Analysis of Generalized Eigenvalue Problems - IWR: · PDF fileAnalysis of Generalized...

Analysis

of

Generalized Eigenvalue Problems

Rike Betten and Eberhard Michel

(1/65)

Topics

A standard eigenvalue problem is of the form

Au = µu in L. (EP)

A generalized eigenvalue problem is of the form

Au = µBu in L. (GEP)

If A is invertible then the generalized eigenvalue problem is of the standard form

λu =(A−1B

)u in L,

for λ 6= 0.

Asymptotics of Non-local Spectral Problems (2/65)

Topics Aims

Aims

1. Description of the spectra of a family of GEPs.

2. Asymptotics of the spectra of a family of GEPs.

Illustration of the ideas using a toy problem:

Simple model for the motion of fuel rods ina core of a nuclear power plant.


Topics Overview of Talks

1. Fresh-Up in Spectral Theory and Functional Analysis: Eberhard

Fundamental Structures.

Parts of the Spectrum and their description.

Characterization of selfadjoint operators.

2. Motion of Tubes in Fluid: Rike

Physical Modeling of the problem.

Mathematical Formulation.

Bloch Wave Method.

Decomposition into independent rods.

3. GEPs and there Asymptotics: Eberhard

Mathematical Structure of the Problem.

Properties and Dependence: Finite number of rods.

Asymptotics of the Problem: Infinite number of rods.


Spectral Theory Contents

Fundamental Structures:

Spectrum σ, Resolvent set , . . . .

(Quadratic) Forms and Associated Operators.

Parts of the Spectrum and their Description:

Discrete Spectrum σd.

Minimax Principle.

Monotonicity of the Eigenvalues.

Characterization of selfadjoint operators:

Functional Calculus.

Spectral Resolution.


Spectral Theory Fundamental structures

1 Fundamental structures

1.1 Spectrum σ, Resolvent set , . . .

A : H ≥ D(A) −→ H a linear operator. Then consider

(λ−A)u = f in H.

Three questions:

Existence: For every f ∈ H exists a u ∈ H . Surjectivity of Aλ := λ−A.

Uniqueness: For every f ∈ H exists at most one u ∈ H . Injectivity of Aλ.

Stability: u depends continuously on f .


Hadamard’s Questions:

Existence: For every f ∈ H exists a u ∈ H

Surjectivity of Aλ.

Uniqueness: For every f ∈ H exists at most

one u ∈ H Injectivity of Aλ.

Stability: u depends continuously on f .

Closed Operators Stability for free.

6-1


Definition 1.1. The Resolvent Set (A) is

(A) =

λ ∈ C∣∣∣Aλ is bijective and Rλ := (Aλ)−1

is continuous

.

The Spectrum σ(A) is given by σ(A) := C \ (A).

If Aλ isn’t injective then λ is an Eigenvalue. The Point Spectrum σp(A) is the

set of eigenvalues.

Remark 1.2. Always σp(A) ⊆ σ(A), but in most cases σ(A) 6= σp(A).

If dim H <∞ then σ(A) = σp(A).

If A is finite or compact then σ(A) = σp(A) ∪ 0.



1.2 Forms and associated Operators

Definition 1.3 (Quadratic Form). H a Hilbert-space. A ∈ Lin(H,H∗) is called aSesquilinear Form, a.k.a Form. The mapping

H ∋ u 7→ A[u] := A[u, u] := Au (u)

is called the Quadratic Form.

Notation 1.4. Form and Quadratic Form determine each other uniquely by (1.3)and the so called Polarization Principle

A[u, v] =1

4

3∑

k=0

ikA[u+ ikv].



Definition 1.5 (Associated Operator). Let A be a form on H , L be an other H-spaces.t. H is dense in L. The inner product on L is denoted by 〈·, ·〉 = 〈·, ·〉L. Then theassociated operator A ( on L ) is given by

Au = Au for u ∈ D(A),

where D(A) =

u ∈ L∣∣∣ u ∈ H and Au ∈ L

. The space H , considered as

subset of L, is denoted by D(A).

Remark 1.6. Actually Au = Au is short for A ιH⊆Lu = ιL⊆H∗ Au. A is themaximal operator that makes the following diagram commutative.

Similarly u ∈ H is short for u ∈ ιH⊆LHand Au ∈ L stands for Au ∈ ιL⊆H∗L.For all practical purposes, each ι is basi-cally the identity.



Example 1.7 (Laplacian). Let L = L2(Ω) and A be formally given by Dirichlet’sIntegral

A[u] :=

∫

Ω

|Du|2 .

Let H := H1(Ω). The associated operator is called the Neumann-Laplacian

on Ω, denoted by −∆ΩN .

If H := H10 (Ω) the associated operator is called Dirichlet-Laplacian on Ω,

referred to as −∆ΩD.

If H := H1per(Ω) the associated operator is called Laplacian on Ω with periodic

boundary conditions, denoted by −∆Ωper.

If L and H are known, every Laplacian is referred to as −∆.



For u ∈ H the Rayleigh Quotient (of A in L) is given by

R[u] :=A[u, u]

〈u, u〉 .

The set of values of the Rayleigh Quotient is called the Numerical Range of A .It is denoted by

W (A) :=

R[u]∣∣∣0 6= u ∈ H

.

If W (A) ⊆

λ ∈ C ∣∣∣ c ≤ Reλ

for some c ∈ R then A is called

semibounded (from below). Short c ≤ ReA. A is called symmetric or realvalued if A[u] ∈ R for u ∈ H . In this case semiboundedness is denoted byc ≤ A.



Theorem 1.8 (Lax-Milgram). If 0 < c ≤ ReA then A is bijective.

Example 1.9 (Laplacians). Let A be given by A[u] :=∫

Ω|Du|2.

Let H := H1(Ω). By construction A[u] ≥ 0. Since A vanishes on constantsC ≤ H , A is not strictly positive. But

0 < c ≤ A in H/C. Let H := H1

0 (Ω). Since C 6≤ H , the form A is strictly positive. Hence,

0 < c ≤ A in H.

Let H := H1per(Ω). Since C ≤ H , A is not strictly positive. But

0 < c ≤ A in H/C.Asymptotics of Non-local Spectral Problems (12/65)

Hausdorff called W (A) Wertevorrat ofAor Wertebereich ofA Halmos, Werner,

. . .

12-1


2 Description of the Spectrum

2.1 Discrete Spectrum

Definition 2.1. Let A be a selfadjoint operator. The discrete spectrum is denoted by

σd(A) =

λ ∈ σp(A)∣∣∣ dim Eigλ(A) <∞ and dist(λ, σ(A) \ λ) > 0

.

The essential spectrum is given by

σe(A) = σ(A) \ σd(A).


Spectral Theory Towards the Minimax Principle

2.2 Minimax Principle

Remark 2.2 (Rayleigh Principle). Let A be a selfadjoint operator on Cn. Then thesmallest and largest eigenvalues are given by

λ1 = min W (A) = min06=φ∈Cn

Aφ · φφ · φ and (2.1)

λn = max W (A) = max06=φ∈Cn

Aφ · φφ · φ , (2.2)

respectively.


Spectral Theory Minimax Principle

Theorem 2.3 (Minimax Principle). Let A be a closed, symmetric, and semiboundedform and A its associated operator. Then the discrete point spectrum σd(A)including multiplicities is enumerated by

λ−k (A) = minU≤H

dim U=k

max06=φ∈U

R[u] = minU≤D(A)dim U=k

max06=φ∈U

R[u]

starting from the left and by

λ+k (A) = max

U≤Hdim U=k

min06=φ∈U

R[u] = maxU≤D(A)dim U=k

min06=φ∈U

R[u]

starting from the right.

λ−1

λ−2

λ−3 . . . λ+

1

λ+2

. . . λ+3


Spectral Theory Minimax Principle

Remark 2.4. The situation depicted occurs rather seldom. More often . . .

The spectrum is unbounded above. All Laplacians.

The spectrum consists only of discrete eigenvalues accumulating at infinity. All operators with compact resolvents, esp. all “our” Laplacians. In this casedenote

λk(A) = λ−k (A) = minU≤H

dim U=k

max06=φ∈U

R[u].

We are interested in the first two eigenvalues (counting multiplicities). Hence,the following equalities are important.

λ1(A) = min06=φ∈H

R[u],

λ2(A) = minU≤H

dim U=2

max06=φ∈U

R[u].


Spectral Theory Monotonicity

2.3 Monotonicity

Notation 2.5. Let A1 and A2 be two forms and A1 and A2 be the associatedoperators.

We write A1 ≤ A2 if D(A1) ⊇ D(A2) and A1[u] ≤ A2[u] for all

u ∈ D(A2).

By abuse of notation we write A1 ≤ A2 instead of A1 ≤ A2.

Corollary 2.6 (Monotonicity). Let A1 ≤ A2 then λ−k (A1) ≤ λ−k (A2).

Proof. By definition, A1 ≤ A2 impliesU ≤ D(A2) ≤ D(A1). Hence, the minimum

for λ−k (A1) is found in a set of subspaces that is at least equal to the set of subspaces

λ−k (A2) is a minimum on.


Spectral Theory Monotonicity

Example 2.7 (Laplacians). Since

H10 (Ω) ≤ H1

per(Ω) ≤ H1(Ω),

for an appropriate set Ω, we have

−∆ΩN ≤ −∆Ω

per ≤ −∆ΩD .

Hence, we get the monotonicity of the eigenvalues

λ−k(−∆Ω

N

)≤ λ−k

(−∆Ω

per

)≤ λ−k

(−∆Ω

D

).


Story from the end

Linear Algebra: Selfadjoint Operators are

Diagonal matrices.

18-1

Spectral Theory Selfadjoint Operators

3 Representation of Selfadjoint Operators

3.1 Functional Calculus

Notation 3.1 (Multiplication Operators). Let Ω ⊆ Rn be an open set anda ∈ C(Ω,R) a continuous function hereon. On the space D(Ma), given by

D(Ma) :=v ∈ L2(Ω)

∣∣ a · v ∈ L2(Ω)

,

we can define an operator by

Mav := a · v.Hence, Mav (ξ) = a(ξ) v(ξ) for v ∈ D(Ma). Such an operator is called aMutiplication Operator.


Spectral Theory Selfadjoint Operators

Mutiplication Operators generalize the notion of diagonal matrix. They have similarproperties:

For all λ ∈ C, the translated operator is given by

λ−Ma = Mλ−a.

The spectrum is given by the image of a:

σ(Ma) = a(ξ)|ξ ∈ Ω.

(λ−Ma)−1 = M(λ−a)−1 .


Every selfadjoint operator is actually a multipli-

cation operator.

20-1

Spectral Theory Functional Calculus: Multiplicator Version

Theorem 3.2 (Functional Calculus). Let A be a selfadjoint operator on the Hilbertspace L. Then A is unitary equivalent to a Multiplication Operator Ma, i.e.There exists a measure space (Ω,B, µ), an unitary mapping U : L→ L2(Ω, µ),and a measurable function a : Ω → R.Let D(Ma) :=

v ∈ L2(Ω)

∣∣ a · v ∈ L2(Ω)

and Mav := a · v for

v ∈ D(Ma). Then the following diagram commutes:

In words:

A = U∗MaU on D(A) = U∗D(Ma)

or equivalently for v ∈ D(Ma)

a · v = (UAU∗) v a.e. in Ω.


Spectral Theory Functional Calculus: Multiplication Version

Example 3.3. 1. If A is a selfadjoint operator on Cn there exist a1, . . . , an ∈ Rsuch, that

AU∼=

a1 0 · · · 0

0 a2 0...

0. . . 0

... 0 an−1 0

0 · · · 0 an

.

Think of a to be a : 1, . . . , n → R.

2. Let A be −∆ in L2(Rn) and U be the Fourier transform. Then for a(ξ) = |ξ|2

−∆u (x)U∼= |ξ|2 v(ξ), in the sense of

−∆U∼= Ma

D(−∆) = u ∈ L∣∣ v = Uu ∈ L2(Rn) satisfies av ∈ L2(Rn).


Spectral Theory Functional Calculus: Motivation

Remark 3.4. Let f : R→ R be continuous.

1. Let A be selfadjoint operator on Cn then define

f(A)U∼=

f(a1) 0 · · · 0

0 f(a2) 0...

0. . . 0

... 0 f(an−1) 0

0 · · · 0 f(an)

.

2. Let A be −∆ in L2(Rn). Then for a(ξ) = |ξ|2,

e∆tu (x)U∼= e−|ξ|2t v(ξ)

defines the fundamental solution of the heat equation, i.e. the solution of

ut = ∆u with initial value u0 ∈ L2(Rn) is given by u(t) = e∆tu0.Asymptotics of Non-local Spectral Problems (23/65)

Spectral Theory Towards the Functional Calculus

Let A ∈ Lin(L) then

A0 = Id, A,A2, . . . , Ak, . . . ∈ Lin(L),

for k ∈ N are well defined. Hence, a linear combination like

p(A) =

grad p∑

k=0

pkAk ∈ Lin(L)

is well defined for every polynomial p(X) =∑grad p

k=0 pkXk ∈ C[X].

For example p(X) = X3 + i yields the operator p(A) = A3 + i Id, or short

A3 + i, resp.

Remark 3.5. Since A is selfadjoint, ||A|| = sup σ(A) = || Id ||C(σ(A)).

For p ∈ R[X] the operator p(A) is selfadjoint.

For p ∈ C[X] the operator p(A) is normal.


Spectral Theory Functional Calculus for polynomials

Theorem 3.6 (Functional Calculus). For each selfadjoint operator A ∈ Lin(L) there

exists a unique Functional Calculus Φ : C[X] → Lin(L), i.e.

Standardization and Normalization: Φ(1) = 1 and Φ(Id) = A.

Algebra Homomorphism: Φ(αp+ q) = αΦ(p) + Φ(q) and Φ(pq) = Φ(p)Φ(q)for α ∈ C and p, q ∈ C[X].

Involutive: Φ(p) = Φ(p)∗ for p ∈ C[X].

Isometry and Continuity: ||Φ(p)|| = ||p||C(σ(A)) for p ∈ C[X].

Hence, a Functional Calculus is a

continuous, 1 − 1, A-normalized algebra homomorphism between the C-algebras(C[X],+, ·, ·, 0, 1) and (Lin(L),+, , ·∗, 0, 1 = Id)

or short ... C[A] ≤ Lin(L) has all the properties of C[X] ≤ Cb(σ(A)).


Spectral Theory Towards a more general Functional Calculus

Remark 3.7 (Extension of the Functional Calculus).

Density C[X] in C(σ(A)) yields Functional Calculus: For f ∈ C(σ(A))define

f(A) = lim pk(A) in Lin(L).

Density of C(σ(A)) in L∞(σ(A)), the space of bounded, measurablefunctions, w.r.t. pointwise limits of uniformly bounded sequences. Forf ∈ L∞(σ(A)) define

f(A)x = lim fk(A)x in L.


Spectral Theory Functional Calculus for bdd meas’able functions

Theorem 3.8 (Functional Calculus).For each bounded, selfadjoint operator A there exists a unique Functional CalculusΦ : L∞(σ(A)) → Lin(L), i.e.

Standardization and Normalization: Φ(1) = 1 and Φ(Id) = A.

Algebra Homomorphism: Φ(αf + g) = αΦ(f) + Φ(g) and Φ(fg) = Φ(f)Φ(g)for α ∈ C and f, g ∈ L∞(σ(A)).

Involutive: Φ(f) = Φ(f)∗ for f ∈ L∞(σ(A)).

Continuity: ||Φ(f)|| = ||f ||L∞(σ(A)) for f ∈ C[X].

So, for short, we get an embedding of L∞(σ(A)) into Lin(L), which respects all the

structures of the commutative C-algebra L∞(σ(A)).


Spectral Theory Functional Calculus: Example Wave Equation

Example 3.9 (Wave Equation). Let A be a non-negative, bounded operator. Thenthe solution of the wave-type equation

utt + Au = 0 in L×R,u(0) = u0 in L, and

ut(0) = v0 in L

is given by

u(t) = Cos(t)u0 + Sin(t) v0,

where the Cosine-Family Cos = Cos(t)t∈R and the Sine-Family

Sin = Sin(t)t∈R are given by

Cos(t) = cos(√

A t)

and Sin(t) =

∫ t

0

Cos(s) ds.


Spectral Theory Functional Calculus: : Example Wave Equation

Proposition 3.10.Let A be a bounded, selfadoint operator. Then the associated cosine family Cos andsine family Sin have the following properties:

1. Sin(t) and Cos(t) are bounded and selfadjoint operators.

2. Sin : R→ Lin(L) and Cos : R→ Lin(L) are differentiable. Moreprecisely:

d

dtSin(t) = Cos(t) for all t ∈ R,

d

dtCos(t) = A Sin(t) for all t ∈ R.


Spectral Theory Towards Spectral Resolution

For each (measurable) set U ⊆ R, the characteristic function

1U (x) =

1 for x ∈ U, and

0 otherwise,

can be used to define a projection

EA(U) := 1U(A) ∈ Lin(L).

Proposition 3.11. EA(U) is a orthogonal projection for each U ⊆ R.

Proof. EA(U) is a projection due to the properties of the functional calculus:

EA(U)EA(U) = 1U (A)1U(A) = (1U1U) (A) = 1U(A) = EA(U).

It is selfadjoint by EA(U)∗ = (1U ) (A)∗ = 1U(A) = 1U(A) = EA(U).Selfadointness is equivalent to orthogonal decomposition of the image and thekernel.


Spectral Theory Spectral Resolution

Theorem 3.12 (Spectral Resolution relative to A.). Let A ∈ Lin(L) be selfadjoint.

Then EA : B (σ(A)) → Lin(L) is a spectral measure, i.e.

1. EA(∅) = 0 and EA(R) = 1.

2. Let Uk ⊆ R be pairwise disjoint, then

EA

(⋃

k∈NUk)

=∑

k∈NEA(Uk).

3. Let U1, . . . , UN ⊆ R be any sets, then

EA

(N⋂

k=1

Uk

)

=N∏

k=1

EA(Uk).

Since 1σ(A) = 1R, considered as functions in L∞(σ(A)), suppEA = σ(A).

Hence, EA is a Lin(L) valued measure with compact support σ(A).


Spectral Theory Consistency

Theorem 3.13 (Spectral Resolution and Functional Calculus are consistent).Let A ∈ Lin(L) be selfadjoint, EA : B (σ(A)) → Lin(L) its spectral measure.

For f ∈ L∞ (σ(A)) and u, v ∈ L

〈f(A)u, v〉 =

∫

f(λ) d 〈EA(λ)u, v〉 .

Remark 3.14. The integral∫f(λ) dEA(λ) is constructed similar to the

Lebesgue-Integral: Integral for simple functions; integration is continuous; simplefunctions are dense in L∞ (σ(A)).


Spectral Theory Generalizations

Remark 3.15.

Decomposition of a normal operator A = S1 + iS2 into two commutingselfadjoint operators generalizes the Functional Calculus and the SpectralResolution to normal, bounded operators.

Every λ ∈ C \R is in the resolvent set (A) of a not necessarily bounded, but

selfadjoint, operator A. Since Rλ = (λ− A)−1is a bounded and normal

operator, all ideas can be generalized to the unbounded case.


Spectral Theory Overview

Finally, we are able to state the following:

Let A be a selfadjoint operator, A be its form with domain D(A). Then the firstand second eigenvalue, counting multiplicities, are given by

λ1(A) = min06=φ∈D(A)

A[φ, φ]

〈φ, φ〉

λ2(A) = minU≤D(A)dim U=2

max06=φ∈U

A[φ, φ]

〈φ, φ〉 .

So it is sufficient to work with the quadratic form domain.

Let S be a selfadjoint and bounded operator. For a continuous function f theoperator f(S) is defined.

A selfadjoint and bounded operator S is characterized uniquely by its SpectralResolution: S =

∫λ dES(λ).


Generalized Eigenvalue Problem Introduction

4 Structure of the Problem

4.1 Framework of the involved Problems.

The considered GEPs are of the following form:

∫

Ω

Dφ • Dψ

︸︷︷︸

=:A[φ,ψ]

= µ∑

α

(∫

γ(α)

φν

)

︸︷︷︸

=: bα(φ)

•(∫

γ(α)

ψν

)

︸︷︷︸

=: bα(ψ)︸︷︷︸

=: B[φ,ψ]

(GEP)

Here A and B are well defined forms on the considered H1−spaces and every sub-

space thereof. Both vanish on constants. Therefore, we identify constant functions

with zero.


Reason for the •: Complex scalar product! In ℓ2.

35-1

Generalized Eigenvalue Problem Involved Sets

γ(α) Y ∗(α)

Γ

τ1

τ2

Y ∗

γ

The complete reference cell is given by

Y = (0, τ1) × (0, τ2).

Removing the tube with smooth boundary γ fromthe cell yields the reference cell Y ∗.For α ∈ Z2 the translated cell/boundary/... isgiven by

Y (α) = Y + (α1τ1, α2τ2)

Y ∗(α) = Y ∗ + (α1τ1, α2τ2)

γ(α) = γ + (α1τ1, α2τ2)

. . . = . . .


Generalized Eigenvalue Problem Involved Sets

The considered domains with a finite and an infinite number of tubes are given by

Ωn = Ω ∩ (−nτ1, (n− 1)τ1) × (−nτ2, (n− 1)τ2)

and

Ω∞ = Int

(⋃

α∈Z2

Y ∗(α)

)

,

respectively.

The domain Ω2.


Generalized Eigenvalue Problem Boundary Conditions

We use the following boundary conditions on each bounded domain:

Neumann-Condition.

Ωn-Periodicity.

ω-Periodicity on each cell.

Dirichlet-Condition on each cell.

In the case of infinite number of tubes we only will pose a bound on the growth of the

functions at infinity: Dφ ∈ L2. These ideas will be modeled through the choice of

spaces.


Generalized Eigenvalue Problem Boundary Conditions

Asymptotic bound on the gradient:

H∞ :=(C∞c (Ω∞)

)∼|·|H1

=φ ∈ L2

loc(Ω∞)∣∣ Dφ ∈ L2(Ω∞)2

/C.

Neumann-Condition: Hn := H1(Ωn)/C.

Ωn-Periodicity: Hn,per := H1per(Ωn)/C.

ω-Periodicity on each cell: Some contain the constants. Hence,For ω 6= (1, 1), Hω := H1

per(ω, Y∗).

For ω = (1, 1), Hω := H1per(Y

∗)/C.

Furthermore, we abbreviate Hα := Hωα .

Each of this spaces is a Hilbert space with inner product A.

Here n ∈ N, ω ∈ SC × SC, and α ∈ Z2.


Generalized Eigenvalue Problem The operators

As we will see, on each of these spaces the forms A and B and the operator b makesense. Their respective versions will have the following names:

Asymptotic Problem on H∞: A∞, B∞, and b∞.

Neumann-Condition on Hn: An, Bn, and bn.

Ωn-Periodicity on Hn,per : An,per, Bn,per, and bn,per.

ω-Periodicity on each cell Hω or Hα: Aω, Bω, and bω or Aα, Bα, and bα.


We will now consider each problem separately:

Rike: Decomposition Theorem

40-1

GEP Properties of the Cell Problem: Hω.

The eigenvalue problem on the cell is of the form: Find 0 6= φω ∈ Hω such that

Aω φω = µ Bω φω in H∗ω.

By construction, A is bijective, since it actually defines the inner product on Hω.Hence, the problem is equivalent to

λφω = (Aω)−1 Bω

︸︷︷︸

=:Cω

φω in Hω.

Thus, to solve our generalized eigenvalue problem, we have to locate the non trivialpart

σp(Cω) \ 0

of the point spectrum of the bounded operator Cω.



For r ∈ C2 we can define an operator Bω,red : C2 → H∗ω

by Bω,redr = r · bω, or equivalently, for φ ∈ Hω

Bω,redr (φ) = r ·∫

γ

φν ∈ C,which enables us to split Bω into bω and Bω,red:

Bω[φω, ψω] = bω(φω) · bω(ψω)= (Bω,redbω(φω)) (ψω) .

For a eigenfunction φω of Cω and rω := bω(φω) we get

λ rω = λ bω(φω) = bω((

(Aω)−1 Bω

)φω)

= bω (A−1ω Bω,red) (bω(φω)) =

(bωA−1

ω Bω,red

)

︸︷︷︸

=:Sω

rω



The main idea now is: The following problems describe the same set of eigenvalues

µ 6= 0 including all multiplicities, where λ = 1µ

.

1. Find 0 6= φω ∈ Hω, such that

Aω φω = µ Bω φω in H∗ω.

2. Find 0 6= φω ∈ Hω, such that

λφω = Cω φω in Hω.

3. Find 0 6= r ∈ C2, such that

λ r = Sωr in C2.

So we reduced the complexity of problem (1) to the simpler form of problem (3).



Proposition 4.1. The operators bωω, given by

bω : Hω → C2 : φ 7→∫

γ

φν,

are linear, continuous, and surjective. Actually, it can be shown, that bω is alreadysurjective if restricted to

H0 :=φ ∈ H1(Y ∗)

∣∣ φ = 0 on ∂Y

.

Proposition 4.2. Aω defines an inner product on Hω, which is equivalent to thestandard inner product.



Proposition 4.3. The operators Sωω are selfadjoint and satisfy λ ≥ Sω ≥ λ, i.e.

λ r · r ≥ Sωr · r ≥ λ r · r

for all r ∈ C2 and ω ∈ SC × SC, where 0 < λ ≤ λ <∞ can be chosenindependent of ω.

Proof. Let rω ∈ C2 and define φω := A−1ω Bω,redrω ∈ Hω. We getR ∋ Aω[φω, φω] = Aω (A−1

ω Bω,redrω) (A−1ω Bω,redrω)

= Bω,redrω (A−1ω Bω,redrω)

= rω · b (A−1ω Bω,redrω)

= rω · Sωrω.

Hence, Sωrω · rω = rω · Sωrω = rω · Sωrω proves symmetry of all Sω andAω[φω, φω] ≥ 0 implies non-negativity.



Idea of the proof of the uniform bounds:

1. The upper bound is given by the largest eigenvalue of Sω. But this eigenvalue isalso the largest eigenvalue of Cω. Therefore it is given by the minimax principleas maximum over Hω. Since Hω ≤ H1(Y ∗)/C the largest eigenvalues are

uniformly bounded by the largest eigenvalue of A−1B on H1(Y ∗)/C. Thisproves the upper bound.

2. The second largest eigenvalue of Sω, which b.t.w. is also the smallesteigenvalue, can be described as the second largest eigenvalue of Cω:

λ+2 (Sω) = λ+

2 (Cω) = maxU≤Hω

dim U=2

min06=φ∈U

〈Cωφ, φ〉Hω

〈φ, φ〉Hω

.

Let

H0 :=φ ∈ H1(Y ∗)

∣∣ φ = 0 on ∂Y

.



This space can be considered as H0 ≤ Hω. Hence, monotonicity in the spacesgives

λ+2 (Sω) = λ+

2 (Cω) ≥ maxU≤H0

dim U=2

min06=φ∈U

〈Cωφ, φ〉Hω

〈φ, φ〉Hω

,

i.e.

λ+2 (Cω) ≥ λ+

2 (C|H0) > 0.

This proves the lower bound, independent of ω.

Let µk(Sω) := 1λk(Sω)

, and let r1ω and r2

ω be the two eigenvectors of Sω. Hence

µkSωrkω = rkω for k, l = 1, 2.



We can assume

rkω · rlω = µl(Sω)δkl for k, l = 1, 2.

Let

φkω := A−1ω Bω,redr

kω for k = 1, 2.

Then for k = 1, 2, we get Aωφkω = Bω,redr

k, and therefore for ψ ∈ Hω

Aω[φkω, ψ] = rkω · b(ψ) = µkSωr

kω · b(ψ)

= µkb(A−1ω Bω,redr

kω

︸︷︷︸

=φkω

)· b(ψ) = µkB[φkω, ψ].

Hence, Aωφkω = µkBφkω for k, l = 1, 2. Furthermore,

Aω[φkω, φ

kω] = µkB[φkω, φ

kω] = µkb

(A−1ω Bω,redr

kω

)· b(A−1ω Bω,redr

lω

)

= µkSωrkω · Sωrlω = µkλkλlr

kω · rlω

= µkλkλlµlδkl = δkl.Asymptotics of Non-local Spectral Problems (48/65)


Let now µ 6= 0 be a generalized eigenvalue with eigenfunction φ ∈ Hω, i.e.

Aωφ = µkBφ. Then define r := bω(φ). Since 0 6= φ = µk (A−1ω Bω,red) r, the

vector r must non trivial. Furthermore, r = µkbω (A−1ω Bω,red) r = µkSωr shows,

that (r, µk) must be a characteristic pair of Sω. Hence r is linear dependent on r1ω

and r2ω and equals µ1 or µ2.

Theorem 4.4. For all ω ∈ SC × SC the complete set of generalized eigenvalues isgiven by

0 < µ ≤ µ1(ω) ≤ µ2(ω) ≤ µ < ∞.

The corresponding eigenfunctions φ1ω and φ2

ω ∈ Hω can be chosen such, that fork, l = 1, 2

⟨φkω, φ

lω

⟩

Hω= A[φkω, φ

lω] = δkl and Aφkω = µk Bφkω.


The bigger problem

49-1

Generalized Eigenvalue Problem Going to Hn,per.

The considered domain with a finite number of tubes is given by

Ωn = Ω ∩ (−nτ1, (n− 1)τ1) × (−nτ2, (n− 1)τ2)

The domain Ω2.


GEP Properties of the periodic Problem: Hn,per.

Let ω = (e2πim , e

2πin ) and denote for α ∈ Z2

ωα = (e2πiα1

m , e2πiα2

n ).

Then the following Orthogonal Decomposition Theorem holds true:

Hn,per =⊕

α∈−n,...,n−12 Hα

For α 6= β and uα ∈ Hα, uβ ∈ Hβ :

uα ⊥L2 uβ, uα ⊥An,peruβ, and uα ⊥Bn,per

uβ.

For uα ∈ Hα

An,peruα ∈ H∗α and Bn,peruα ∈ H∗

α.

Hence, An,per and Bn,per can be considered as

An,per =⊕

Aα and Bn,per =⊕

Bα.



Therefore we get the following formulations of the eigenvalue problems:

An,per φn = µ Bn,per φn in H∗n,per,

λ φn = Cn,per φn in Hn,per,

λ r = Sn,perr in ℓ2n(C2).

Here, the space ℓ2n(C2) is given by

ℓ2n(C2) := ℓ2(−n, . . . , (n− 1)2;C2).

Let Ln := 4n2, i.e. the number of tubes, then ℓ2n(C2) is just a strange way, to

describe the space C2Ln with the standard inner product. For r, s ∈ ℓ2n(C2) theproduct is denoted by r · s. The decomposition theorem provides us with compatibledecompositions. We can consider the problems in each space Hα. Reformulatingthis approach and adding up all the results, we get the main theorem.



Theorem 4.5. For every n ∈ N each generalized eigenvalue problem

∫

Ω

Dφ · Dψ = µ∑

α

(∫

γ(α)

φν

)

·(∫

γ(α)

ψν

)

, (GEP)

posed in H1per(Ωn)/C, has exactly 8n2 = 2×Number of holes solutions, counting

all multiplicities. They satisfy independently of n

0 < µ ≤ µ1 ≤ µ2 ≤ . . . ≤ µ8n2−1 ≤ µ8n2 ≤ µ < ∞.

The corresponding eigenfunctions φk, for k, l = 1, . . . , 8n2, chosen such, that

⟨φk, φl

⟩

Hn,per= A[φk, φl] = δkl and An,perφ

k = µk Bn,perφk,

are called Bloch Waves (of the GEP).


GEP Properties of the Problem: Hn.

Like in the periodic case, the eigenvalue problems

An φn = µ Bn φn in H∗n,

λ φn = Cn φn in Hn,

λ r = Snr in ℓ2n(C2),

are considered.

Contrary to the periodic case, we lack a decomposition theorem. Nevertheless, everySn has the following properties, which can be proved similarly to the case Sω.

Proposition 4.6. For all n ∈ N the operators Sn are selfadjoint and uniformly strictlypositive:

Snrn · rn ≥ λ rn · rn for all rn ∈ ℓ2n(C2) and n ∈ N.Asymptotics of Non-local Spectral Problems (54/65)

GEP Properties of the Problem: Hn.

Since Sn is a selfadjoint operator, it possesses a system r(k)n of eigenfunctions.

These can be transferred into eigenfunction of Cn like in the periodic case, using

φ(k)n = A−1

n Bn,red r(k)n . Finally we get

Theorem 4.7. For all n ∈ N each generalized eigenvalue problem

∫

Ω

Dφ · Dψ = µ∑

α

(∫

γ(α)

φν

)

·(∫

γ(α)

ψν

)

for all ψ ∈ Hn,

posed in Hn, has exactly 8n2 = 2×Number of holes solutions, counting allmultiplicities. They satisfy independently of n

0 < µ ≤ µ1 ≤ µ2 ≤ . . . ≤ µ8n2−1 ≤ µ8n2 .

The corresponding eigenfunctions φ(k)n , for k, l = 1, . . . , 8n2, can be chosen

orthonormal in Hn.


GEP Properties of the Asymptotic Problem: H∞.

We want these diagrams

to work.

To prove, that similar ideas also work in this case, the infinitenumber of tubes case needs some consideration.

Proposition 4.8. Denote ℓ2(C2) := ℓ2(Z2;C2). The operator

b∞ : H∞ → ℓ2(C2) : φ 7→(∫

γ(α)

φν

)

α∈Z2

is well-defined and continuous.

Proof. For φ ∈ H∞ we have on each cell

∣∣∣∣

∫

γ(α)

φν

∣∣∣∣

2

≤ ||φ||2L2(γ(α)) ||ν||2L2(γ(α))2

≤ c||Dφ||2L2(Y ∗(α)),

since on H1(Y ∗(α))/C the norm is given by ||D ·||L2(Y ∗(α)).


GEP Properties of the Asymptotic Problem: H∞.

This constant is independent of α. Thus,

|b∞(φ)|2 =∑

α

∣∣∣∣

∫

γ(α)

φν

∣∣∣∣

2

≤ c∑

α

||Dφ||2L2(Y ∗(α))

= c∑

α

||Dφ||2L2(Y ∗(α)) = c||Dφ||2L2(Ω∞)

= c||φ||2H∞

.

Hence, b∞ is well defined and continuous. The continuity of b∞ yields the continuity of B∞ and of B∞,red. So we obtain

Proposition 4.9. The operator

S∞ : ℓ2(C2) → ℓ2(C2) : r 7→ b∞A−1∞ B∞,red r

is well-defined, continuous, and selfadjoint.


GEP Asymptotics.

Remark 4.10 (Extension of Sn to ℓ2(C2)). Every operator Sn can be considered as

an operator on ℓ2(C2) by restriction and extension:

An element rn ∈ ℓ2n(C2) can be extended by zero to an element r∞ ∈ ℓ2(C2).

This defines an embedding ℓ2n(C2) ≤ ℓ2(C2).

An element r∞ ∈ ℓ2(C2) can be restricted to an element rn ∈ ℓ2n(C2).

Therefore, Snr∞ ∈ ℓ2(C2) is well defined for r∞ ∈ ℓ2(C2).

Theorem 4.11 (Extension of Hn to H∞). There exists a sequence of operatorsEn ∈ Lin(Hn, H∞), that is uniformly continuous: For φ ∈ Hn

Enφ(x) = φ(x) for x ∈ Ωn,

||Enφ||H∞≤ c||φ||Hn

, and

Enφ = 0 on γ(α) if α 6∈ −n, . . . , n− 12,

where c is independent of n and φ.


GEP Asymptotics.

Now we are able to state the first asymptotic result.

Theorem 4.12. For all r ∈ ℓ2(C2) we have

Snr → Sr in ℓ2(C2) for n→ ∞.

Especially,

Sr · r ≥ λ r · r for all r ∈ ℓ2(C2),

where λ is the uniform positivity constant of the operators Sn.

Let Sinn and Cosn be the sine and cosine families generated by Sn, and Sin∞ and

Cos∞ be the families generated by S∞. All these operator families can be considered

as families of operators in ℓ2(C2).


GEP Asymptotics.

For example, the cosine family is given by

Cosn : R→ Lin(ℓ2(C2)

): t 7→ cos(

√

Sn t)

For r ∈ ℓ2(C2), we have Cosn r : R→ ℓ2(C2) : t 7→ cos(√Sn t) r.

For r, s ∈ ℓ2(C2), Cosn r · s : R→ C : t 7→ cos(√Sn t)r · s.

Theorem 4.13. All operators are considered as operators on ℓ2(C2). Then for

r, s ∈ ℓ2(C2)

Sinn r · s ∗ Sin∞ r · s in L∞(R) for n→ ∞ and

Cosn r · s ∗ Cos∞ r · s in L∞(R) for n→ ∞.


GEP Asymptotics.

Finally, we are able to state and prove the main theorem.

Theorem 4.14. or r, s ∈ ℓ2(C2)

E√Snr · s −→ E√

Snr · s in S(R)∗ for n→ ∞.

Proof. For r, s ∈ ℓ2(C2) the convergence

Cosn r · s ∗ Cos∞ r · s in L∞(R) for n→ ∞

implies

Cosn r · s −→ Cos∞ r · s in S(R)∗ for n→ ∞.


GEP Asymptotics.

By consistency of Functional Calculus and Spectral Resolution we have

Cosn(t) r · s =

∫ ∞

λ

cos(√λ t) dESn

(λ)r · s

=

∫ ∞

√λ

cos(λ t) dE√Sn

(λ)r · s

=1

2

∫ ∞

√λ

eiλ t + e−iλ t dE√Sn

(λ)r · s

=1

2

∫ ∞

√λ

eiλ t d(E√

Sn(λ) − E√

Sn(−λ)

)r · s

︸︷︷︸

=: µn(λ)

.


GEP Asymptotics.

Similarly, we define the measure µ∞, and obtain

Cosn(t) r · s =1

2

∫

eiλ t dµn(λ) and

Cos∞(t) r · s =1

2

∫

eiλ t dµ∞(λ).

Therefore, convergence of Cosn r · s to Cos∞ r · s in S(R)∗ is equivalent to

Fµn −→ Fµ∞ in S(R)∗ as n→ ∞.

Inverting the Fourier transform in S(R)∗ proves

µn −→ µ∞ in S(R)∗ as n→ ∞.


GEP Asymptotics.

Since, in the sense of measures,

µn(λ) =

E√Sn

(λ)r · s for λ ≥ λ

0 −λ < λ < λ

−E√Sn

(−λ)r · s for λ ≤ −λ,

we can reformulate this result and get

E√Snr · s −→ E√

S∞

r · s in S(R)∗ as n→ ∞.


References

Freddy Aguirre and Carlos Conca, (1988). Eigenfrequencies of a Tube Bundle Immersed in

a Fluid, Appl. Math. Optim 18, pp 1–38, Springer Verlag.

M. or Carlos Conca, (1988). A Spectral Problem arising in Fluid-Solid Structures, Computer

Methods in Applied Mechanics and Engineering 69, pp 215–242, North-Holland.

Carlos Conca, J. Planchard, and M. Vanninathan, (1995). Fluids and Periodic Structures.

P.D. Hislop and I.M. Sigal, (1996). Introduction to Spectral Theory.

Tosio Kato, (1995). Perturbation Theory of Linear Operators.

Dirk Werner, (1995). Funktionalanalysis.

Kosaku Yosida, (1995). Functional Analysis.


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