’�� is

a 1-periodic vector potential taken from

L2 ˝ L2�T;C�� L2�T;C�; T � R=Z:

A detailed exposition containing proofs of the facts provided here can be found in [22].

The periodic spectrum of L�’� is defined with respect to the dense domain

Dper ˝nf 2 H1c : f�0� � �f�1�

o;

where H1c � H1��0;1�;C� �H1��0;1�;C� denotes the standard Sobolev space of complex

2-vector valued functions, which together with their derivative are L2-integrable. The

periodic spectrum is pure point and can be obtained as the zero set of an entire function,

determined by the trace of the fundamental solution of L�’�. The periodic spectrum

more precisely [22, Proposition 6.7] consists of a sequence of pairs of complex eigenvalues

��n�’� and ��n�’�, k 2 Z, listed with multiplicities, such that

��n�’� � n� � ‘2n;

locally uniformly in ’. Here, ‘2n denotes a generic ‘2-sequence, i.e.

Xk2Zj��n �n�j2 < M;

where M can be chosen locally uniformly in ’.

If we employ a lexicographical ordering on the complex numbers by

a ä b :a

8>>>>>><>>>>>>:Rea < Reb

or

Rea � Reb and Ima à Imb;


Appendix G: Spectra

then the periodic spectrum of L�’� can be represented as a bi-infinite sequence of com-

plex eigenvalues

: : : ä ��n�1 ä ��n ä ��n ä �

�n�1 ä : : : ;

counting them with their algebraic multiplicities, where equality or inequality may occur

in every place.

b

b

b

b

bb bb

b

b nπλ−

n

λ+n

γn

γn+1γn−1

. . .. . .τn

τn−1τn+1

Figure 5.1: The periodic eigenvalues and their spectral gaps for jnj sufficiently large.

The nth spectral gap length is defined by

n ˝ ��n � ��n;

and

�n ˝ ��n � ��n�=2

denotes the mid-point of the nth gap. According to [22, Lemma 12.3], �n and 2n are analytic

functions for jnj sufficiently large, and their L2-gradients satisfy

@�n � ‘2n; @ 2n � ‘1n

locally uniformly in ’.

We now turn to the Dirichlet and Neumann spectra. Those are most transparently de-

fined in Akns-coordinates [1]. For this reason, we write ’ � �q � ip; q � ip� to obtain the

equivalent representation

L�q;p� � �11

!d

dx� q pp �q

!

of L�’�, called the Akns-system. The operators L�’� and L�q;p� are unitary equivalent

and satisfy

L�q;p� � T�1L�’�T ; T � 1 i1 �i

!:


Appendix G: Spectra

When ’ is of real type, q and p can be chosen as real functions, so L�q;p� is real. Hence,

L�’� may be viewed as its complexification when q and p are allowed to be complex. In

these coordinates the Dirichlet and Neumann spectra of L are defined with respect to the

dense domains

ADir ˝nf 2 H1c : f��0� � 0 � f��1�

o;

ANeu ˝nf 2 H1c : f��0� � 0 � f��1�

o:

Using the transformation T they are mapped onto the L�’� domains

DDir ˝ng 2 H1c : �g� � g��

��0 � 0 � �g� � g��

��1

o;

DNeu ˝nh 2 H1c : �h� � h��

��0 � 0 � �h� � h��

��1

o:

Similar to the periodic case, both spectra are pure point and may be obtained as the zero

set of an entire function. The Dirichlet eigenvalues �n and the Neumann eigenvalues

�n are given as bi-infinite unbounded sequences, which can be ordered lexicographically,

such that

: : : ä �n�1 ä �n ä �n�1 ä : : : ; : : : ä �n�1 ä �n ä �n�1 ä : : : :

They share the same asymptotics as the periodic eigenvalues, i.e.

�n; �n � n� � ‘2n;

locally uniformly in ’ – see [22, Lemma 5.3 & 5.5]. When ’ is of real type, we even have

the following relation

��n à �n; �n à ��n; n 2 Z:

Thus all Dirichlet and Neumann eigenvalues are simple in the real case. Still, �n and �n

are simple for an arbitrary potential when jnj is sufficiently large. Consequently, these

eigenvalues are simple roots of their characteristic functions. It follows from the implicit

function theorem that they are analytic functions, and by [22, Corollary 7.6] their L2-gradients satisfy

@�n � 12�e��2n � e��2n�� ‘2n; @�n � �12�e

��2n � e��2n�� ‘2n;

locally uniformly in ’.


Appendix H: Auxiliary gap lengths

b

b

b

b

b

b b

b

b

b

b

b

b

nπλ−

n

λ+n

γn+1γn−1

. . .. . .

νn−1

µn−1 νn

µn µn+1

νn+1

Figure 5.2: Periodic, Dirichlet and Neumann spectra for sufficiently large jnj.

H Auxiliary gap lengths

When’ is an arbitrary potential inHw , the spectral gap lengths �’�may all vanish and

thus may not provide any information about ’. In chapter 2 we thus consider addition

spectral data, more precisely, a family of auxiliary gap lengths ��’� � ��n�n2Z of ’,

characterized by the following three properties:

(a) �n is continuously differentiable on some neighbourhood U of ’,

(b) �n vanishes whenever ��n � ��n has a geometric multiplicity of two, and

(c) there are real numbers ��n , ��n and an integer NU Æ 1, such that

k@�n � tnko à 1=16; tn � 12�e2� in��n��; e�2� in��n��

�;

uniformly on U for jnj Æ NU .

Note that the factor 1=2 in the definition of tn is due to the usual normalization of

the Dirichlet and Neumann eigenfunctions used to express the gradient @�n in the case

�n � �n � �n and �n � �n � �n, respectively. Now, we make up the actual verification of

properties (a)-(c) for these two choices of auxiliary gap lengths.

Proposition H.1 For each w 2 M? the quantities �n � �n � �n, with �n the nth Dirichlet

or Neumann eigenvalue, form a set of auxiliary gap lengths on Bwm. ˇ

Proof. Since w is assumed to be monotone and unbounded, the space Hw is compactly

embedded into L2 by Rellich’s Theorem C.3. This allows us to translate results holding

locally uniformly on L2 into uniform results on bounded subsets of Hw . In particular,

the balls Bwm are compact in L2. Consequently, the Dirichlet and Neumann eigenvalues

satisfy

�n; �n � n� � ‘2n;


Appendix I: Three auxiliary estimates

uniformly on Bwm. Thus we can choose some N� Æ 1, such that �n and �n are simple for

all jnj Æ N�, hence their L2-gradients exists in every ’ 2 Bwm, and uniformly satisfy

@�n � 12�e��2n � e��2n�� ‘2n; @�n � �12�e

��2n � e��2n�� ‘2n:

Eventually increasing N�, the same applies to the mid-points �n. For jnj sufficiently large

they are analytic functions with

@�n � ‘2n;

uniformly on Bwm. Thus �n � �n � �n is analytic on Bwm which gives (a).

Moreover, the gradient of �n � �n � �n satisfies

@�n � @�n � @�n � 12�e2n; e�2n�� ‘2n;

uniformly on Bwm. So we can choose ��n � ��n � 0 and eventually increase N� to effect that

k@�n � tnko à 1=16; jnj Æ N�;

which gives (c) for the Dirichlet case. In a similar fashion one checks property (c) for the

Neumann case.

Suppose ��n � ��n � �n with geometric multiplicity two. Then we can choose g as a

nontrivial linear combination of functions from the two-dimensional periodic eigenspace

of �n such that

�g2 � g1��0 � 0 � �g2 � g1�

��1:

Thus g is a Dirichlet eigenfunction and �n � �n � �n. The Neumann case is treated

similarly, which gives (b) and completes the proof. v

I Three auxiliary estimates

Many calculations presented here involve the Dirichlet seriesXnÆ1

1ns; s > 1:

To make our choice of constants and thresholds convenient, we provide a simple estimate.

Lemma I.1 Let N Æ 1 and s > 1, thenXnÆN

1nsà�1� 1

s � 1�1Ns�1

;Xn>N

1nsà

1�s � 1�Ns�1 : ˇ



Proof. With ’�x� � �x �N � 1��s we may write the sum as

XnÆN

1ns�XnÆ1’�n�:

The function ’ is positive and monotone decreasing on the interval �1;1�. Hence,

XnÆN

1nsà ’�1��

Z11’�x�dx � 1

Ns� 1�s � 1�Ns�1 : v

To be consistent with the definition of w-norms for 2-periodic functions we redefine

the w-norm of a given complex valued sequence �uk� in this appendix as follows

k�uk�k2w ˝Xk2Zw2k jukj2:

Thus we have

u �Xk2Zukek 2 Hw� a �uk� 2 hw :

Moreover, the i-shifted w-norm of �uk� is given by

k�uk�k2w;i �Xk2Zw2k�ijukj2:

Lemma I.2 Let a � �ak� and x � �xk� be real positive sequences and

hk ˝Xl�n

ak�ljn� kj

x�ljn� lj ; k � n:

Then for any monotone weight w 2M we have

khk1;w;�n à 4 kakwpjnj � kRnakwwn

!kxkw;�n

à8wnkakwkxkw;�n; wn ˝min

�qjnj;wn

�: ˇ

Proof. Our aim is to estimate

khk1;w;�n �Xk�nwk�nhk �

Xk;l�n

wk�nak�ljn� kj

x�ljn� lj :

To proceed, we split up the latter sum into two parts �[ � �\ defined by the index sets

I[ ˝ fk; l � n : jn� kj > jnj=2 or jn� lj > jnj=2g ;I\ ˝ fk; l � n : jn� lj à jnj=2 and jn� kj à jnj=2g � Ic[:



Using Hölder’s inequality together with the submultiplicity of the weight w, we obtain

for the first term

�[ �XI[


x�ljn� lj

à XI[

1jn� kj2

1jn� lj2

!1=2 Xk;lw2k�la

2k�lw

2�l�nx

2�l

!1=2

à4pjnjkakwkxkw;�n;

where Lemma I.1 with s � 2 was applied in the final step.

Furthermore, for �k; l� taken from I\ we have 1 à jn� kj; jn� lj à jnj=2, hence

jk� lj Æ 2jnj � jn� kj � jn� lj Æ jnj Æ jn� kj; jn� lj Æ jnj:

Thus wk�nwn à wk�lw�l�n by the monotonicity of the weight w, and consequently

�\ �XI\


x�ljn� lj

à1wn

0@ Xk;l�n

1jn� kj2

1jn� lj2

1A1=20@XI\

w2k�la2k�lw

2�l�nx

2�l

1A1=2

à4wnkRnakwkxkw;�n: v

Lemma I.3 Let a � �ak�, b � �bk� and cn � �ck;n�, for any n 2 Z, be real positive se-

quences, with a uniform bound kcnkw;n à M for every n. Define

h2n ˝Xk;l�n

an�ljn� lj

bl�kjn� kjck;n:

Then for any monotone weight w 2M and for any N Æ 1, we haveXjnj>N

w22nh22n à

28

w2Nkak2wkbk2wM2; wN ˝min

�pN;wN

�: ˇ

Proof. To estimate the w-norm of h, we investigate the weighted coefficients given by

w2nh2n �Xk;l�n

w2nan�ljn� lj

bl�kjn� kjck;n:

For that purpose we define three index sets

Ink � f�k; l� : jn� kj > jnj=2g ;Inl � f�k; l� : jn� lj > jnj=2g ;In\ � f�k; l� : 1 à jn� kj; jn� lj à jnj=2g ;



and consider the above expression on each index set separately with the restrictions of

the series labeled �k;n, �l;n and �\;n, respectively. In contrast to the situation of Lemma

I.2, this time n is not fixed, hence we explicitly note the dependence on n of the � terms.

First, Hölder’s inequality together with the submultiplicity and symmetry of w gives

�2k;n � XInk

w2nan�ljn� lj

bl�kjn� kjck;n

!2

à4jnj2

XInk

w2n�kc2k;n

jn� lj2! X

k;lw2n�ka

2n�lb

2l�k

!

à16jnj2 kcnk

2w;nkak2wkbk2w :

Second, we find for �l;n after two applications of Hölder’s inequality,

�2l;n � XInl

w2nan�ljn� lj

bl�kjn� kjck;n

!2

à4jnj2

Xkw2n�kc

2k;n

! Xk�n

1jn� kj2

�Xlwn�kan�lbl�k

�2!

à4jnj2 kcnk

2w;n

Xk�n

1jn� kj2

�Xlw2n�la

2n�l

��Xlw2l�kb

2l�k

�!

à16jnj2 kcnk

2w;nkak2wkbk2w :

Together with Lemma I.1 and the uniform bound of kcnkw;n we thus obtain

Xjnj>N

��2k;n � �2l;n

�à 32kak2wkbk2wM2

Xjnj>N

1jnj2 à

64Nkak2wkbk2wM2:

It remains to consider the term �\;n. We start by an application of Hölder, which gives

�2\;n � XIn\

w2nan�ljn� lj

bl�kjn� kjck;n

!2

à Xkwn�kc2k;n

! X1àjn�kjàjnj=2

1jn� kj2

X1àjn�ljàjnj=2

wn�kan�lbl�kjn� lj

!2!:

Note that for �k; l� with 1 à jn� kj; jn� lj à jnj=2 we have

jk� lj Æ 2jnj � jn� kj � jn� lj Æ jnj; jn� lj Æ jn� kj:



Thus by the monotonicity of w, it follows that wn�kwn à wn�lwl�k, hence

�2\;n à1w2nkck2w;n

Xk�n

1jn� kj2

X1àjn�ljàjnj=2

w2n�la2n�l

jn� lj2! X

lw2l�kb

2l�k

!!

à4w2nkck2w;nkbk2w

X1àjmjàjnj=2

w22n�ma22n�m

jmj2!;

where we changed parameters n�l,m. Since j2n�mj Æ N for jnj Æ N and jmj à jnj=2,

we finally get

XjnjÆN

X1àjmjàjnj=2

w22n�ma22n�m

jmj2 àXj�jÆN

Xjmj�0

w2�a2�jmj2 à 4kak

2w :

Consolidating the estimates for �k;n, �l;n and �\;n, we arrive at

XjnjÆN

w22nh22n à 3

XjnjÆN

��2k;n � �2l;n � �2\;n

�

à28

w2Nkak2wkbk2wM2: v

For the sake of brevity and clarity we restricted ourselves to monotone weights. Now

we consider the general situation. Given a potential ’ 2 H v with an arbitrary weight

w 2 M, we can by Lemma C.1 always find an unbounded, monotone weight w such that

k’kw�v <1. To suppress the product structure in our notation we introduce the notion

v Ý w :a v � w � w for some w 2M:

In particular v Ý w implies that v Æ w. It turns out that Lemma I.2 and Lemma I.3 can be

adapted to fit into this situation.

Addendum to Lemma I.2 Let a � �ak� and x � �xk� be real positive sequences and

hk ˝Xl�n

ak�ljn� kj

x�ljn� lj ; k � n:

Then for any weight v Ý w with w monotone, we have

khk1;v ;�n à 4 kakvpjnj � kRnakvwn

!kxkv ;�n

à8wnkakvkxkv ;�n; wn ˝min

�qjnj;wn

�: ˇ



Proof. We proceed as in the proof of Lemma I.2 and split up the sum into two parts �[

and �\. We did not make use of the monotonicity to estimate �[ hence by exactly the

same arguments

�[ à4pjnjkakvkxkv ;�n:

By assumption v Ý w, so vw�1 is submultiplicative and symmetric, thus

vn�k � wn�k vn�kwn�kà wn�k

vn�lvl�kwn�lwl�k

:

Moreover, for �k; l� 2 I\ we have jl�kj à jnj and jn�lj Æ jn�kj. Hence the monotonicity

of w gives

vn�k à1wk�l

wn�kwn�l

vn�lvl�k à1wnvn�lvl�k:

Thus we can proceed as in the original proof to obtain

�\ à4wnkRnakvkxkv ;�n: v

Addendum to Lemma I.3 Let a � �ak�, b � �bk� and cn � �ck;n� be, for any n 2 Z, real

positive sequences, with an uniform bound kcnkv ;n à M for every n. Define

h2n ˝Xk;l�n

an�ljn� lj

bl�kjn� kjck;n:

Then for any weight v Ý w with w monotone and for any N Æ 1, we have

Xjnj>N

v22nh22n à

28

w2Nkak2vkbk2vM2; wN ˝min

�pN;wN

�: ˇ

Proof. We proceed as in the proof of Lemma I.3 by considering the terms �k;n, �l;n and

�\;n. To estimate the first two sums we did not make any use of monotonicity, hence we

obtain in a similar fashion

XjnjÆN

��2k;n � �2l;n

�à64Nkak2vkbk2vM2:

To estimate the last term �\;n, we argument as in the proof of the preceding addendum

to obtain for �k; l� 2 In\

vn�k à1wnvn�lvl�k:



Using this, we can proceed as in the proof of Lemma I.3 to obtain,

�2\;n à kckv ;n

Xk�n

1jn� kj2

� X1àjn�ljàjnj=2

v2n�kan�lbl�kjn� lj

�2!

à4w2nkckv ;nkbkv

X1àjmjàjnj=2

v22n�ma22n�m

jmj2!:

The conclusion now follows by exactly the same arguments. v


The End

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Eidesstattliche Erklärung

Hiermit erkläre ich, dass ich die vorliegende Arbeit selbständig und ohne fremde Hilfe

bzw. unerlaubte Hilfsmittel angefertigt, andere als die angegebenen Quellen und Hilfsmit-

tel nicht benutzt und die den benutzten Quellen wörtlich oder inhaltlich entnommenen

Stellen als solche kenntlich gemacht habe.

Stuttgart, den 19. Dezember 2011

Jan-Cornelius Molnar


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