Non-standard boundary conditions for Sturm-Liouville operators in the limit point case

AB CD EF

GH

Non-standard boundary

conditions for Sturm-Liouville operators in the limit point case

A.E. Sterk

Department of

Preface and acknowledgements

During the spring term of 2005 I participated in a small research project con- cerning non-standard boundary conditions for Sturm-Liouville differential equa- tions.This project eventually resulted in a small joint paper with Seppo Hassi, Henk de Snoo and Henrik Winkler.The present thesis is an extension and generalization of the results in that paper.

Several people supported me in the process of doing research and writing this thesis.First of all, I would like to express my gratitude to my supervisor Henk de Snoo for the interesting subject of this thesis and for giving me the opportunity to get involved in doing research.At times I got stuck again, his suggestions were of great importance to me.

I am also grateful to Henrik Winkler (TU Berlin, Germany).He was always willing to help me and during our discussions I obtained a better understanding of what I was doing.

In May 2006 I have visited Seppo Hassi (University of Vaasa, Finland).I thank him for his kind hospitality and the interesting discussions we had during the ten days of my visit.I am also grateful for his remarks which have improved the text of this thesis significantly.

I thank Jussi Behrndt (TU Berlin, Germany) for the many discussions we had and for organizing with Henk de Snoo the seminar Boundary tiplets and Weyl functions at the University of Groningen during the spring term of 2006.

Finally, I thank Ineke Kruizinga for taking care of the administrative tasks which came along with finishing this thesis.

Groningen, August 2006, Alef Sterk

Contents

Preface and acknowledgements i

Contents ii

1 Introduction 1

2 Preliminaries 3

2.1 Nevanlinna families ... 3 2.2 Boundary triplets and Weyl functions ... 4 2.3 Symmetric extensions with compressed Weyl functions ... 5 2.4 Selfadjoint exit space extensions and generalized resolvents . . . 5 2.5 Finite-dimensional relations . . . 7 2.6 Finite-dimensional restrictions of selfadjoint relations ... 8

3 A class of Sturm-Liouville operators 12

3.1 Singular Sturm-Liouville operators ... 12 3.2 Defect numbers: limit point and limit circle ... 14 3.3 The limit point case . . . 16

4 Selfadjoint boundary value problems 19

4.1 Standard boundary conditions ... 19 4.2 Non-standard boundary conditions ... 19 4.3 Examples . . . 21 5 The spectral matrix and generalized Fourier transform 23 5.1 The Kre˘ın-Naimark formula and generalized Fourier transform .23 5.2 The generalized Fourier transform onR . . . 27 5.3 An example . . . 30

A Linear relations in Hilbert spaces 32

Bibliography 33

Chapter 1

Introduction

Let q be a real-valued, locally integrable function on the interval [0,∞), such that the Sturm-Liouville differential equation

(1.1) −y

(x) + q(x)y(x) = λy(x), x∈ [0, ∞),

is in the limit-point case at ∞.This means that for each λ ∈ C \ R the linear subspace ofH = L²(0,∞) of eigensolutions of (1.1) is one-dimensional. With a fundamental system (u₁(·, λ) : u₂(·, λ)) of (1.1), fixed by the initial conditions

u₁(0, λ) = 1, u
₁(0, λ) = 0, u₂(0, λ) = 0, u
₂(0, λ) = 1.

there exists a function m(λ) defined on C \ R such that for all λ ∈ C \ R the function

ω(·, λ) = u1(·, λ)m(λ) − u2(·, λ)

belongs to H.The standard selfadjoint boundary value problems associated with this equation are determined by boundary conditions of the form

(1.2) y
(0) = sy(0), s∈ R ∪ {∞},

with the interpretation that y(0) = 0 for s = ∞.The corresponding spectral theory involving the so-called Titchmarsh-Weyl coefficient m(λ) is well known, see for instance [8], [16], and [22].An extension to λ-depending boundary conditions of the form (1.2) is also well known.

With the underlying Sturm-Liouville operator−D²+q one can also associate boundary value problems different from (1.1) and (1.2). In the present paper it is shown for the case of the Sturm-Liouville operator how the spectral theory for non-standard selfadjoint boundary value problems can be obtained via the abstract extension theory of not necessarily densely defined symmetric operators or relations, in particular, via the theory of boundary triplets and their Weyl functions, as originally worked out by V.A. Derkach and M.M. Malamud, cf.

[12], [13], and [9].

The main idea is to consider non-standard boundary value problems as graph perturbations of a fixed standard boundary value problem determined by the boundary condition (1.2). The use of finite-dimensional subspaces to restrict symmetric relations to enlarge the classes of boundary value problems goes back to [6], [7], and [14].Restricting selfadjoint relations is more general: the symmetric restriction need not be part of the minimal operator associated with the Sturm-Liouville operator.

The spectral theory is given by means of generalized Fourier transforms.The corresponding spectral matrix will be given explicitly in terms of the two Weyl functions associated with the fixed standard boundary value problem and its perturbation.The present work is a continuation of [18] where only canonical extensions of one-dimensional perturbations were considered; now, more gener- ally, exit space extensions of finite-dimensional perturbations will be considered.

The contents of this report are as follows.Chapter 2 contains some prelim- inary results.Boundary triplets and their Weyl functions are introduced and the Kre˘ın-Naimark formula for generalized resolvents will be stated.Finally, the framework for finite-dimensional graph perturbations of selfadjoint relations will be given in detail.In Chapter 3 the formal Sturm-Liouville operator −D²+ q will be realized as a symmetric operator in the Hilbert space H = L²(0,∞).

Since the Titchmarsh-Weyl coefficient forms the basic motivation for the con- cept of a Weyl function, Weyl’s alternative will be proved in detail along the lines of [8].Chapter 4 shows how non-standard boundary value problems can be obtained by means of graph perturbations and examples will be given.The spectral matrix will be derived in Chapter 5.Furthermore, some properties of the generalized Fourier transform will be given.For the convenience of the reader some facts concerning linear relations in Hilbert spaces will be stated in Appendix A.

A paper based on the results in this report has been submitted, see [19].

Chapter 2

Preliminaries

2.1 Nevanlinna families

A family of linear relations M (λ), λ ∈ C \ R in a Hilbert space H is called a Nevanlinna family if:

(i) for every λ∈ C+ (resp. λ∈ C−) the relation M (λ) is maximal dissipative (resp.accumulative);

(ii) M (λ)^∗= M (¯λ), λ∈ C \ R;

(iii) for some (and hence for all) µ∈ C₊(resp.C₋) the operator family (M (λ) + µ)⁻¹∈ [H]

is holomorphic for all λ∈ C+(resp.C−).

If the multi-valued part of M (λ) is nontrivial, then it is independent of λ ∈ C \ R, c.f., for instance, [9]. Hence, each Nevanlinna family M(·) can be decom- posed as

M (λ) = M_s(λ)⊕ M_∞, M_∞={0} × mul M(λ), λ ∈ C \ R,

where the orthogonal operator parts M_s(λ) form a Nevanlinna family of densely defined operators in the Hilbert spaceH  mul M(λ).

Denote by
R(H) the class of all Nevanlinna families in a Hilbert space H.

The following subclasses of
R(H) will be useful:

R[H] = { M(·) ∈
R(H) : dom M(λ) = H for all λ ∈ C \ R}, R^s[H] = { M(·) ∈ R[H] : ker Im M(λ) = {0} for all λ ∈ C \ R}, R^u[H] = { M(·) ∈ R^s[H] : 0 ∈ ρ(Im M(λ)) for all λ ∈ C \ R}.

The Nevanlinna families that belong to R[H] are called bounded Nevanlinna functions; the Nevanlinna functions in R^s[H] and R^u[H] are called strict and uniformly strict, respectively.

If M (·) ∈ R[H], then it admits an integral representation of the form M (λ) = A + Bλ +



R

 1

t− λ− t t²+ 1

 dΣ(t),

where A = A^∗ ∈ [H], 0 ≤ B = B^∗ ∈ [H], and the [H]-valued family Σ(·) is nondecreasing and satisfies



R

1

t²+ 1d(Σ(t)h, h) <∞,

for all h ∈ H.Moreover, the family Σ(·) can be obtained from M(·) via the Stieltjes-Livˇsic formula:

Σ(t₂)− Σ(t₁) = lim

ε→0

1 π

 _t2

t1

Im M (s + iε)ds.

For a proof, see, for instance, [20].

2.2 Boundary triplets and Weyl functions

Let S be a closed symmetric relation with defect numbers (n, n) in a Hilbert space H with inner product (·, ·).Let Nλ(S^∗) = ker (S^∗− λ) be the defect subspace of S, and define

Nλ(S^∗) :={ f_λ={fλ, λf_λ} : fλ∈ Nλ}, λ ∈ C, so that Nλ(S^∗)⊂ S^∗.

A boundary triplet Π ={H, Γ0, Γ₁} for S^∗ consists of an auxilliary Hilbert spaceH with dim H = n and two boundary mappings Γ0 and Γ₁ from S^∗ toH with the following properties:

(i) the abstract Green’s identity (f
, g)− (f, g
) =



Γ₁f , Γ ₀g

H−

Γ₀f , Γ ₁g

H, holds for all elements f ={f, f
}, g = {g, g
} ∈ S^∗;

(ii) the mapping Γ : f → {Γ0f , Γ ₁f} from S^∗into H × H is surjective.

When S is densely defined, then the mappings Γ₀and Γ₁are interpreted as being defined on dom S^∗instead on S^∗, i.e., in that case one speaks of Γ₀f and Γ₁f when f ={f, f
} ∈ S^∗.

The mapping f → {Γ₀f ,−Γ₁f} induces a one-to-one correspondence be- tween the closed extensions H of S which are intermediate (i.e., which satisfy S⊂ H ⊂ S^∗) and the closed linear relations τ inH, via

(2.1) H :={ f ∈ S^∗: {Γ₀f ,−Γ₁f} ∈ τ },

cf.[3], [12], and [13]. Moreover, (2.1) establishes a one-to-one correspondence between all selfadjoint extensions A = A^∗ of S and all selfadjoint relations τ in H.In particular, the special choices τ = {0} × H and τ = H × {0} show that A₀:= ker Γ₀and A₁:= ker Γ₁define two selfadjoint extensions of S.

Associated to the boundary triplet Π are two operator functions: the Weyl function M (λ), defined by

(2.2) M (λ) ={ {Γ0f_λ, Γ₁f_λ} : f_λ∈ Nλ(S^∗)},

the graph of a bounded linear operator inH, and the γ-field γ(λ) defined by (2.3) γ(λ) ={ {Γ0f_λ, f_λ} : f_λ∈ Nλ(S^∗)},

the graph of a bounded linear operator fromH to N_λ(S^∗).For all λ, µ∈ ρ(A₀) the γ-field satisfies the identity

(2.4) γ(λ) = (I + (λ− µ)(A₀− λ)⁻¹)γ(µ),

which, in particular, shows that γ(λ) is a holomorphic operator function for λ ∈ C \ R.The Weyl function M(λ) and the γ-field γ(λ) are related via the identity

(2.5) M (λ)− M(µ)^∗

λ− ¯µ = γ(µ)^∗γ(λ),

which holds for all λ, µ ∈ ρ(A₀).Moreover, it can be shown that each Weyl function belongs to the class R^u[H], cf.[13].

2.3 Symmetric extensions with compressed Weyl functions

Let S be a closed symmetric relation with equal defect numbers and let Π = {H, Γ₀, Γ₁} be a boundary triplet for S^∗.Denote by M (·) and γ(·) the associated Weyl function and γ-field respectively.Assume that the spaceH is decomposed as H = H₁⊕ H₂ and denote by P_i, i = 1, 2 the orthogonal projection ontoHi. Decompose the Weyl function M (·) as

M (·) = (Mij(·))²_i,j=1, where M_ij(·) = PiM (·)
Hj.

Let τ ={0} × H2, then it is clear that τ^∗=H1× H so that τ is a symmetric relation in H.It follows by (2.1) that the relation H1 defined by

(2.6) H₁={ f ∈ S^∗ : {Γ0f ,−Γ1f} ∈ τ} = { f ∈ S^∗ : Γ₀f = P ₁Γ₁f = 0 } is a symmetric extension of S.The next proposition gives a boundary triplet for H₁^∗ such that the corresponding Weyl function coincides with M₁₁(·).

Proposition 2.1. A boundary triplet for H₁^∗ is given by Π₁={H1, Γ^r₀, P₁Γ^r₁} where Γ^r_j = Γ_j
H₁^∗ and the γ-field and Weyl function associated with Π₁ are given by

γ₁(λ) = γ(λ)
H1, M₁(λ) = P₁M (λ)
H1. Moreover, the identity ker Γ₀= ker Γ^r₀ holds.

The proof can be found in [3], see also [10].

2.4 Selfadjoint exit space extensions and generalized resolvents Let S be a closed symmetric relation with equal defect numbers in a Hilbert space H.A selfadjoint relation A in a larger Hilbert space H ⊕ K is called a selfadjoint exit space extension of S if S ⊂ A.The space K is called the exit space; whenK = {0}, the relation A is called a canonical selfadjoint extension of S.The compression of the resolvent (A− λ)⁻¹ to the Hilbert spaceH defined by

R_λ:= P_H(A− λ)⁻¹
H, λ ∈ ρ(A), is called a generalized resolvent of S.

Theorem 2.2. Let S be a closed symmetric relation with equal defect numbers and let Π ={H, Γ0, Γ₁} be a boundary triplet for S^∗, and let M (·) and γ(·) be the corresponding Weyl function and γ-field. The generalized resolvents of S are in one-to-one correspondence with the Nevanlinna families τ (·) ∈
R(H) via the Kre˘ın-Naimark formula:

(2.7) R_λ= (A₀− λ)⁻¹− γ(λ)(M(λ) + τ(λ))⁻¹γ(¯λ)^∗, where A₀= ker Γ₀.

Conversely, it can be shown that for each generalized resolvent defined by (2.7) there exists a selfadjoint exit space extension A_τ of S such that R_λ= P_H(A_τ− λ)⁻¹
H.Moreover, A_τ can be chosen to be minimal, i.e.

H ⊕ K = span { (I + (Aτ− λ)⁻¹)H : λ ∈ ρ(Aτ)}.

In special cases, the exit spaceK and the extensions Aτ can be given explic- itly.The following example gives an explicit formula in the case when τ (·) is a rational Nevanlinna function.

Example 2.3 (Rational Nevanlinna functions, c.f. [2]). Let S be a closed symmetric relation in a Hilbert spaceH and let {H, Γ0, Γ₁} be a boundary triplet for S^∗.For j = 0, . . . , m pick operators A_j, B_j ∈ [H] such that Aj = A^∗_j and B_j> 0.Define

τ (λ) = A₀+ λB₀+

m j=1

B_j^1/2(A_j− λ)⁻¹B_j^1/2,

then it readily follows that τ (·) is a uniformly strict Nevanlinna function.In particular, it follows that τ (·) is the Weyl function of a symmetric relation H in a Hilbert spaceK, see [9].In this particular example this can be made completely explicit.Let K = H^m+1 and let

H =

























 0 h₁

... h_m





,





 h
₀ A₁h₁

... A_mh_m

















∈ K², : h
₀=

m j=1

B₀^−1/2B_j^1/2h_j









 .

Clearly, H is a symmetric relation inK and its adjoint is given by the relation

H^∗=

























 h₀ h₁ ... h_m





,







h
₀

A₁h₁+ B₁^1/2B₀^−1/2h₀ ...

A_mh_m+ B^1/2_m B^−1/2₀ h₀

















∈ K² : h
₀, h₀, . . . , h_m∈ H









 .

Define the mappings χ₀, χ₁: H^∗→ H by χ₀h₀⊕ · · · ⊕ hm:= B₀^−1/2h₀,

χ₁h₀⊕ · · · ⊕ hm:= A₀B₀^−1/2h₀+ B^1/2₀ h
₀−

m j=1

B_j^1/2h_j.

It easily follows that {H, χ₀, χ₁} is a boundary triplet for H^∗ and the corre- sponding Weyl function is given by τ (·).Finally, the relation

A_τ =

f⊕ h₀⊕ · · · ⊕ hm∈ S^∗⊕ H^∗ : Γ₀f− χ₀h = Γ₁f + χ ₁h = 0

is a selfadjoint exit space extension for S and the generalized resolvent R_λ = P_H(A_τ−λ)⁻¹
H is given by (2.7), see [11]. The question whether this realization is minimal will not be treated.

It will be convenient to consider only generalized resolvents determined by Nevanlinna families τ (·) such that mul τ(λ) = {0}.The following observation will be useful.

Lemma 2.4. Let τ (·) ∈
R(H) and let P be the projection from H onto dom τ(λ).

Then the identity

(M (λ) + τ (λ))⁻¹= P (P M (λ)P + τ_s(λ))⁻¹P holds for any Nevanlinna function M (·) ∈ R^u[H].

For a proof, consult [21].

In particular, it follows from Lemma 2.4 that the generalized resolvent de- termined by τ (·) can be written as

(2.8) R_λ= (A₀− λ)⁻¹− γ(λ)P (P M(λ)P + τs(λ))⁻¹(γ(¯λ)P )^∗.

The right hand side of (2.8) can be interpreted as the generalized resolvent of a symmetric relation H₁ extending S.To see this, let H1 = dom τ (λ) and H2= mul τ (λ) so thatH = H1⊕H2.Let H₁be a canonical symmetric extension of S induced by τ ={0} × H2 via (2.6). By Proposition 2.1 it follows that the generalized resolvent of H₁determined by the Nevanlinna family τ_s(·) coincides with the right hand side in (2.8). Therefore, by extending the symmetry S in the way indicated, it is always possible to consider only the generalized resolvents induced by Nevanlinna families with a trivial multi-valued part.

The next proposition will be useful in connection with boundary value prob- lems.

Proposition 2.5. Let the assumptions be as in Theorem 2.2, then for every h ∈ H the vector f = Rλh is a solution of the following ‘abstract boundary value problem’ with the paramater τ (·) in the ‘boundary conditions’:

(2.9)

f
− λf = h, f = {f, f
} ∈ S^∗, {Γ0f ,−Γ1f} ∈ τ(λ).

For a proof, see [12], [13], and [10].

2.5 Finite-dimensional relations

LetH be a Hilbert space and let Z and T be linear relations in H with dim T = dim Z = n <∞.Fix a basis for the n-dimensional spaces T and Z by

T = span{ {e₁, e
₁}, . . . , {en, e
_n} }, and

Z = span{ {ϕ1, ψ₁}, . . . , {ϕn, ψ_n} }.

Lemma 2.6. Define the n× n matrix G by

G_ij={e_j, e
_j}, {ϕ_i, ψ_i}, i, j = 1, . . . , n.

Then the kernel of G is given by

ker G =



(c₁, . . . , c_n)^ ∈ Cⁿ :

n j=1

c_j{ej, e
_j} ∈ Z^∗



. In particular, ker G ={0} if and only if T ∩ Z^∗={0, 0}.

Proof. A vector (c₁, . . . , c_n)^∈ Cⁿ belongs to ker G if and only if

n j=1

{ej, e
_j}, {ϕi, ψ_i}cj= 0, i = 1, . . . , n,

or, equivalently, if 

j=1

c_j{ej, e
_j} ∈ Z^∗. This completes the proof.

2.6 Finite-dimensional restrictions of selfadjoint relations

Let A₀= A^∗₀be a selfadjoint relation in a Hilbert spaceH and let Z be a finite- dimensional subspace ofH × H such that A₀∩ Z = {0, 0}.Define the relation S = A₀∩ Z^∗.Clearly, S is a closed symmetric relation inH and the adjoint S^∗ is given by the componentwise sum inH × H:

(2.10) S^∗= A₀ + Z.

Fix a basis for Z by

Z = span{ {ϕ1, ψ₁}, . . . , {ϕn, ψ_n} },

so that dim Z = n.The defect spacesNλ(S^∗) of S can be expressed in terms of the elements of Z.

Lemma 2.7. The defect space Nλ(S^∗) of the symmetric relation S is spanned by the elements

(2.11) χ_j(λ) = ϕ_j+ (A₀− λ)⁻¹(λϕ_j− ψ_j), j = 1, . . . , n.

In particular, the symmetric relation S has defect numbers (n, n).

Proof. Observe that

{χj(λ), λχ_j(λ)} = {ϕj, ψ_j}

+{(A₀− λ)⁻¹(λϕ_j− ψj), (I + λ(A₀− λ)⁻¹)(λϕ_j− ψj)} (2.12)

which shows that{χj(λ), λχ_j(λ)} ∈ S^∗ for all j = 1, . . . , n.Hence, span{χ₁(λ), . . . , χ_n(λ)} ⊂ Nλ(S^∗).

To prove the reverse inclusion, let f ∈ Nλ(S^∗) so that {f, λf} ∈ S^∗.By (2.10) it follows that

{f, λf} = {h, h
} + c1{ϕ1, ψ₁} + · · · + cn{ϕn, ψ_n}, where{h, h
} ∈ A0 and c₁, . . . , c_n∈ C.Clearly,

h
= λh +

n j=1

c_j(λϕ_j− ψj),

so that{h, h
} ∈ A0 implies that

h =

n j=1

c_j(A₀− λ)⁻¹(λϕ_j− ψj)

Hence,

f =

n j=1

c_j(ϕ_j+ (A₀− λ)⁻¹(λϕ_j− ψj)) =

n j=1

c_jχ_j(λ).

This completes the proof.

Two finite-dimensional subspaces Z and
Z of H × H are called equivalent (with respect to A₀) if

(2.13) A₀∩ Z = A₀∩
Z ={0, 0} and A₀ + Z = A ₀ +
 Z.

Obviously, (2.13) gives rise to an equivalence relation on the set of finite- dimensional subspaces inH × H and two equivalent subspaces necessarily must have the same dimension.The next lemma shows that each equivalence class contains at least one symmetric representative.

Lemma 2.8. Let the selfadjoint relation A₀ and the n-dimensional space Z satisfy A₀∩ Z = {0, 0}. Then there exists an n-dimensional symmetric space
Z which is equivalent to Z.

Proof. Since dim A₀/S = n there exists an n-dimensional relation T ⊂ A0 such that A₀ = S + T and S∩ T = {0, 0}, or, equivalently, T ∩ Z^∗ ={0, 0}.Fix a basis for T by

T = span{ {e1, e
₁}, . . . , {en, e
_n} }.

Define for i = 1, . . . , n the vectors b⁽ⁱ⁾ ∈ Cⁿ of which the components are given by b⁽ⁱ⁾_j = (ϕ_i, ψ_j) and let α⁽ⁱ⁾ ∈ Cⁿ be the unique vector such that Gα⁽ⁱ⁾ = b⁽ⁱ⁾, where G is the matrix of Lemma 2.6. Define the elements

(2.14) {
ϕ_i,
ψ_i} = α⁽ⁱ⁾₁ {e1, e
₁} +· · · + α⁽ⁱ⁾_n {en, e
_n} +{ϕi, ψ_i}.

Clearly, the space
Z spanned by the elements in (2.14) is equivalent to Z.There- fore it suffices to check that
Z is symmetric.By a straightforward calculation it follows that

{
ϕ_i,
ψ_i}, {
ϕ_j,
ψ_j} = (Gα⁽ⁱ⁾− b⁽ⁱ⁾)_j− (Gα^(j)− b^(j))_i= 0, for all i, j = 1, . . . , n.This completes the proof.

It follows from Lemma 2.8 that in the definition of S = A₀∩ Z^∗with A₀∩ Z = {0, 0} the subspace Z can always be assumed to be symmetric.Clearly, Z is symmetric if and only if A₁ := S + Z is a selfadjoint extension of S in which case the selfadjoint extensions A₀and A₁are transversal, i.e.

S = A₀∩ A1, S^∗= A₀+ A ₁.

By (2.10) it follows that each element{f, f
} ∈ S^∗= A₀ + Z can be decom- posed as

(2.15) {f, f
} = {h, h
} + c₁{ϕ₁, ψ₁} + · · · + cn{ϕn, ψ_n},

where {h, h
} ∈ A0 and c_j ∈ C for all j = 1, . . . , n.Moreover, it follows from the assumption A₀∩ Z = {0, 0} that the decompostion in (2.15) is unique.

Define the mappings Γ₀, Γ₁: S^∗→ Cⁿ by

(2.16) Γ₀{f, f
} :=



 c₁

... c_n



 , Γ1{f, f
} :=





{h, h
}, {ϕ1, ψ₁}

...

{h, h
}, {ϕn, ψ_n}



 ,

where{f, f
} is as in (2.15).

Proposition 2.9. Assume that A₀∩ Z = {0, 0} and that Z is symmetric. Let S = A₀∩ Z^∗ and let the mappings Γ₀, Γ₁: S^∗→ Cⁿ be as in (2.16). T hen:

(i) Π ={Cⁿ, Γ₀, Γ₁} is a boundary triplet for S^∗= A₀ + Z;

(ii) the corresponding γ-field is given by the linear operator fromCⁿ toH by

(2.17) χ(λ) =















 c₁

... c_n



 ,

n j=1

c_jχ_j(λ)





 : c₁, . . . , c_n∈ C





,

where the χ_j(λ) are given by (2.11);

(iii) the corresponding Weyl function is given by the matrix function Q(λ) of which the components are given by

Q_ij(λ) = (χ_j(λ), ¯λϕ_i− ψi), i, j = 1, . . . , n.

Moreover, ker Γ₀= A₀ and ker Γ₁= S + Z.

Proof. (i) Let f ={f, f
}, g = {g, g
} ∈ S^∗ then

{f, f
} = {h, h
} + c1{ϕ1, ψ₁} + · · · + cn{ϕn, ψ_n}, {g, g
} = {k, k
} + d₁{ϕ₁, ψ₁} + · · · + dn{ϕn, ψ_n}.

It follows from a straightforward calculation that



Γ₁f , Γ ₀g

Cⁿ−

Γ₀f , Γ ₁g

Cⁿ=

n j=1

d_j{h, h
}, {ϕj, ψ_j}

+

n j=1

c_j{ϕj, ψ_j}, {k, k
}.

On the other hand (f
, g)− (f, g
) =

n j=1

d_j{h, h
}, {ϕj, ψ_j} +

n j=1

c_j{ϕj, ψ_j}, {k, k
}

+

n i,j=1

c_id_j{ϕ_i, ψ_i}, {ϕ_j, ψ_j} + {h, h
}, {k, k
}.

The facts that A₀ is selfadjoint and Z is symmetric imply that the last two terms vanish.Hence, the abstract Green’s identity is satisfied.

To show that Γ = {Γ₀, Γ₁} : S^∗ → C²ⁿ is surjective it suffices to show that Γ₀ and Γ₁ are surjective mappings from S^∗ toCⁿ, since{h, h
} ∈ A₀ and (c₁, . . . , c_n)^∈ Cⁿ can be chosen independently in (2.15). Clearly, the map Γ₀ is surjective.To show that Γ₁ is surjective let (α₁, . . . , α_n)^ ∈ Cⁿ belong to (ran Γ₁)^⊥.Then

n j=1

α_j{h, h
}, {ϕj, ψ_j} = 0,

for all{h, h
} ∈ A0.This implies that

α₁{ϕ₁, ψ₁} + · · · + α_n{ϕ_n, ψ_n} ∈ A^∗₀= A₀.

Since A₀∩ Z = {0, 0} it follows that α_j = 0 for all j = 1, . . . , n so that ran Γ₁= Cⁿ.

(ii) & (iii) Every element f_λ∈ Nλ can be written as

f_λ={fλ, λf_λ} = c₁{χ₁(λ), λχ₁(λ)} + · · · + cn{χn(λ), λχ_n(λ)}, from which it easily follows that

Γ₀{fλ, λf_λ} =



 c₁

... c_n



 , Γ1{fλ, λf_λ} =





_n

j=1c_j(χ_j(λ), ¯λϕ₁− ψ1) ...

_n

j=1c_j(χ_j(λ), ¯λϕ_n− ψn)





The proof is completed by applying the definitions (2.2) and (2.3).

Chapter 3

A class of Sturm-Liouville operators

3.1 Singular Sturm-Liouville operators

Consider the following eigenvalue problem on the positive half-axis:

(3.1) (L− λ)y(x) = f(x), x ∈ (0, ∞), λ ∈ C where L is the formal differential expression

L =− d²

dx² + q(x).

Assume that the function q(x) to be real valued and locally integrable on the in- terval [0,∞).The formal operator L is said to be a singular differential operator with a regular left end point.

Let f be a measurable function on (0,∞).A function y is called a solution of (3.1) if y, y
∈ ACloc(0,∞) and (3.1) holds almost everywhere in the interval (0,∞).The eigenvalue problem (3.1) is said to be homogeneous if f = 0;

otherwise it is said to be inhomogeneous.The eigenvalue problem (3.1) together with the Cauchy condition

y(b) = β₁, y
(b) = β₂, 0≤ b < ∞,

has a unique solution; see, for instance, [23].In particular, there exists a funda- mental system{u1(·, λ), u2(·, λ)} of the homogeneous eigenvalue problem, fixed by the following initial conditions:

u₁(0, λ) = 1, u
₁(0, λ) = 0, u₂(0, λ) = 0, u
₂(0, λ) = 1.

The functions u₁(·, λ) and u2(·, λ) are continuous as functions in (x, λ) and entire in λ for a fixed x∈ [0, ∞).For a proof of this fact, consult, for instance, [8].

Define the linear space

D := {y ∈ L²(0,∞) : y, y
∈ ACloc(0,∞), Ly ∈ L²(0,∞)}.

For functions y₁, y₂∈ D we have Green’s identity:

(3.2)

 _b

a

(Ly₁)(t)y₂(t)dt−

 _b

a

y₁(t)(Ly₂)(t)dt = [y₁, y₂](b)− [y1, y₂](a), where

[y₁, y₂](x) := y₁(x)y₂
(x)− y₁
(x)y₂(x).

Clearly, the limits for a→ 0 and b → ∞ exist in (3.2), so that (3.3) (Ly₁, y₂)− (y₁, Ly₂) = [y₁, y₂](∞) − [y₁, y₂](0), for all functions y₁, y₂∈ D.

The linear spaceD is the largest subspace of H = L²(0,∞) on which the formal operator L is well-defined as an operator inH.Define the linear relation (3.4) T_max={ {y, Ly} : y ∈ D} ,

so that, in fact, T_maxis (the graph of) an operator.In general, the right hand side of (3.3) is nonzero, which implies that T_maxis not a symmetric operator.

Define the subspace

D0:={y ∈ D : y(0) = y
(0) = 0, y vanishes outside a bounded interval}.

The following observation will be useful.

Lemma 3.1. Let 0 < b <∞ and let f ∈ L²(0,∞) be zero outside the interval (0, b). Then the equation Ly = f has a solution satisfying the conditions

y(0) = y
(0) = y(b) = y
(b) = 0

if and only if f is orthogonal to the functions u₁(·, 0) and u2(·, 0). In addition, such a solution belongs to D0.

Proof. Let ϕ be the unique solution of

Ly = f, y(b) = y
(b) = 0,

in the interval (0, b).Extend ϕ outside the interval (0, b) by setting ϕ equal to zero.Clearly, ϕ∈ D0.By Green’s identity (3.2) it follows that

ϕ(0) =−

 _b

0

u₁(t, 0)f (t)dt, ϕ
(0) =

 _b

0

u₂(t, 0)f (t)dt.

This completes the proof.

Define the linear relation

(3.5) T_min:= clos{ {y, Ly} ∈ Tmax : y∈ D₀}.

Proposition 3.2. The adjoint of the linear relation T_min is given by T_min^∗ = T_max. In particular, the linear relation T_min is symmetric.

Proof. Define the linear relation

T₀:={ {ϕ, Lϕ} ∈ Tmax : ϕ∈ D₀},

so that T_min= clos T₀.Clearly, it suffices to show that T₀^∗= T_max. Let{y, Ly} ∈ Tmax, then, by Green’s identity (3.3),

(Ly, ϕ)− (y, Lϕ) = [y, ϕ](∞) − [y, ϕ](0) = 0, for all elements{ϕ, Lϕ} ∈ T₀.This proves the inclusion T_max⊂ T₀^∗.

Conversely, let{h, k} ∈ T₀^∗.Denote by h₀a solution of the equation Ly = k.

Let 0 < b <∞ and let f ∈ L²(0,∞) be such that f is zero outside the interval (0, b) and f is orthogonal to the functions u₁(·, 0) and u₂(·, 0).By Lemma 3.1 there is a function ϕ∈ D₀ such that Lϕ = f and

ϕ(0) = ϕ
(0) = ϕ(b) = ϕ
(b) = 0.

Then

(f, h) = (Lϕ, h) = (ϕ, k),

where the last equality follows from the fact that{ϕ, Lϕ} ∈ T₀and{h, k} ∈ T₀^∗. On the other hand, by Green’s identity (3.3),

(f, h₀) = (Lϕ, h₀) = (ϕ, Lh₀) = (ϕ, k) This implies that

(f, h− h0) = 0.

By choice of the function f it follows that h = h₀+ u in the interval (0, b) where u is a solution of Ly = 0.Hence, h, h
∈ ACloc(0, b) and Lh = k almost everywhere in the interval (0, b).Since b is arbitary, it follows that h∈ D and Lh = k almost everywhere.This proves the inclusion T₀^∗⊂ Tmaxand completes the proof.

It follows from Proposition 3.2 that T_min is (the graph of) a densely defined operator.Hence, the operator T_max, as an extension of T_min, is also densely defined.In addition, the operator T_max is closed.

3.2 Defect numbers: limit point and limit circle By definition, the defect numbers of T_min are given by

n_±= dim ker (T_max− ¯λ), λ ∈ C \ R.

Since the potential q(x) is real valued, it follows that the defect numbers are equal.Indeed, let ϕ be a solution of the equation Ly = λy for λ∈ C \ R.Clearly,

¯

ϕ is a solution of the equation Ly = ¯λy and ϕ belongs to D if and only if ¯ϕ belongs to D.Consequently the defect numbers are equal, i.e.n− = n₊ = n.

In particular, it follows that for all λ ∈ C \ R there are precisely n linearly independent solutions of the equation Ly = λy that belong to L²(0,∞).

The following theorem gives a description of the defect spaces of T_min. Theorem 3.3 (Weyl’s alternative). Let λ∈ C \ R. The solution ω(·, λ) = u₁(·, λ)
m− u₂(·, λ) satisfies the boundary condition

(3.6) cos β y(b) + sin β y
(b) = 0, 0≤ β < π,

at 0 < b <∞ if and only if
m lies on a circle C_b(λ) in the complex plane. Then, independent of λ∈ C \ R, precisely one of the following alternatives holds:

(i) the limit circle case: C_b(λ)→ C∞(λ), a limit circle with positive radius, as b→ ∞, in which case every solution of Ly = λy belongs to L²(0,∞);

(ii) the limit point case: C_b(λ) → m(λ), a limit point, as b → ∞, in which case only the scalar multiples of the solution

ω(·, λ) = u1(·, λ)m(λ) − u2(·, λ), belong to L²(0,∞).

Proof. Clearly, the solution ω(·, λ) = u1(·, λ)
m− u2(·, λ) satisfies the boundary condition (3.6) if and only if

(3.7) m =
cot β u₂(b, λ) + u
₂(b, λ) cot β u₁(b, λ) + u
₁(b, λ).

As λ, b, β vary, m becomes a function of these arguments, i.e.
m =
m(λ, b, β).
If z = cot β and λ, b are held fixed, (3.7) may be written as

(3.8) m =
Az + B

Cz + D, where

(3.9) A = u₂(b, λ), B = u
₂(b, λ), C = u₁(b, λ), D = u
₁(b, λ),

are fixed while z varies over the real line if β varies from 0 to π.The image of the real axis under the mapping (3.8) is a circle C_b(λ) in the complex plane.

Therefore, ω satisfies (3.6) if and only ifm lies on the circle C
_b(λ).

From (3.8) it follows that the image of the real axis, Im z = 0, becomes (3.10) (Dm
− B)(C
m− A) − (D
m− B)(C
m− A) = 0,

which is the equation for C_b(λ).A straightforward calculation shows that the center and the radius of C_b(λ) are given by

ξ_b(λ) = BC− AD

CD− CD, r_b(λ) = |AD − BC|

|CD − CD|,

respectively. From (3.9) and (3.10) it follows that the equation for C_b(λ) is

(3.11) [ω, ω](b) = 0,

and that

BC− AD = [u1, u₂](b), CD− CD = [u1, u₁](b), AD− BC = −1, so that

(3.12) ξ_b(λ) = [u₁, u₂](b)

[u₁, u₁](b), r_b(λ) = 1

|[u₁, u₁](b)|.

Since the coefficient of |
m|² in (3.11) equals [u₁u₁](b), the interior of C_b(λ) is given by

(3.13) [ω, ω](b)

[u₁, u₁](b) < 0.

By Green’s identity,

[u₁, u₁](b) = 2iIm λ

 _b

0 |u₁(x, λ)|²dx, and

[ω, ω](b) = 2iIm λ

 _b

0 |ω(x, λ)|²dx + [ω, ω](0).

Since [ω, ω](0) =−2iIm
m, it follows from (3.11) and (3.13) that a point m is
on or inside C_b(λ) if and only if

(3.14)

 _b

0 |ω(x, λ)|²dx≤Imm
Im λ.

A point m is on C
_b(λ) if and only if equality holds in (3.14); otherwisem is in
the interior of C_b(λ).The radius of C_b(λ) is determined by

(3.15) r_b(λ)⁻¹= 2|Im λ|

 _b

0 |u1(x, λ)|²dx.

Now let 0 < a < b <∞.Then if
m is inside or on C_b(λ)

 _a

0 |ω(x, λ)|²dx <

 _b

0 |ω(x, λ)|²dx≤ Imm
Im λ,

and thereforem is inside C
_a(λ).This shows that the circles C_b(λ) are nested:

the interior of C_a(λ) contains C_b(λ) for 0 < a < b <∞.Thus for a given λ such that Im λ= 0, as b → ∞ the circles Cb(λ) converge either to a circle C_∞(λ) or to a point m(λ).

In the case that the C_b(λ) converge to a circle C_∞(λ), let m be a point on
C_∞; otherwise letm = m(λ), the limit point.In both cases
m is inside C
_b(λ) for all b > 0.Hence,

 _b

0 |u1(x, λ)m
− u2(x, λ)|²dx < Im m Im λ,

and letting b → ∞ on sees that u₁(·, λ)
m− u₂(·, λ) ∈ L²(0,∞).Therefore, if Im λ= 0, there always exists a solution of Ly = λy that belongs to L²(0,∞).

If the circles C_b(λ) converge to a circle C_∞(λ), then its radius r_∞(λ) = lim r_b(λ) is strictly positive, and from (3.15) it follows that u₁(·, λ) ∈ L²(0,∞).

In the case that the C_b(λ) converge to a point m(λ), it follows that lim r_b(λ) = 0 so that u₁∈ L/ ²(0,∞) in this case.This completes the proof.

Theorem 3.3 implies that the defect numbers of T_min are either (2, 2), the limit circle case, or (1, 1), the limit point case.In the limit circle case the defect space Nλ = ker (T_max − λ) is spanned by the functions u1(·, λ) and u₂(·, λ).In the limit point case the defect space is spanned by the function ω(·, λ) = u₁(·, λ)m(λ) − u₂(·, λ).

3.3 The limit point case

In all forthcoming sections it will be assumed that the formal operator L is in the limit point case at infinity.For instance, L is in the limit point case when q(x) satisfies the inequality q(x)≥ −kx²where k is a positive constant, see [8].

In particular, for all nonnegative constant functions q(x) the formal operator L is in the limit point case.

Proposition 3.4. Suppose that the operator L is in the limit point case at infinity. Then [y₁, y₂](∞) = 0 for all y₁, y₂∈ D. In particular, Green’s identity reduces to

(3.16) (Ly₁, y₂)− (y₁, Ly₂) =−[y₁, y₂](0), for all y₁, y₂∈ D.