A BRIEF FORAY INTO THE REGULAR STURM LIOUVILLEPROBLEM

RAPHAEL STUHLMEIER

1. Introduction

1.1. A first overview. The Sturm Liouville equation is a second order linear ordinarydifferential equation of the form

(1) (p(x)y(x)) + (q(x) w(x))y(x) = 0for some C, x I = [a, b], and y C2(I). It was first introduced in a 1837publication [7] by the eminent French mathematicians Joseph Liouville (1809 - 1882)and Jacques Charles Francois Sturm (1803 - 1855).

At this point, our initial questions might be: why is this problem important? Whatcan we say about the structure of this equation? How about the solutions? We will tacklea few basic results and hope to provide a little insight into each of these questions.

First we note that the Sturm Liouville eqn. (1) can easily be transformed into afirst order system, which will give us a better overview of its structure. We note that(1) y(x) = (q(x)w(x))y(x)p(x)y(x)p(x) from which we see that the first order systemhas the form

y(x) =z(x)p(x)

z(x) = (q(x) w(x))y(x)(2)

We can see from the form (2) we need an assumption to guarantee that p(x) 6= 0 inorder to avoid a singularity. We know that a unique solution to our problem exists aslong as p1(x), q(x), and w(x) are continuous in I.

Hence we will assume the following:

w, q C0([a, b],R), p C1([a, b],R), p(x) > 0, w(x) > 0, for x [a, b]We call the Sturm Liouville equation satisfying these assumptions regular, while otherSturm Liouville equations are referred to as singular. 1 We note furthermore that theseconditions on the coefficients could be weakened considerably, e.g. to r = 1p , q, w L1(I,R), while preserving the existence and uniqueness of solutions for the associatedinitial value problem.

1Some authors classify singular Sturm Liouville problems to include those where I is not a finiteinterval, where others do not. The hallmark of the singularity is that p(x) vanishes at one or both of thebounds.

1

2 RAPHAEL STUHLMEIER

1.2. Boundary conditions for the Sturm Liouville Problem. At this point, weneed to give some thought to formulating appropriate boundary conditions for the SturmLiouville problem. The knowledge of how an operator acts is intimately connected towhere it acts. Thus it is important to realize that the form of our boundary conditionswill determine the action of our Sturm Liouville operator, in particular (as we shall seebelow) whether or not it is self-adjoint.The general form of the aptly named self-adjoint boundary conditions can be given asfollows:

(3) AY (a) +BY (b) = 0 Y =(

ypy

)where A,B M2(C), AEA = BEB, rank(A : B) = 2 and E =

(0 11 0

). We

note that these boundary conditions are invariant under multiplication by a nonsingularmatrix. (It may be remarked in passing that this leads to the question of whether theeigenvalues of our operator depend continuously on the boundary conditions - for theseparated boundary conditions (see below) the answer is negative, see [3] [4].)It can be shown that the above general case characterizes all self-adjoint realizations ofthe Sturm-Liouville equation (py) + qy = wy, I = (a, b), a < b subjectto our conditions for the coefficients, i.e. r = 1p , q, w L1(I,R), w > 0.2

This general form (3) is equivalent to exactly one of the following cases:1) Separated self-adjoint boundary conditions:

BCa(y) = A1y(a) +A2(py)(a), A1, A2 R, (A1, A2) 6= (0, 0)BCb(y) = B1y(b) +B2(py)(b), B1, B2 R, (B1, B2) 6= (0, 0)

or in the equivalent parametrized form:

BCa(f) = cos()f(a) sin()p(a)f (a) BCb(f) = cos()f(b) sin()p(b)f (b)

2) Real coupled self-adjoint boundary conditions:

Y (b) = KY (a), Y =(

ypy

)with K SL2(R)

3) Complex coupled self-adjoint boundary conditions:

Y (b) = eiKY (a), Y =(

ypy

)with K SL2(R), (pi, 0) or (0, pi)

We will consider only the case of separated boundary conditions, but keep in mind thatthis is not the only way to end up with a self-adjoint operator. We may occasionallyrefer to Dirichlet boundary conditions, or von Neumann boundary conditions at a i.e.f(a) = 0 and f (a) = 0 respectively.

2The weight function w changing sign leads to problems with nonreal eigenvalues further down theroad.

A BRIEF FORAY INTO THE REGULAR STURM LIOUVILLE PROBLEM 3

1.3. Remark on the Liouville normal form. The Sturm Liouville equation (1) canbe transformed into an equation satisfying w = p = 1 as follows: Given the SturmLiouville equation (py) + qy = wy we rearrange the coefficients to rewrite it asfollows

y +p

py +

(w qp

)y = 0

This can be written in the so called Liouville normal form

d2

d2+ (+ ()) = 0

by means of the substitution () = (x)y(x), = xa

w(t)p(t) dt

(x) =(w(x)p(x)

) 14

exp(

12

xa

p(t)p(t)

dt)

2. The Sturm Liouville Equation in Practice

2.1. The One Dimensional Wave Equation. The standard model for an oscillatingstring of length 1 with fixed endpoints is given by the one dimensional wave equationUtt Uxx = 0 which allows the well known separation ansatz into two functions of onevariable, position x and time t respectively.

Taking U(x, t) = u(x)v(t) we see that the following ordinary differential equationholds for u:

u(x) = u(x), u C2[0, 1]u(0) = 0

u(1) = 0

Solutions of the above boundary value problem are c sinx for a constant c. Hence the

BVP has nontrivial solutions if and only if sin = 0 i.e. if = pi2k2 where k = 1, 2, . . ..

These values of are eigenvalues, and the corresponding solutions, eigenfunctions of theproblem, are

k(x) =

2 sin kpix (k = 1, 2, . . .)

(cf. [8], [2]). Indeed this is just the Fourier sine series which forms a basis of L2[1, 1]odd 'L2[0, 1]

Comparing this problem with the general Sturm-Liouville equation (1) above, wesee that it is simply the case w = 1, p = 1, q = 0, and boundary values BCa(u) =u(0), BCb(u) = u(1). This makes it attractive to analyze eigenvalues and eigenfunctionsin the more general case, and see whether similarly nice results hold.

Before we do this, however, we briefly sketch a few less trivial settings in which theSturm Liouville equation turns up - to show that its analysis is really worthwhile.

4 RAPHAEL STUHLMEIER

2.2. The Sturm Liouville equation in exotic places... The Sturm-Liouville equa-tion turns up in a number of places, even if this isnt always obvious at first glance:The Ince equation from Floquet theory:

(1 + cos(2t))y + (sin(2t))y + (+ cos(2t))y = 0

is just a Sturm Liouville equation in disguise: (py) + qy = wy with coefficientsp(t) := e

R t0

sin(2s)1+cos(2s)

ds

q(t) := p(t) cos(2t)1+cos(2t)w(t) := e

2cos(2t) if = 0

w(t) := (1 + cos(2t))1/(2) if 6= 0

The Legendre equation on (1, 1)d

dx

((1 x2)dy

dx

)+ y = 0

is also of Sturm Liouville type with p(t) = 1x2, q(t) = 0, w(t) = 1, as is the hydrogenatom equation on (0,):

y +(k

t+h

t2

)y y = 0

with p(t) = 1, q(t) = kt +ht2, w(t) = 1, k, h R. This gives the two parameter version

of the classical one-dimensional equation for modeling the hydrogen atom in quantumtheory.These and many other examples (Bessel equation, Heun equation - for the latter seeKing, Kong et al [1]) make it clear that the Sturm Liouville problem is of broad inter-est, and that our results will yield applications in diverse domains of pure and appliedmathematics.

3. The Main Results

Theorem 1. The Sturm Liouville problem (1) has a countable number of eigenvalues0, 1, 2, . . . with

|0| |1| |2| . . . as well as a corresponding sequence of eigenfunctions u0(x), u1(x), u2(x), . . . C2[a, b]which (after appropriate normalization) form a complete orthonormal system of theHilbert space L2[a, b] with the scalar product

f |gw := baf(t)g(t)w(t)dt

Additionally, any function u C2 which fulfills our boundary conditions BCa(u) =BCb(u) = 0 has a unique generalized Fourier series

(4) u =k=0

u|ukwuk

A BRIEF FORAY INTO THE REGULAR STURM LIOUVILLE PROBLEM 5

This theorem looks quite promising already, and we will prove it at length below.There are even more amazing properties of the eigenvalues of the Sturm Liouville prob-lem, however, as demonstrated by

Theorem 2. All eigenvalues of the Sturm Liouville problem are simple and monotoni-cally increasing, i.e.

0 < 1 < 2 < . . . Furthermore, the eigenfunction uk to eigenvalue k has exactly k zeroes in the interval(a, b); between any two zeroes of uk there lies exactly one zero of uk+1

Moreover, any u C1 which fulfills the boundary conditions can be written in theform (4).

Unfortunately the proof of this theorem is beyond the scope of this Brief Foray intothe Regular Sturm Liouville Problem - for more on this strange oscillatory behavior ofthe eigenfunctions see for example [10].

Now we really start our examination of the Sturm Liouville operator associated with(1), L : x 7 (px) + qx. Our main interest will be in the eigenvalues and eigenfunc-tions of 1wL :

1wLx = x. As we noted in section 1.3 we can simplify our calculations by

setting w 1 and using the standard scalar product on L2.The first question to ask is what manner of operator L is. Of course, given the pow-

erful tools we have in Hilbert spaces, we would like to apply this setting to studying theSturm Liouville operator - in the best case, we would establish that the Sturm Liouvilleoperator is compact and self adjoint, and apply the powerful spectral theorem for suchoperators. It is easily seen, however, that our operator L is unbounded on the spaceL2 (consider the operator T = d2

dx2on I = (0, pi) with f(x) = sin(nx) satisfying the

Dirichlet boundary conditions, cf. [5, 2.28.iv)]).

We might attempt to salvage the situation by turning to the less desirable case of aBanach space (recall that kth-order differential operators are bounded on the spaces Ck

when these are equipped with an appropriate norm || || :=k

i=0 ||f (i)||).

The natural setting for differential operators, however, is between subspaces of aHilbert space, such that an operator A has its domain D(A) as well as its range R(A)contained in the Hilbert space H. We call such an operator symmetric (often simplyreferred to as self-adjoint in the ODE literature) if D(A) is dense in H and

Au|v = u|Av u, v D(A)Example. Consider the operator H given formally by

(5) Hf := f on the following alternative domains in the Hilbert space L2(a, b). To treat Dirichletboundary conditions we take the domain LD consisting of all functions C2[a, b] satisfy-ing the Dirichlet boundary conditions, and similarly the von Neumann conditions will betreated on the analogous domain LN . Because of the two different domains, equation 5determines two different operators we denote by HD and HN . We have seen eigenvaluesof this very operator above, and it can be easily established that 0 is an eigenvalue of

6 RAPHAEL STUHLMEIER

HN but not of HD. This should demonstrate the impact that a change of domain canhave on the spectrum of an operator - though for the Laplacian this is more significantin higher dimensions.

We still hold out hope for the strong structural results associated with the spectraltheorem in the compact self-adjoint case. While differential operators may have a numberof undesirable properties, we can attempt to establish certain facts about the spectrumindirectly. The way we go about doing this, similar to the standard proof of the existenceand uniqueness theorem for ordinary differential equations, is by considering the muchnicer inverse operator. The integral operator that is the inverse of L will in fact live ona Hilbert space.

In consideration of our boundary values, for our general Sturm Liouville operator wedefine a suitable subspace of the space C2[a, b] as follows:

D := {x : [a, b] C| x C2[a, b], BCa(x) = BCb(x) = 0}Indeed L is defined and bounded on D L2 dense.Remark 3. In order to assure invertibility of the Sturm Liouville operator L, we requireit to be injective, i.e. the sole solution of Lx = 0, BCa(x) = BCb(x) = 0 is x 0.Theorem 4. The inverse operator of L is given by T : C[a, b] D,

(6) (Tf)(t) = baG(t, s)f(s)ds (f C[a, b])

with the kernel of the integral operator G C([a, b] [a, b]) defined below.Remark 5. i) In particular this means, L : D C[a, b] is surjective (T is a continuousFredholm operator cf. [5, 2.28.ix)] and T continuous L open L surjective cf. [5,4.49)]. Thus the boundary value problem Lu = g is (uniquely) solvable for any continuousg.ii) Equation (6) above defines a Fredholm operator T1 : L2[a, b] D

Now we have the Hilbert space setting we were hoping for, and a very well behavedintegral operator at that. It remains to provide the following:

Definition 6 (Greens Function). The Greens function which is the kernel of the Fred-holm operator in (6) is defined as follows:Let u,v (both 6= 0) be solutions of the differential equation Ly = 0 such that BCa(u) =0, BCb(v) = 0. Then we define

(7) G(t, s) :=

{u(t)v(s)

c if t sv(t)u(s)

c if t sRemark 7. G is continuous and, more importantly, symmetric, i.e. G(s, t) = G(t, s)The constant c is established in the following calculation:i) We establish that u, v are linearly independent. Assume u = v, then R2(u) =R2(v) = 0 = u D,Lu = 0 = u = 0, a contradiction.ii) We see that

(uu

)and

(vv

)are linearly independent for all t, thus det

(u vu v

)=

A BRIEF FORAY INTO THE REGULAR STURM LIOUVILLE PROBLEM 7

uv uv 6= 0 for all t.iii) thus (p(uvuv)) = (pu)v+puvu(pv)puv = ((pu)v+quv)+((pv)u+quv) = uLv vLu = 0 = p(uv uv) = const. =: c 6= 0Remark 8. Of course this Greens function depends on the boundary conditions, whichwe have restricted to the separated self-adjoint case. The explicit derivation of Greensfunction in general requires some tools from complex analysis, and is beyond the scopeof this paper.For s 6= t, G(t, s) as defined above is twice continuously differentiable; for s = t howeverwe have:

tG(t+, t) =

u(t)v(t)c

tG(t, t) = v(t)u

(t)c

}=

tG(t+, t)

tG(t, t) = uv

uvc

= 1p(t)

Hence we may remark (without giving a rigorous treatment)

LtG(t, s) ={

0 t 6= s(pG) + qG t = s

}= (t s)

= (LTf)(t) = L G(t, s)f(s)ds = LtG(t, s)f(s)ds = (t s)f(s)ds = f(t)We will give a rigorous (but slightly more involved) proof of the fact that LTf = f

below.

Example We look at a relatively simple Sturm Liouville operator: Lu = u on[0, 1] and consider Greens function. As in definition 6, u, v are posited as solutions ofthe equation Lx = 0 with u(0) = 0, v(1) = 0; hence u, v are necessarily of the format+ b = u(t) = t, v(t) = 1 tc = p(uv uv) = 1(1(1 t) t(1)) = 1 = G(t, s) =

{t(1 s) (t s)s(1 t) (t s)

Proof (Of Theorem 4)i) LT = idC[a,b]:

(Tf)(t) =v(t)c

tau(s)f(s)ds+

u(t)c

btv(s)f(s)ds

= (Tf) = v

c

tauf +

v

c(uf) +

u

c

btvf u

c(vf) =

v

c

tauf +

u

c

btvf

(p(Tf )) =(pv)

c

tauf +

pvufc

+(pu)

c

btvf pu

vfc

= LTf = (p(Tf)) + qTf =

((pv) + qv) Lv=0

1c

tauf+((pu) + qu)1

c

btvf + pf

uv uvc

= f

ii) TL = idD LTLu = Lu , L injective = TLu = u

8 RAPHAEL STUHLMEIER

We recall that, although L as an operator is only densely defined, we have a substitutefor self-adjointness in the form of symmetry.

Theorem 9. We establish a few properties of the Sturm-Liouville operator L.i) L is symmetric on the domain Dii) All eigenvalues of L are realiii) The eigenfunctions of L to distinct eigenvalues are pairwise orthogonal

Proof of Theorem 9:

i) u|Lv Lu|v = bauLv (Lu)v =

ba

(p(uv uv)) = (p(uv uv))|ba = 0ii) Let be an eigenvalue of L, i.e. Lx = x for some x, then self adjointness implies( )x|x = 0 for some eigenfunction x. Since x|x > 0 it follows that = .iii) Let 1, 2 be distinct eigenvalues with eigenfunctions x1, x2 respectively. Then

Lx1|x2 x1|Lx2 = (1 2)x1|x2 = x1|x2 = 0 We go on to prove our main theorem :Proof of Theorem 1:Case 1: L is injective

We consider operator T resp. T1 as defined in (7) resp. Remark 5. We first gather someproperties of the operator T1 : L2[a, b] D L2[a, b]. Recall that D C2[a, b] suchthat the boundary conditions hold. Because of the symmetric Greens function definedin (7), the operator T is an integal operator with symmetric kernel, hence self-adjoint(cf. [6, 5.54.iii)]). Further we know from [6, 5.14.i)] that T1 is a compact operator, andfurther that it is of Hilbert-Schmidt type ([6, 6.59]). All that remains is to apply thespectral theorem for compact self-adjoint operators on a Hilbert space [6, 6.26 ff], andcollect the results:

T1 has (possibly finitely many) real eigenvalues k 6= 0 such that|0| |1| |2| . . . 0

and continuous pairwise orthogonal eigenfunctions u0(x), u1(x), u2(x), . . ..

Furthermore, since

kuk = Tuk = baG(t, s) uk(s)

continuous

ds

we have kuk im T thus uk im T = D C2 The eigenfunctions uk form a complete orthonormal system

im T = D is dense in L2[a, b], thus im T1 im T is likewise dense in L2. SincekerT1 = (im T1) = (im T1)

[6,5.41.ii)]= (L2) = {0}

hence T1 is injective. Owing to T1 0 on (lin uk), it follows that (lin uk) ={0}, hence

lin uk = (lin uk) = {0} = L2

A BRIEF FORAY INTO THE REGULAR STURM LIOUVILLE PROBLEM 9

showing that (uk)k is complete. This implies too, that the sequence of eigenvaluescannot be finite. Tf = f is equivalent to f = Lf , i.e. Lf = 1f

Defining k := 1k , L has countably many real eigenvalues k 6= 0 such that|0| |1| |2| . . .

with the eigenfunctions uk Every f im T1, in particular every f im T = D, i.e. every u C2[a, b] such

that BCa(u) = BCb(u) = 0 has a uniformly convergent expansion

u = Tf =

kf |ukuk =f |kukuk =

u|ukuk

Case 2 : L not injectiveWe know from Theorem 9 that the eigenfunctions to different eigenvalues form an or-thonormal system on the pre-Hilbert space D, a dense subspace of L2, hence there can beat most countably many eigenvalues and eigenfunctions. For any R not eigenvalueof L it is clear that L = L (Lx = (px) + (q )x)) is injective on D. Hence wecan apply Case 1 above to get eigenvalues k and eigenfunctions uk of L. (k + ) arethus eigenvalues of L with eigenfunctions uk.

References

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[2] Earl A. Coddington and Norman Levinson. Theory of ordinary differential equations. Internationalseries in pure and applied mathematics. McGraw-Hill, New York, 1955.

[3] W.N. Everitt, M. Moeller, and A. Zettl. Discontinunous dependence of the n-th sturm-liouville eigen-value. In General Inequalities 7, volume 123, pages 145150. Proceedings of the 7th InternationalConference at Oberwolfach, Nov. 13-19, 1995, 1997.

[4] W.N. Everitt, M. Moeller, and A. Zettl. Sturm-liouville problems and discontinuous eigenvalues.Proc. Roy. Soc. Edinburgh, 129A:707716, 1999.

[5] Roland Steinbauer. Funktionalanalysis 1. Lecture Notes, 2007.[6] Roland Steinbauer. Functional analysis 2. Lecture Notes, 2008.[7] C. Sturm and J. Liouville. Extrait dun memoire sur le developpement des fonctions en series dont

les differents terms sont assujettis a` satisfaire a` une meme equation differentielle lineaire, contenantun parame`tre variable. Journal de Mathematiques Pures et Appliquees, 1837.

[8] Gerald Teschl. Ordinary differential equations and dynamical systems. Lecture Notes, March 2006.[9] Dirk Werner. Funktionalanalysis. Springer, 2007.

[10] Anton Zettl. Sturm-Liouville theory, volume v. 121 of Mathematical surveys and monographs. Amer-ican Mathematical Society, Providence, R.I., 2005.