On Spectra of Non-Self-Adjoint Sturm-Liouville Operators ...

On Spectra of Non-Self-Adjoint Sturm-Liouville Operators

Sergio Albeverio, Rostyslav Hryniv, Yaroslav Mykytyuk

no. 254

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs-

gemeinschaft getragenen Sonderforschungsbereiches 611 an der Univer-

sität Bonn entstanden und als Manuskript vervielfältigt worden.

Bonn, Dezember 2005

ON SPECTRA OF NON-SELF-ADJOINT STURM–LIOUVILLEOPERATORS

SERGIO ALBEVERIO∗ ROSTYSLAV HRYNIV†‡ YAROSLAV MYKYTYUK‡

Abstract. The spectra of non-self-adjoint Sturm–Liouville operators with distribu-tional potentials belonging to the space W−1

2 (0, 1) are studied. In particular, it isshown that any sequence of complex numbers obeying a specified asymptotics co-incides with the spectrum of some non-self-adjoint Sturm–Liouville operator of theclass under consideration. The inverse spectral problem of reconstructing an operatorfrom two spectra or from one spectrum and suitably defined norming constants is alsosolved, and a complete description of the spectral data for the operators consideredis given.

1. Introduction

The main aim of the present paper is to answer the following question: What spectracan non-self-adjoint Sturm–Liouville operators

(1.1) − d2

dx2+ q(x)

on the interval [0, 1] have? The inverse spectral theory for Sturm–Liouville operatorsdeveloped in the 1950-ies by Gelfand, Krein, Levitan, and Marchenko [7, 18, 20] gavea complete answer to this question in the self-adjoint case. A typical result (Theo-rem 3.4.1 in the book by Marchenko [21]) reads

Theorem A. Two sequences λ1 < λ2 < . . . resp. µ1 < µ2 < . . . of real numbersare Dirichlet resp. Neumann–Dirichlet spectra of a Sturm–Liouville differential expres-sion (1.1) with a potential q ∈ L2(0, 1) if and only if these sequences interlace, i.e.,µn < λn < µn+1 for all n ∈ N, and obey the asymptotics

λn = π2n2 + A+ an,(1.2)

µn = π2(n− 12)2 + A+ bn,(1.3)

with some A ∈ R and some `2-sequences (an)n∈N and (bn)n∈N.

If different constants A are allowed in (1.2) and (1.3), then the λn and µn are gen-erated respectively by the Dirichlet boundary conditions and Robin–Dirichlet (i.e., of

Date: December 20, 2005.2000 Mathematics Subject Classification. Primary 34A55; Secondary 34B24, 47A56, 47E05.Key words and phrases. Inverse spectral problems, Sturm–Liouville, non-self-adjoint operators,

singular potentials.∗ Institut fur Angewandte Mathematik, Universitat Bonn, Wegelerstr. 6, D–53115, Bonn, Germany;

SFB 611 and IZKS, Bonn, Germany; BiBoS, Bielefeld, Germany; IZKS; CERFIM, Locarno, Switzer-land; and Accademia di Architettura, Mendrisio, Switzerland; e-mail: [email protected].

† Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova st., 79601 Lviv,Ukraine; e-mail: [email protected].

‡ Dept. of Mechanics and Mathematics, Lviv National University, 1 Universytetska st., 79602 Lviv,Ukraine; e-mail: [email protected].

1

2 S. ALBEVERIO, R. HRYNIV, AND YA. MYKYTYUK

the form y′(0) − hy(0) = y(1) = 0 with some nonzero h ∈ R) ones, cf. [5]. More-over, there is an efficient algorithm reconstructing the real-valued potential q from thetwo spectra. Similar complete descriptions of the spectra in terms of interlacing andasymptotics are known for other types of boundary conditions, e.g., for the Robin–Robin ones [15, 16], [19, Ch. II.10], [21, Problem 3.4.3] and for potentials from, e.g.,W n

2 (0, 1), n ∈ N [21, Corollary of Theorem 3.4.1]. Completely solved is also the inversespectral problem of reconstructing a Sturm–Liouville operator from its spectrum andthe norming constants [19, 24]. Recent developments of the inverse spectral theoryof self-adjoint Sturm–Liouville operators have included the case of singular potentialsfrom W−1

2 (0, 1) [2, 4, 11, 12, 25, 27, 28] and resulted, in particular, in a completecharacterisation of possible spectra for such operators.

Spectra of non-self-adjoint Sturm–Liouville operators have in general been studiedconsiderably less. To the best of our knowledge, the strongest result here is due toTkachenko [29], and it reads as follows:

Theorem B. Let Λ := (λn)n∈N be a sequence of complex numbers that is symmetricwith respect to the real line (i.e., Λ contains λ along with each λ ∈ Λ and, moreover,both are repeated in Λ the same number of times) and obeys the asymptotics of (1.2) forsome A ∈ R and (an)n∈N ∈ `2. Then there exists q ∈ L2(0, 1) such that Λ coincides withthe Dirichlet spectrum (counting multiplicity) of the Sturm–Liouville operator (1.1).

Here and hereafter, we say that a sequence (cn)n∈N of complex numbers coincides withthe spectrum S(T ) of an operator T in L2(0, 1) if S(T ) =

⋃n∈Ncn and the algebraic

multiplicity of every c ∈ S(T ) equals the number of occurrences of c in (cn)n∈N. Weshall always label eigenvalues in such a way that equal values are adjacent and theirmoduli do not decrease. We shall also assume that zero is not an eigenvalue, whichcan always be achieved by adding a suitable constant to the potential q.

Theorem B implies [29, Proposition 1] that for every finite set λ1, . . . , λn of pairwisedistinct complex numbers and every set m1, . . . ,mn of natural numbers there exists aSturm–Liouville operator (1.1), for which λ1, . . . , λn are points of the Dirichlet spectrumof multiplicities m1, . . . ,mn respectively.

Observe that the assumptions of Theorem B (the symmetry and the asymptoticsof Λ) force the numbers λn to be real for all n large enough. Even if one takes intoaccount that a sufficiently small perturbation of a Dirichlet spectrum is a Dirichletspectrum for another Sturm–Liouville operator, neither Theorem B nor its corollarymentioned above imply that any sequence (λn)n∈N of complex numbers obeying (1.2)with A ∈ C and (an)n∈N ∈ `2 is the Dirichlet spectrum of a Sturm–Liouville opera-tor (1.1) with some q ∈ L2(0, 1).

The aim of the present article is two-fold. First, we would like to show thatthe spectra of non-self-adjoint Sturm–Liouville operators, apart from obeying spec-ified asymptotics, can indeed be arbitrary. Secondly, we also want to consider awider class of Sturm–Liouville operators—namely, those with distributional potentialsfrom W−1

2 (0, 1). Such potentials may have singularities of, e.g., a Dirac delta-typeor a Coulomb 1/x-type that have been widely used in quantum mechanics to modelinteractions in the molecules and atoms, see, e.g., [1, 3].

The precise definition of a Sturm–Liouville operator (1.1) with distributional poten-tial q ∈ W−1

2 (0, 1) goes as follows [27]. We take σ to be the distributional primitiveof q of zero mean and denote by T the operator in L2(0, 1) given by

Ty = l(q)y := −(y′ − σy)′ − σy′

SPECTRA OF NON-SELF-ADJOINT OPERATORS 3

on the domain

domT = y ∈ W 11 (0, 1) | y[1] := y′ − σy ∈ W 1

1 (0, 1), l(q)y ∈ L2(0, 1).

We denote by T (q,∞) the restriction of T by the Dirichlet boundary conditions y(0) =y(1) = 0 and recall [13, 27] that T (q,∞) has discrete spectrum accumulating at +∞and that the sequence (λn)n∈N of its eigenvalues obeys the asymptotics

(1.4) λn = (πn+ λn)2

with some `2-sequence (λn). Since a function from the domain of T needn’t have acontinuous derivative, the classical Neumann or Robin boundary condition has no sense;however, the quasi-derivative y[1](x) = y′(x) − σ(x)y(x) of y ∈ domT is continuous.For any h ∈ C, we can thus define the restriction T (q, h) of T by the Robin–Dirichletboundary conditions

(1.5) y[1](0)− hy(0) = y(1) = 0.

The operator T (q, h) has a discrete spectrum and its eigenvalues (µn)n∈N obey therelation [13, 27]

(1.6) µn =(π(n− 1

2) + µn

)2, (µn) ∈ `2.

Let us point out that in this singular case the “Neumann” (h = 0) and Robin (h 6= 0)boundary conditions at zero in (1.5) are in fact undistinguishable since taking σ+h asa primitive of q does not change l(q) and makes the Robin condition the “Neumann”one.

It turns out that the asymptotics of (1.4) or of (1.6) is the only condition that thespectra of the corresponding singular Sturm–Liouville operators must satisfy.

Theorem 1.1. For any sequence (λn)n∈N of complex numbers satisfying (1.4) thereis a Sturm–Liouville operator (1.1) with a complex-valued potential from W−1

2 (0, 1),whose Dirichlet spectrum coincides with (λn).

Similarly, for any sequence (µn)n∈N of complex numbers satisfying (1.6) there are acomplex-valued distribution q ∈ W−1

2 (0, 1) and a number h ∈ C such that the spectrumof T (q, h) coincides with (µn).

In the non-self-adjoint case there is no simple analogue of Marchenko’s Theorem A.Although there are necessary and sufficient conditions for two sequences to be theDirichlet and Robin–Dirichlet spectra of a Sturm–Liouville operator (known for poten-tials from L2(0, 1) [29] and established in Section 3 for potentials from W−1

2 (0, 1), seeTheorem 3.6), they cannot be formulated in terms of geometric properties of the verysequences alone. In this regard the following result seems of interest.

Theorem 1.2. Assume that sequences (λn)n∈N and (µn)n∈N of complex numbers verifyconditions (1.4) and (1.6) respectively. Then for every ε > 0 there is a sequence (µn)n∈Nsuch that

(a) the index set I := n ∈ N | µn 6= µn is finite and |µn−µn| < ε for every n ∈ I;(b) there are q ∈ W−1

2 (0, 1) and h ∈ C such that the sequences (λn)n∈N and (µn)n∈Nare spectra of the Sturm–Liouville operators T (q,∞) and T (q, h) respectively.

We remark that this theorem does not answer the question whether for any twodisjoint finite sequences in C there are q ∈ W−1

2 (0, 1) and h ∈ C such that the firstsequence is in the spectrum of T (q,∞) and the second in that of T (q, h).


Another classical inverse spectral problem consists in the reconstruction of a Sturm–Liouville operator from one spectrum and the sequence (αn)n∈N of the correspondingnorming constants. In the self-adjoint case we have

αn :=

∫ 1

0

|y(x, λn)|2 dx,

where y(·, λ) is the solution to the equation −y′′ + qy = λ2y satisfying the initial

conditions y(0) = 0 and y[1](0) =√λ if h = ∞ and the initial conditions y(0) = 1

and y[1](0) = h if h is a finite number. Positivity and correct asymptotics are the onlyrequirements that the norming constants must satisfy in the self-adjoint case, as thefollowing statement conveys (see [11] and also [19, 24] for the case of regular potentialsfrom L2(0, 1)).

Theorem C. For any sequences of real numbers (λn)n∈N and (αn)n∈N such that λn

strictly increase and obey (1.4) and αn are positive and αn = 12+αn for some `2-sequence

(αn)n∈N there is a real-valued q ∈ W−12 (0, 1) such that λn and αn are respectively the

eigenvalues and norming constants of the Sturm–Liouville operator T (q,∞).Similarly, if in the above assumptions (λn) is replaced by a sequence (µn) obey-

ing (1.6) instead of (1.4), then the conclusion holds with a Sturm–Liouville operatorT (q, h) for some q ∈ W−1

2 (0, 1) and h ∈ R.

In the non-self-adjoint case, non-simple eigenvalues are possible, and the notion ofthe norming constant should be revised. Assume that λ is an eigenvalue of T (q, h)of algebraic multiplicity m; then λ = λn = λn+1 = · · · = λn+m−1 for some n ∈ N.Introduce the functions yn, . . . , yn+m−1 via

yn+j(x) :=1

j!

∂jy(x, z)

∂zj

∣∣∣z=λn

, j = 0, 1, . . . ,m− 1.

Then it is readily seen that yn is an eigenfunction of T (q, h) corresponding to theeigenvalue λn and that yn+j ∈ domT (q, h) and

T (q,∞)yn+j = λnyn+j + yn+j−1,

for j = 1, . . . ,m − 1—in other words, that the sequence yn, yn+1, . . . , yn+m−1 forms achain of eigen- and associated functions of T (q, h) corresponding to the eigenvalue λn.In this way we construct the sequence (yk)k∈N, in which yk is an eigen- or associatedfunction of T (q, h) corresponding to the eigenvalue λk. It is known [13, 27] that thesequence (yk)k∈N forms a Riesz basis of L2(0, 1).

We now introduce the bilinear (not sesquilinear!) form 〈 ·, · 〉 in L2(0, 1) by

〈f, g〉 =

∫ 1

0

f(t)g(t) dt

and put

akl := 〈yk, yl〉.The Gram matrix A = (akl) is known [21, Lemma 1.3.4] to have a block-diagonal form,namely akl = 0 if λk 6= λl. Moreover, the sub-matrix on the diagonal corresponding toan eigenvalue λn = λn+1 = · · · = λn+m−1 of multiplicity m is a Hankel lower-triangularmatrix of size m, i.e.,

akl =

0, k, l = n, . . . , n+m− 1, k + l < 2n+m− 1,

αk+l−(n+m−1), k, l = n, . . . , n+m− 1, k + l ≥ 2n+m− 1.


We observe that αn 6= 0, as otherwise the function yn would be orthogonal to yl for alll ∈ N; therefore this Hankel matrix is non-singular. We call the number αk, k ∈ N, thenorming constant corresponding to the eigenvalue λk. If k is such that λk is a simpleeigenvalue (i.e., if m = 1), then this definition gives αk =

∫ 1

0y2

k(x) dx, which agreeswith the above definition in the self-adjoint case. Observe also that the αk have thesame asymptotics as in the self-adjoint case, see Corollary 2.5.

We stress that the structure of the matrix A (i.e., the size of its blocks on thediagonal) is uniquely determined by the sequence (λn) and that A is in a one-to-onecorrespondence with the sequence of norming constants (αn). Therefore knowing (λn)and (αn) amounts to knowing (λn) and the corresponding Gram matrix A.

Although in the non-self-adjoint case Theorem C has no direct analogue, it still holdsgenerically—in the following sense—and immediately implies Theorem 1.1.

Theorem 1.3. Assume that sequences (λn)n∈N and (αn)n∈N of complex numbers satisfy

the asymptotics λn = (πn+ λn)2 and αn = 12+ αn for some `2-sequences (λn) and (αn).

Fix an arbitrary ε > 0. Then finitely many numbers αn can be changed at most by εso that (λn) becomes the spectrum and the sequence of changed αn the sequence ofthe corresponding norming constants of a Sturm–Liouville operator T (q,∞) with someq ∈ W−1

2 (0, 1).A similar statement holds if instead of the asymptotics (1.4) that of (1.6) is taken,

resulting in a Sturm–Liouville operator T (q, h) with some h ∈ C.

Combining the above statements with the criterion on solubility of the inversespectral problem for Sturm–Liouville operators with complex-valued potentials in thespace L2(0, 1) [29] (cf. also [19, 21] in the self-adjoint case), we get the following de-scription of the spectra of such operators.

Corollary 1.4. Assume that sequences (λn)n∈N, (µn)n∈N, and (αn)n∈N of complex num-bers are such that

λn = π2n2 + A+ an,

µn = π2(n− 12)2 +B + bn,

αn =1

2+βn

n

for some complex A and B and some complex `2-sequences (an), (bn), and (βn). Thenthe conclusions of Theorems 1.1–1.2 hold true with some complex-valued q ∈ L2(0, 1).

The organisation of the paper is as follows. In Section 2 we study some properties ofthe Gram matrix A generated by the norming constants and establish the resolutionof identity for the operator T (q, h). This is used in Section 3 to derive the Gelfand–Levitan–Marchenko equation and establish two criteria for the solubility of the inversespectral problem. In Sections 4 and 5 Theorems 1.3 and 1.2 are proved and, finally,in Appendix A one auxiliary result on the zero set of a function of two variables isestablished.

In the sequel, we always take the branch of the square root that is defined as follows:if z = reiθ with θ ∈ (−π, π], then

√z =

√reiθ/2.

2. Preliminaries

Assume that q ∈ W−12 (0, 1) is a complex-valued distribution and h ∈ C an arbi-

trary complex number. We denote by (λn)n∈N and (µn)n∈N the eigenvalue sequences


of the operators T (q,∞) and T (q, h) that are defined in the Introduction and assumethroughout that none of λn and µn vanishes. In what follows, mk shall always denotethe multiplicity of λk; we shall write k ∼ l if and only if λk = λl.

We denote by y(·, λ) and y+(·, λ) the solutions to the differential equation l(q)y = λy

satisfying respectively the initial conditions y(0) = 0 and y[1](0) =√λ and the terminal

conditions y(1) = 0 and y[1](1) =√λ.

The numbers λn and µn are zeros of the functions y+(0, λ) and y[1]+ (0, λ)− hy+(0, λ)

respectively. Since these functions become entire of exponential type 12

after division by√λ, they coincide up to a constant factor with the corresponding canonical products

for λn and µn. More exactly, we put

(2.1) s(z) =∞∏

k=1

λk − z

π2k2, c(z) =

∞∏k=1

µk − z

π2(k − 12)2

;

the following holds then, cf. [12, 29]:

Proposition 2.1. For z ∈ C, we have√zs(z) ≡ y+(0, z),

√zc(z) ≡ y

[1]+ (0, z)− hy+(0, z).

Lemma 2.2. Let ν be a (nonzero) eigenvalue of T (q,∞) of multiplicity m ≥ 1. Thenthe spectral projector Pν of T (q,∞) corresponding to ν is an integral operator withkernel

(2.2) resz=ν

[c(z)

zs(z)y(x, z)y(t, z)

].

Proof. Recall [17, Ch. III.6.4] that the spectral projector Pν is the operator

Pν = − 1

2πi

∫γ

(T (q,∞)− λ

)−1dλ,

where γ is a closed contour in the resolvent set of T (q,∞) surrounding ν but no other

points of the spectrum of T (q,∞). The resolvent(T (q,∞) − λ

)−1of the operator

T (q,∞) at the point λ is an integral operator with the kernel

G(x, t, λ) :=y(x ∨ t, λ)y+(x ∧ t, λ)

W (λ)

where x ∨ t = minx, t, x ∧ t = maxx, t, and

W (λ) = y[1](x, λ)y+(x, λ)− y(x, λ)y[1]+ (x, λ),

is the Wronskian of y and y+. Evaluation at x = 0 gives W (λ) =√λy+(0, λ). We also

have

y+(·, λ) = y+(0, λ)y−(·, λ) +y

[1]+ (0, λ)− hy+(0, λ)√

λy(·, λ),

where y−(·, λ) is a solutions of the equation lσy = λy (λ 6= 0) satisfying the initialconditions y(0) = 1 and y[1](0) = h; therefore

G(x, t, λ) =y(x ∨ t, λ)y−(x ∧ t, λ)√

λ+y

[1]+ (0, λ)− hy+(0, λ)

λy+(0, λ)y(x ∨ t, λ)y(x ∧ t, λ).

Recalling Proposition 2.1, we conclude that the projector Pν is an integral operatorwith kernel

resz=ν

G(x, t, z) = resz=ν

[ c(z)zs(z)

y(x, z)y(t, z)]


as claimed.

Since the system of root subspaces of T (q,∞) forms a Riesz basis of subspaces [13,27], the spectral projectors Pν1 , Pν2 , . . . corresponding to all pairwise distinct eigenval-ues ν1, ν2, . . . perform the resolution of identity

s-limn→∞

n∑k=1

Pνk= I.

Substituting in this relation formula (2.2) for Pk and taking into account multiplicities,we get

(2.3) s-limn→∞

n∑k=1

resz=λk

[c(z)

mkzs(z)〈 ·, y(·, z)〉y(·, z)

]= I.

The residues in (2.3) can be calculated explicitly. Recalling the definitions of thefunctions yn and of the equivalence relation ∼ , we can recast (2.3) in the form

(2.4) s-limn→∞

n∑k=1

∑l∼k

blk〈 ·, yl〉yk = I.

The matrix B := (bkl)k,l∈N has the following properties: first, bkl = 0 if k 6∼ l, i.e., Bhas a block-diagonal form of the same structure as the Gram matrix A. Next, if λn =λn+1 = · · · = λn+m−1 is an eigenvalue of multiplicity m and l, k ∈ n, . . . , n +m − 1(so that l ∼ k), then

blk =

βk+l−n, k + l < 2n+m,

0, k + l ≥ 2n+m,

with

(2.5) βn+j =m!

(m− 1− j)!

∂m−1−jc(z)

∂zm−1−j

∣∣∣z=λn

/∂mzs(z)

∂zm

∣∣∣z=λn

for j = 0, . . . ,m − 1. In other words, the corresponding block is an upper-triangularHankel matrix of size m, with βn in the upper-left corner, βn+1 on the second skewdiagonal, and so on, with βn+m−1 on the main skew diagonal, and with zeros below it.Since λn is a zero of order m for s and c(λn) 6= 0, we see that βn+m−1 6= 0 and thusthis Hankel matrix is non-singular.

Lemma 2.3. The numbers βn satisfy the relation βn = 2 + βn, where the βn form an`2-sequence.

Proof. There is n0 ∈ N such that λn with n ≥ n0 have multiplicity 1, and βn =c(λn)/(λns(λn)) for such n. According to [12], there are functions f and g in L2(0, 1)such that

s(z) =sin

√z√

z+

∫ 1

0

f(t)sin

√zt√z

dt,

c(z) = cos√z +

∫ 1

0

g(t) cos√zt dt,(2.6)

whence

λns(λn) =1

2

[cos

√λn +

∫ 1

0

tf(t) cos√λnt dt

].


Since cos√λnxn≥n0 is a Riesz basic system in L2(0, 1) [10], it follows that the numbers∫ 1

0

tf(t) cos√λnt dt,

∫ 1

0

g(t) cos√λnt dt

form `2-sequences. Since also cos√λn = (−1)n +O(λ2

n), we conclude that the sequence(βn − 2)n∈N belongs to `2 as required.

Lemma 2.4. The matrix B is invertible and its inverse is the Gram matrix A.

Proof. We have shown above that all diagonal blocks of B are non-singular Hankelmatrices. We remark that the diagonal blocks of B are of size 1 starting from someindex n0, i.e., B becomes diagonal with βn on the diagonal after removing the top n0−1rows and the left n0− 1 columns. Since such βn are nonzero and obey the asymptoticsof Lemma 2.3, B is boundedly invertible.

Upon applying (2.4) to yj and recalling the definition of the Gram matrix A we findthat

yj = s-limn→∞

n∑k=1

∑l∼k

blk〈yj, yl〉yk =∞∑

k=1

[∑l∼k

ajlblk

]yk.

Since the system ynn∈N is a Riesz basis of L2(0, 1), we conclude that∑l∼k

ajlblk =∞∑l=1

ajlblk = δjk,

δjk being the Kronecker delta, which implies that B−1 = A.

Corollary 2.5. The norming constants αn obey the asymptotics αn = 12+ αn for some

`2-sequence (αn).

3. Criteria on solubility of the inverse spectral problem

The main aim of this section is to give necessary and sufficient conditions on twosequences in order that they could be Dirichlet and Robin–Dirichlet spectra for someSturm–Liouville expression with a complex potential in W−1

2 (0, 1). Similar results forregular non-self-adjoint operators were obtained in [26, 29].

Assume that q is an arbitrary complex-valued distribution from W−12 (0, 1) and that

h is an arbitrary complex number and denote by (λn)n∈N and (µn)n∈N the sequences ofeigenvalues of the operators T (q,∞) and T (q, h) respectively. We always repeat eacheigenvalue according to its multiplicity and enumerate the eigenvalues so that theirmoduli do not decrease and coinciding eigenvalues are adjacent. Also, mk denotes themultiplicity of λk in the sequence (λn).

It is well known that the Dirichlet and Robin-Dirichlet spectra of a Sturm–Liouvilleexpression do not intersect and must satisfy the asymptotics of (1.4) and (1.6) respec-tively [12, 27]. There is another condition that the two spectra must satisfy, whichcomes from the solubility of the related Gelfand–Levitan–Marchenko equation andwhich we are going to derive.

As earlier, y(·, λ) stands for the solution of the equation l(q)y = λy satisfying the

initial conditions y(0) = 0 and y[1](0) =√λ. It is known [13] that there exists an

integral Hilbert–Schmidt operator K with a lower-triangular kernel k such that, for allλ ∈ C,

(3.1) y(x, λ) = sin√λx+

∫ x

0

k(x, t) sin√λt dt =: (I +K) sin

√λx.


The operator I+K maps the solution sin√λx of the unperturbed equation −y′′ = λ2y

(with q ≡ 0) into the solution y(·, λ) of the perturbed one and is called the transfor-mation operator.

Moreover, K belongs to the special operator algebra G consisting of all integraloperators G in L2(0, 1) whose kernels g are measurable functions on [0, 1]× [0, 1] withthe following properties:

(a) for each fixed x ∈ [0, 1], the functions g(x, ·) and g(· , x) are in L2(0, 1);(b) the mappings from [0, 1] into L2(0, 1) given by x 7→ g(x, ·) and x 7→ g(· , x) are

continuous.

It follows from definition that every operator in G belongs to the Hilbert–Schmidt classand, in particular, is bounded in L2(0, 1).

Definition 3.1. Denote by G+ (resp., by G−) the subalgebra of G consisting of alloperators in G with lower-triangular (resp., upper-triangular) kernels. We say that anoperator I+G with G ∈ G has a special factorisation in G (or that I+G is factorisablein G) if there are operators R+ ∈ G+ and R− ∈ G− such that

I +G = (I +R+)−1(I +R−)−1.

Substituting (3.1) into the resolution of identity (2.3), we derive the relation

(3.2) (I +K)U(I +K>) = I,

where

(3.3) U := s-limn→∞

n∑k=1

resz=λk

[c(z)

mkzs(z)〈 ·, sin

√zt〉 sin

√zx

]and K> is the conjugate operator with kernel (k>)(x, t) := k(t, x). Our next aim is toestablish some properties of the operator U .

Lemma 3.2. The operator F := U − I belongs to G and has the kernel f(x, t) =φ(|x− t|)− φ(x+ t), where

(3.4) φ(x) :=∞∑

n=1

res

z=λk

(c(z) cos√zx

2mkzs(z)

)− cos πkx

is a function in L2(0, 2).

Proof. We use the relation

I = s-limn→∞

n∑k=1

2( · , sin πkt) sinπkx

and observe that the k-th summand is an integral operator with kernel

hk(x, t) := cos πk(x− t)− cos πk(x+ t).

Similarly the k-th summand in (3.3) is an integral operator with kernel gk(x − t) −gk(x+ t), where

gk(x) := resz=λk

[c(z) cos√zx

2mkzs(z)

].

Choose n0 ∈ N such that λn is a simple zero of s for each n ≥ n0; then for such n wehave gn(x) = 1

2βn cosλnx with βn := c(λn)/(λns(λn)). In view of Lemma 2.3 and the


asymptotics of λn the series ∑n≥n0

(βk

2cosλkx− cos πkx

)converges in L2(0, 2) [11]. Thus F is an integral operator with the kernel f as stated,and so belongs to G. The proof is complete.

Definition 3.3. For a given sequence Λ := (λn)n∈N, we denote by SΛ, resp. by CΛ, thesequences (sn), resp. (cn), of functions defined as follows: if λn = λn+1 = · · · = λn+m−1

has multiplicity m in Λ, then

sn+j(x) :=1

j!

∂j

∂zjsin

√zx|z=λn , cn+j(x) :=

1

j!

∂j

∂zjcos

√zx|z=λn

for j = 0, . . . ,m− 1.

We observe that if Λ obeys the asymptotics of (1.4) or (1.6), then the systems SΛ

and CΛ form Riesz bases of L2(0, 1) [10].It is easily seen that if Λ is the eigenvalue sequence of the operator T (q, h), then

sn = (I +K)−1yn, where yn are defined in the Introduction. Using (2.4) and (3.2), weget the representation of U in the form

(3.5) U = s-limn→∞

n∑k=1

∑l∼k

blk〈 ·, sl〉sk.

Lemma 3.4. Denote by D = (dkl) the operator in `2 that is inverse to B = (bkl). Then

〈U−1sk, sl〉 = dkl.

Proof. As the system (sn)n∈N forms a Riesz basis in L2(0, 1) [10], there is a sequence(sn) that is biorthogonal to (sn) with respect to the bilinear form 〈 ·, · 〉, i.e., such that〈sk, sl〉 = δkl. This implies that the inverse operator U−1 can be written explicitly as

U−1 = s-limn→∞

n∑i=1

∑j∼i

dji〈 ·, sj〉si,

and the required formula follows.

Returning to our analysis, we see from the above relations that the operator I + Fpossesses a special factorisation in G in the sense of Definition 3.1, namely

(3.6) I + F = (I +K)−1(I +K>)−1.

Applying I+K to both sides of this equality and writing the resulting relation in termsof kernels, we arrive at the Gelfand–Levitan–Marchenko (GLM) equation,

(3.7) k(x, t) + f(x, t) +

∫ x

0

k(x, s)f(s, t) dt = 0, x > t.

It is known that, conversely, if an integral operator F ∈ G with kernel f is such that theGLM equation (3.7) has a solution k generating an integral operator K in G, then (3.6)holds and I + F is factorisable in G.

The well-known criterion on factorisablity in classical algebras of integral operatorsover (0, 1) [8, 9] (or, which is the same, solubility of the GLM equation (3.7)) wasshown in [22, 23] to hold also in G, namely:


Proposition 3.5. Let G be an integral operator in G with kernel g. Then I + G isfactorisable in G if and only if for every a ∈ (0, 1] the equation

h(x) +

∫ a

0

g(x, t)h(t) dt = 0

has only a trivial solution in L2(0, 1).

These considerations establish the necessity part of the following theorem.

Theorem 3.6. Sequences (λn)n∈N and (µn)n∈N of complex numbers are eigenvalues ofcorresponding multiplicities of Sturm–Liouville operators T (q,∞) and T (q, h) respec-tively for some q ∈ W−1

2 (0, 1) and h ∈ C if and only if the following two conditions aresatisfied:

(i) the sequences (λn) and (µn) have no common elements (i.e., λn = µm for no

n,m ∈ N) and satisfy the asymptotics√λn = πn+ λn and

√µn = π(n− 1

2)+ µn

with some `2-sequences (λn) and (µn);(ii) let the entire functions s and c be defined via (2.1) and let f(x, t) = φ(|x− t|)−

φ(x+ t) with φ of (3.4); then for every a ∈ (0, 1] the equation

h(x) +

∫ a

0

f(x, t)h(t) dt = 0


Proof of sufficiency. We assume that (λn)n∈N and (µn)n∈N are sequences of complexnumbers satisfying the properties (i) and (ii) and we construct the functions s, c, φ,and f as explained. The proofs of Lemmata 2.3 and 3.2 only used the asymptotics ofthe λn and µn and thus apply to this situation, so that the integral operator F withkernel f belongs to G.

Assumption (ii) in view of Proposition 3.5 means that the operator I+F is factoris-able in G. Since F is self-adjoint, this factorisation is of the form (3.6), with K anintegral operator in G+. By [13], the operator I + K is the transformation operatorcorresponding to a potential q ∈ W−1

2 (0, 1) with a primitive

(3.8) σ(x) := −2φ(2x)− 2

∫ x

0

k(x, s)f(s, x) ds+ C ∈ L2(0, 1),

C being chosen so that∫σ(x) dx = 0. We shall prove that λn are eigenvalues of the

operator T (q,∞) of required multiplicities and that µn are those of T (q, h), for someh ∈ C.

Since I +K is the transformation operator, the function

u(x, λ) := sin√λx+

∫ x

0

k(x, t) sin√λt dt

is a solution to the equation l(q)y = λy satisfying the initial condition y(0) = 0, sothat λ is an eigenvalue of T (q,∞) of multiplicity m if and only if λ is a zero of theentire function u(1, z)/

√z of the same multiplicity. We observe that

(3.9)

∫ 1

0

u(x, µ)u(x, λ) dx =⟨(I +K>)(I +K) sin

√µx, sin

√λx

⟩=

⟨(I + F )−1 sin

√µx, sin

√λx

⟩.

Reverting the arguments of the proof of Lemma 3.2, we see that U := I + F is givenby formula (3.3).


Define the sequence of functions (un) as follows: if λ = λn = λn+1 = · · · = λn+m−1

has multiplicity m in the sequence (λn), then

un+j(x) :=1

j!

∂j

∂zju(x, z)|z=λn , j = 0, . . . ,m− 1,

and let sn be the functions of Definition 3.3. Since the proof of Lemmata 2.3 and 3.4only used the properties of the functions s and c, we find that U is given also by (3.5),in which B := (bkl) is a block-diagonal matrix of sizes equal to the multiplicities of theλn in Λ, and with non-singular upper-triangular Hankel matrices on the diagonal. Theasymptotics of the numbers βn of (2.5) implies that B is invertible, and we denote byD = (dkl) its inverse. It is easily seen that the matrix D is of block-diagonal form, andeach block on the diagonal is a non-singular lower-triangular Hankel matrix. Now, byLemma 3.4,

(3.10) 〈un, uk〉 = 〈U−1sn, sk〉 = dnk.

Taking λ = λn and µ = λk in (3.9) such that n 6∼ k (i.e., λn 6= λk), we conclude fromthe Lagrange identity that

0 =

∫ 1

0

u(x, λn)u(x, λk) dx =u[1](1, λk)u(1, λn)− u(1, λk)u

[1](1, λn)

λn − λk

.

If u(1, λn) 6= 0 for some n ∈ N, then there would exist a number h ∈ C such thatu[1](1, λk) = hu(1, λk) for all sufficiently large eigenvalues λk. Thus such λk wouldbe eigenvalues of a Sturm–Liouville operator T (q, h), but the latter have a differentasymptotics of (1.6). The derived contradiction shows that u(1, λn) = 0 for all n ∈ N,which means that λn are indeed eigenvalues of T (q,∞). In fact, the relation∫ 1

0

u(x, λ)u(x, λn) dx = u[1](1, λn)u(1, λ)

λ− λn

together with (3.10) and the properties of the (dnk) show that u(1, λ) has a zero oforder mn at λn, so that λn is an eigenvalue of T (q,∞) of multiplicity mn. Since thesystem u(x, λn)n∈N, being a homeomorphic image of the Riesz basis snn∈N, is itselfa Riesz basis, T (σ,∞) has no other eigenvalues.

We take now h := u[1](1, µ1)/u(1, µ1) and denote by µn the eigenvalues of T (σ, h), by

c(z) the function of (2.1) constructed for µn instead of µn, and by F the corresponding

integral operator. Since both I+F and I+F are factorized as (I+K)−1(I+K>)−1—thefirst by construction, and the second by the results at the beginning of this section, theycoincide. It follows from Lemma 3.2 that the meromorphic functions c(z) cos zx/s(z)and c(z) cos zx/s(z) have the same main parts, so that the function c− c has zeros atthe points λn of order at least mn. Since c has the representation

c(z) = cos z +

∫ 1

0

g(t) cos zt dt

for some g ∈ L2(0, 1) and c has the representation (2.6), we conclude that g − g isorthogonal to cosλnx as well as to their derivatives in λ up to the order mn − 1, i.e.,that g − g is orthogonal to the system CΛ of Definition 3.3 with Λ := (λn). As thesystem CΛ is complete in L2(0, 1), we conclude that g ≡ g, i.e., that µn are eigenvaluesof T (q, h). The theorem is proved.

We observe that, in passing, we established the following criterion for two sequencesto be the sequences of eigenvalues and norming constants for some operator T (q,∞).


For given sequences (λn) and (αn) we construct the block-diagonal matrix A as ex-plained in the Introduction. Assume that A is invertible in `2, denote by B = (bkl) itsinverse, determine the sequence βn generating B, and put

(3.11) φ(s) :=∞∑

n=1

(βn

2cn(x)− cos πnx

),

with the functions cn constructed in Definition 3.3.

Theorem 3.7. Sequences (λn)n∈N and (αn)n∈N of complex numbers are respectivelysequences of eigenvalues and norming constants of a Sturm–Liouville operators T (q,∞)for some q ∈ W−1

2 (0, 1) if and only if the following two conditions are satisfied:

(i) the sequences (λn) and (αn) satisfy the asymptotics√λn = πn + λn and αn =

2 + αn with some `2-sequences (λn) and (αn);(ii) the matrix A is invertible, and if the function φ is defined via (3.11) and

f(x, t) = φ(|x− t|)− φ(x+ t), then for every a ∈ (0, 1] the equation

g(x) +

∫ a

0

f(x, t)g(t) dt = 0


There is an analogue of Theorem 3.7 for Sturm–Liouville operators T (q, h) withh ∈ R, but we leave the details to the reader.

4. Proof of Theorem 1.3

For the sake of definiteness, we shall consider in detail only the Dirichlet case h = ∞.The proof of the case h ∈ C of Theorem 1.3 requires only minor modifications, whichare left to the reader.

We denote by Pa, a ∈ (0, 1], the operator in L2(0, 1) of multiplication by the char-acteristic function of the interval [0, a]. Then Proposition 3.5 claims that an operatorI+G with G ∈ G is factorisable in G if and only if the operators I+FPa are invertiblein L2(0, 1) for all a ∈ (0, 1], and condition (ii) of Theorems 3.6 and 3.7 requires that theoperator I + F , with F ∈ G being an integral operator with kernel f , be factorisablein G. This motivates the following definition.

Definition 4.1. A number a ∈ (0, 1] is called critical for an operator V ∈ B∞ if theoperator I + V Pa is not invertible, i.e., if ker(I + V Pa) 6= 0. The set of all criticalpoints of the operator V will be denoted by Cr(V ).

In this language, the claims of Theorems 1.2 and 1.3 mean that it is always possibleto change a little the spectral data so that the corresponding operator F will have nocritical points.

Assume that Λ = (λn) is a sequence of complex numbers satisfying asymptotics (1.4).We keep the usual convention about the labelling of λn in Λ and write l ∼ k if andonly if λl = λk. We also denote by mk the multiplicity of λk in Λ, i.e., the number ofindices l such that l ∼ k, and by Mk the smallest index l with l ∼ k.

Let also (βn) be a sequence of complex numbers obeying the asymptotics βn = 2+ βn

with an `2-sequence (βn) and such that the βk do not vanish for all indices k of theform Ml + ml − 1 (in particular, βk 6= 0 if mk = 1). This sequence generates aninvertible block-diagonal matrix B, with upper-triangular Hankel matrices of size mk

on the diagonal as in Section 2.


Asymptotics (1.4) forces λn to be simple (i.e., to have multiplicity 1) starting fromsome n0 ∈ N. We choose n0 so that in addition the sets λnn<n0 and πnn≥n0 do notintersect and consider the kernel

fw(x, t) :=∑k<n0

[ wmk

∑l∼k

βl

(l −Ml)!

∂l−Ml

∂zl−Ml

(sin

√zx sin

√zt

)∣∣∣z=λk

− 2 sinπkx sin πkt]

that depends on w ∈ C.

Lemma 4.2. Let Fw be the integral operator with kernel fw. Then there exists anonzero w ∈ C such that Fw has no critical points.

Proof. The definition of Fw implies that the operators I+F0Pa, a ∈ [0, 1), are invertible.Indeed,

(I + F0)f =∑n≥n0

2(f, sin πnt) sinπnx,

so that ((I + F0Pa)f, Paf

)=

((I + F0)Paf, Paf

)=

∑n>n0

2|(Paf, sin πnx)|2.

If (I + F0Pa)f = 0 for some nontrivial f ∈ L2(0, 1) and some a ∈ [0, 1), then Paf isa trigonometric polynomial that is an identical zero on (a, 1). This is impossible, thusI + F0Pa are invertible for all a ∈ (0, 1).

We also observe that the operator I + F1 has the form

(I + F1)f =∑k<n0

∑l∼k

blk〈f, sl〉sk +∑k≥n0

2(f, sin πkt) sinπkx

with the usual definition of sn and with the matrix B := (bkl) determined by βk.Since the system snn<n0 ∪ sin πnxn≥n0 forms a Riesz basis of L2(0, 1) and the

matrix B is invertible, the equality (I + F1)f = 0 is only possible when f = 0. Theoperator F1 being compact, this proves that I + F1 is invertible. As the operator-function I +Fw is holomorphic in C, it follows that the set of w ∈ C, for which I +Fw

is not invertible, is discrete.Put for w ∈ C and t ∈ [0, 1]

φ(w, t) := det(I + FwPt).

Since Fw is of finite rank, it follows that

φ(w, t) =

n0−1∑k=0

wkφk(t)

with φk ∈ W 12 (0, 1) for all k = 0, . . . , n0− 1. By the above, φ(0, t) 6= 0 for all t ∈ (0, 1).

Take an open set G in C such that φ(w, 1) 6= 0 for all w ∈ G. Applying Lemma A.1to the above function φ and the sets G and J := (0, 1), we find w0 ∈ G such thatφ(w0, t) 6= 0 for all t ∈ (0, 1) and hence Cr(Fw0) = ∅.

Lemma 4.3. There are w ∈ C and n1 > n0 such that the integral operator Fw withthe kernel

fw(x, t) =∑k<n0

[ wmk

∑l∼k

βl

(l −Ml)!

∂l−Ml

∂zl−Ml

(sin

√zx sin

√zt

)∣∣∣z=λk

− 2 sin πkx sin πkt]

+∑k>n1

[βk sin

√λkx sin

√λkt− 2 sin πkx sin πkt

]


has no critical points.

Proof. Take w ∈ C as in Lemma 4.2 such that Cr(Fw) = ∅, i.e., such that I + Fw

is factorisable. The difference Fz − Fz can be made arbitrarily small in the Hilbert–Schmidt norm by choosing n1 large enough. Since the set of factorisable operators isopen in the Hilbert–Schmidt operator topology [22], the claim follows.

We shall use the following two auxiliary results, the first of which is straightforwardto verify.

Proposition 4.4. If V is an integral operator in L2(0, 1) with kernel v(x, t) = y(x)y(t),y ∈ L2(0, 1), then

Cr(V ) =a ∈ [0, 1] | 1 +

∫ a

0

y2(t) dt = 0.

Moreover, for a ∈ Cr(V ),

ker(I + V Pa) = ζy | ζ ∈ C,

ran(I + V Pa) = f ∈ L2(0, 1) |∫ a

0

y(t)f(t) dt = 0.

Lemma 4.5. Under the assumptions of Proposition 4.4, take y to be a nontrivialsolution of the equation l(q)y = λy. Then the set Cr(V ) of critical points of thecorresponding operator V is finite.

Proof. In view of Proposition 4.4, we have to show that the function

h(x) := 1 +

∫ x

0

y2(t) dt

has at most finitely many zeros on [0, 1]. To this end it suffices to show that every zeroof h (if any) is isolated.

Since y ∈ W 12 (0, 1), the function h belongs to W 2

2 (0, 1) and thus it is continuouslydifferentiable; moreover, h′(x) = y2(x). Let x0 be a zero of h. If y(x0) 6= 0, thenh′(x0) 6= 0 and we are done. Assume therefore that y(x0) = 0, in which case y[1](x0) 6= 0due to uniqueness of solutions to the equation l(q)y = λy. Using the Cauchy inequalityin the relation

y(x) =

∫ x

x0

y′(t) dt,

we find that y(x) = o(|x− x0|1/2) as x→ x0 and then similarly that∣∣∣∫ x

x0

σ(t)y(t) dt∣∣∣ ≤ max

t∈[x0,x]|y(t)|

∫ x

x0

|σ(t)| dt = o(|x− x0|).

Hence

y(x) =

∫ x

x0

y[1](t) dt+

∫ x

x0

σ(t)y(t) dt = y[1](x0)(x− x0) + o(|x− x0|)

as x→ x0, which by de l’Hopital’s theorem yields

limx→x0

h(x)

(x− x0)3=

(y[1](x0))2

36= 0.

Thus x0 is an isolated zero of h.


Lemma 4.6. Assume that Λ := (λn)n∈N is the spectrum of a Sturm–Liouville operatorT (q,∞). If ζ ∈ Λ is a simple eigenvalue of T (q,∞) and ζ0 6= ζ is its nonzero resolventpoint, then there is a distribution q0 ∈ W−1

2 (0, 1) such that Λ0 := Λ ∪ ζ0 \ ζ is thespectrum of the Sturm–Liouville operator T (q0,∞).

Proof. Denote by U the operator of (3.3) corresponding to T (q,∞) and put

U0 := U − α−1〈 · , sin√ζt〉 sin

√ζx,

where α is the norming constant corresponding to the eigenvalue ζ. Then U0 is notinvertible, while U0 + β0〈 · , sin

√ζ0t〉 sin

√ζ0x is invertible for any nonzero β0; both

statements easily follow from the fact that the systems SΛ and SΛ0 (see Definition 3.3)form Riesz bases of L2(0, 1). We claim that there exist a natural number l ∈ N, complexnumbers β0, . . . , βl and simple eigenvalues ζ1 < · · · < ζl of T (q,∞) different from ζsuch that the operator

U1 := U0 +l∑

j=0

βj〈 · , sin ζjt〉 sin ζjx

is factorisable. By Theorem 3.6, there exists then a distribution q0 ∈ W−12 (0, 1) such

that the spectrum of the Sturm–Liouville operator T (q0,∞) coincides with the set Λ0.It remains to find U1 of the above form that is factorisable. Let

U = (I +K)−1(I +K>)−1

be the factorisation of U and put

U0 = (I +K)U0(I +K>), U1 = (I +K)U1(I +K>).

Then

U0 = I − α−1〈 · , y(·, ζ)〉y(·, ζ), U1 = U0 +l∑

j=0

βj〈 · , vj〉vj,

where y(x, λ) := (I +K) sin√λx is the solution of the differential equation l(q)y = λy

satisfying the initial condition y(0) = 0 and vj := y(·, ζj) for j = 0, 1, . . . , l. By

Lemma 4.5 the set Cr(U0 − I) of the critical points of the operator U0 − I is finite. As

U0 is not invertible along with U0, 1 is a critical point of U0 − I; thus there are l ∈ Nand a1, . . . , al ∈ (0, 1) such that a1 < · · · < al < al+1 := 1 and Cr(U0 − I) = akl+1

k=1.We choose simple eigenvalues ζ1 < · · · < ζl of T (q,∞) so that

(4.1)

∫ ak

0

y(t, ζ)y(t, ζk) dt 6= 0, k = 1, . . . , l.

This is possible since, for any n ≥ n0 and any a ∈ (0, 1), the system y(·, λm)m>n

is complete in L2(0, a). This follows from completeness in L2(0, a) of the systemsin

√λmxm>n and the fact that the operator (I + K)Pa is a homeomorphism of

L2(0, a).Put

pk(w) :=l+1∏j=0

′w − j

k − j, k = 1, . . . , l + 1,


prime meaning that the factor with j = k is omitted, and denote by Vw, w ∈ C, theoperator

Vw := −α−1〈 · , y(·, ζ)〉y(·, ζ) +l∑

j=1

pj(w)〈·, vj〉vj + pl+1(w)〈·, v0〉v0.

Since Vw is of finite rank, the function φ(w, t) := det(I + VwPt) of w ∈ C and t ∈ [0, 1]is well defined and is easily seen to be of the form

φ(w, t) =N∑

k=0

wkφk(t)

with some N ∈ N and some φk ∈ W 12 (0, 1), k = 0, . . . , N . By the definition of Vw,

Cr(V0) = t ∈ [0, 1] | φ(0, t) = 0 = a1, . . . , al+1,so that φ(0, t) 6= 0 if t ∈ [0, 1]\akl+1

k=1. We claim that φ(k, ak) 6= 0 for k = 1, . . . , l+1,i.e., that the operators I + VkPak

are invertible.Indeed, vk 6∈ ran(I + V0Pak

) for k = 1, . . . , l in view of Proposition 4.4 and thecondition (4.1). Therefore the equality

0 = (I + VkPak)f = (I + V0Pak

)f + vk

∫ ak

0

f(t)vk(t) dt

would yield f ∈ ker(I + V0Pak) and thus f = cy(·, ζ) by Proposition 4.4; it would then

follow that

0 = cvk(x)

∫ ak

0

y(t, ζ)vk(t) dt,

which is only possible for c = 0 due to the choice of vk = y(·, ζk). Thus I + VkPakhas

a trivial kernel (whence is invertible) for k = 1, . . . , l. Next, as the operator

(I +K)−1(I + Vl+1)(I +K>)−1 = U0 + 〈 · , sin√ζ0t〉 sin

√ζ0x

is invertible, the same is true of I + Vl+1. In particular, the polynomial φ(w, 1) is notidentically zero.

Applying Lemma A.1 and noticing that φ(w, 1) has at most N zeros, we concludethat the set

Ω0 := w ∈ C | ∀t ∈ (0, 1] φ(w, t) 6= 0is not empty. For some fixed w0 ∈ Ω0, we can take now βj := pj(w0), j = 1, . . . , l andβ0 := pl+1(w0) to complete the proof.

We remark that as a result of the transformation of the above lemma the eigenvalue ζ0gets the norming constant β−1

0 , the norming constant αj of the eigenvalue ζj, j =1, . . . , l, becomes (α−1

j + βj)−1, and all the other norming constants do not change.

Proof of Theorem 1.3. As was mentioned at the beginning of this section, only theDirichlet case h = ∞ will be treated.

Assume that sequences Λ = (λn)n∈N and (αn) satisfy the assumptions of the theo-rem. The corresponding block-diagonal matrix A can be made boundedly invertible bychanging very little finitely many of the αn. The inverse matrix B determines then inthe way explained in Section 2 the sequence (βn) satisfying the properties formulatedat the beginning of this section. We find now indices n0 and n1 and a complex numberw as in Lemmata 4.2 and 4.3 so that the corresponding operator Fw has no criticalpoints. Then I + Fw is factorisable and generates a Sturm–Liouville operator T (q,∞)


for some q ∈ W−12 (0, 1), whose spectrum coincides with a sequence Λ′ = (λ′n)n∈N, where

λ′n = λn if n < n0 or n ≥ n1 and λ′n = π2n2 for n = n0, n0 + 1, . . . , n1 − 1.Applying the procedure of Lemma 4.6, we can replace the eigenvalue λ′n0

by λn0 notchanging the other eigenvalues. Repeating this procedure for the eigenvalues λ′n withn = n0 + 1, . . . , n1 − 1, we end up with a Sturm–Liouville operator T (q0,∞) havingthe required spectrum Λ.

We now keep track of the change of the norming constants (or, rather their inversesβn). The set of w ∈ C for which the operator Fw of Lemma 4.2 has no critical points, iseverywhere dense by [22], so that w within this set can be chosen arbitrarily close to 1,to ensure ε-proximity of the related norming constants αn, n < n0, to the prescribedones. Next we choose n1 as in Lemma 4.3 with w just fixed and observe that the oper-ator T (q,∞) generated by Fw has the required βn for all n ≥ n1. Repeated applicationof the procedure of Lemma 4.6 will only change norming constants corresponding tofinitely many simple eigenvalues with indices not less than n0. Since for a fixed n ≥ n0

there is an everywhere dense set of values of βn for which the corresponding opera-tor F has no critical points, those finitely many of the βn which are not sufficientlyclose to the required values α−1

n can be made close enough without changing the otherβ’s. This shows that the required proximity of the norming constants can always beachieved.

5. Proof of Theorem 1.2

The general strategy of the proof of Theorem 1.2 consists in the following. Given twosequences of complex numbers (λn) and (µn), we construct the constants βn of (2.5)and then the function φ of (3.4). Theorem 1.3 implies that finitely many of the βn

can be changed very little so that for the changed φ condition (ii) of Theorem 3.7holds. Thus there exists a potential q ∈ W 1

2 (0, 1) such that T (q,∞) has spectrum(λn). There is also h ∈ C such that the spectrum (µn) of T (q, h) differs little from thesequence (µn). Finally, we prove that there is k ∈ N such that for the two sequences(λn) and (µn)n<k∪ (µn)n≥k condition (ii) of Theorem 3.6 holds, so that these sequencesare Dirichlet and Robin–Dirichlet eigenvalues of the Sturm–Liouville operator (1.1) foran appropriate choice of q ∈ W−1

2 (0, 1) and the Robin boundary condition at x = 0.To perform this program, we have to specify the topologies under which the related

mappings are continuous.We start with the observation that, without loss of generality, we may assume that

the numbers µn are pairwise distinct (otherwise we can replace every µ repeated mtimes by m pairwise distinct numbers from an ε/2-neighbourhood of µ and replace εby ε/2 in the conclusion of the theorem).

According to (2.5), we have

1

(m− 1− j)!

∂m−1−jc(z)

∂zm−1−j

∣∣∣z=λn

=βn+j

m!

∂mzs(z)

∂zm

∣∣∣z=λn

for j = 0, . . . ,m − 1, where m is the multiplicity of λn and, moreover, n is such thatthere is no k < n with k ∼ n. On the other hand, the functions s and c of (2.1) have


also integral representations of the form

s(z) =sin

√z√

z+

∫ 1

0

f(t)sin

√zt√z

dt,

c(z) = cos√z +

∫ 1

0

g(t) cos√zt dt,(5.1)

for some f, g ∈ L2(0, 1). Introducing the functions cn as in Definition 3.3, we concludethat

1

(m− 1− j)!

∂m−1−jc(z)

∂zm−1−j

∣∣∣z=λn

= cn+m−j(1) +

∫ 1

0

cn+m−j(t)g(t) dt,

so that the sequences (λn) and (βn) determine the scalar products of g and the func-tions cn. We claim that the function g is thus uniquely determined by the sequence(βn) and, moreover, depends continuously on (βn) in a certain sense.

To be more precise, we denote by B the set of all complex sequences (γn) of theform 2 + γn with an `2-sequence (γn). The set B becomes a metric space if we definethe distance between two elements (γn) and (γ′n) by

ρB

((γn), (γ′n)

):= ‖(γn − γ′n)‖`2 .

Lemma 5.1. The function g ∈ L2(0, 1) depends continuously on (βn) ∈ B. Con-versely, the sequence (βn) ∈ B depends continuously on the function g ∈ L2(0, 1).

Proof. Since the sequence (cn) forms a Riesz basis of L2(0, 1), it suffices to show thatthere is a homeomorphism between the sequence (en) ∈ `2 of Fourier coefficients of gwith respect to this Riesz basis (and taken in the pseudo-scalar product 〈 ·, · 〉) and thesequence (βn) ∈ B. Since λn are simple if n is sufficiently large, say n ≥ n0, for suchn we find that

en = 〈g, cn〉 = βnλns(λn)− cn(1).

It remains to recall that, for n ≥ n0, λns(λn) = (−1)n/2 + δn with an `2-sequence (δn)by Lemma 2.3 and that cn(1) = cos

√λn = (−1)n + δ′n with some (δ′n) ∈ `2, due to the

asymptotics of λn.

In a similar manner one can prove the following.

Lemma 5.2. The function φ of (3.11) depends continuously in L2(0, 2) on (βn) ∈ B.

Denote by M the set of all sequences (νn) of complex numbers such that the numbersνn :=

√νn − π(n − 1

2) form a sequence from `2. This establishes a bijection between

M and `2, and thus makes M a metric space. For a given f ∈ L2(0, 1), we form thefunction

Gf (z) := cos√z +

∫ 1

0

f(t) cos√zt dt.

and consider the sequence (zn(f)) of its zeros taken with multiplicities and enumeratedin some way. The mapping f 7→ (zn(f)) is multi-valued since it clearly depends on theenumerations of the zeros. However, under certain conditions it is possible to singleout a single-valued and locally continuous branch of this mapping.

Lemma 5.3. Assume that the numbers µn are pairwise distinct and g ∈ L2(0, 1) is thefunction in the integral representation (5.1) of the corresponding canonical product c.Then there is a neighbourhood O of g in L2(0, 1) and a continuous mapping ψ from Ointo `2, ψ : f 7→ ψ(f) =: (ψn(f)) such that

zn(f) =(π(n− 1

2) + ψn(f)

)2


are all zeros of Gf .

The inverse mapping is also continuous, as the following lemma claims.

Lemma 5.4. Assume that the entries of a sequence (µ0n) ∈ M are pairwise distinct.

Then there exists a neighbourhood O′ of (µ0n) in M in which the mapping O′ 3 (µn) 7→

g ∈ L2(0, 1) is continuous. Here g is the function in the integral representation of thecanonical product c corresponding to (µn).

Statements similar to those in Lemmata 5.3 and 5.4 have already been establishedin [14] (even in a more general form) and thus we omit their proofs here.

Finally, we denote by Φ the set of all φ ∈ L2(0, 2), for which the integral operatorsin L2(0, 1) with kernel φ(x − t) − φ(x + t) have no critical points (see Definition 4.1)and recall the following statement.

Proposition 5.5 ([22]). The set Φ is open in L2(0, 2).

Having established the above properties, we can now complete the proof of Theo-rem 1.2 as follows. We denote by (β0

n) and g0 the corresponding objects constructedfor given sequences (λn) and (µ0

n). We choose δ > 0 such that, in the notations ofLemma 5.3, ρM

((zn(f)), (zn(g0))

)< ε/2 whenever ‖f − g0‖L2 < δ. Due to Lemma 5.1,

we can take δ1 so that g remains in the δ-neighbourhood of g0 if (βn) varies in the open

ball of B of radius δ1 centered at (β0n). Now we find a perturbation (βn) of (β0

n) within

this ball that generates a function φ of (3.11) in Φ (cf. the proof of Theorem 1.3).

Since βn are of the form (2.5) for c := Gg, where g is uniquely determined by (βn) asexplained before Lemma 5.1, Theorem 3.6 gives unique q ∈ W−1

2 (0, 1) and h ∈ C suchthat (λn) is the spectrum of T (q,∞) and (zn(g)) is the spectrum of T (q, h).

Now we have µ0n = zn(g0) by the definition of g0 and |µ0

n − zn(g)| < ε/2 by theconstruction of g. The function φ depends continuously in L2(0, 1) on (µn) ∈ M in

view of Lemmata 5.1, 5.2, and 5.4. Since the elements (µ(k)n ) of M , with µ

(k)n = zn(g)

if n < k and µ(k)n = µ0

n otherwise, converge in M to (zn(g)) and the set Φ is openin L2(0, 2) by Proposition 5.5, we conclude that there is an index k such that the

function φ corresponding to (µ(k)n ) belongs to Φ. Theorem 3.6 now states that (λn)

and (µ(k)n ) are the spectra of the operators T (q,∞) and T (q, h) respectively, for some

q ∈ W−12 (0, 1) and h ∈ C, which completes the proof of Theorem 1.2.

Acknowledgements. The authors express their gratitude to DFG for financial sup-port of the project 436 UKR 113/84. The research of the second author was partiallysupported by the Alexander von Humboldt Foundation, which is gratefully acknowl-edged. The second and the third authors thank the Institute for Applied Mathematicsof Bonn University for the warm hospitality.

Appendix A. Auxiliary result

In this appendix, we prove one result that was used several times in Section 4.

Lemma A.1. Assume that G is an open connected domain in C, that J is an openinterval, and that ϕ : G× J → C is a function of the form

ϕ(w, t) =n∑

k=0

wkϕk(t),


where the functions ϕ0, ϕ1, . . . , ϕn are absolutely continuous on J . Assume further thatfor every t ∈ J there exists w ∈ G with ϕ(w, t) 6= 0. Then the Hausdorff dimension ofthe set

Γ(G, J) := Γ(G, J, ϕ) := z ∈ G | ∃t ∈ J ϕ(z, t) = 0is at most 1.

Proof. We start with observing that if⋃i∈I

Gi × Ji = G× J

is an open covering of G× J , then⋃i∈I

Γ(Gi, Ji) = Γ(G, J).

In particular, if for every i ∈ I the set Γ(Gi, Ji) is of Hausdorff dimension not greaterthan 1, then such is G × J [6, Ch. 1]. This follows from the fact that every opencovering of G×J contains a countable subcover and that the union of countable manysets of Hausdorff dimension ≤ 1 has Hausdorff dimension ≤ 1.

Since for every w ∈ G and t ∈ J there is k ∈ N such that

∂kϕ

∂wk(w, t) 6= 0,

the above consideration implies that it suffices to prove the statement under the fol-lowing assumptions (Ck), k = 1, . . . , n:

(Ck) ∀w ∈ G ∀t ∈ J it holds that∂kϕ

∂wk(w, t) 6= 0.

Assuming that the statement of the lemma is already proved under (Ck) for somek ∈ N, we show that it then holds under (Ck+1). Indeed, if the function ϕ verifies(Ck+1), then ϕ1 := ∂ϕ/∂w verifies (Ck) and thus the set

Γ1 := Γ(G, J, ϕ1)

has Hausdorff dimension ≤ 1. It is clear that for every pair (ζ, τ), with ζ ∈ G1 := G\Γ1

and τ ∈ J , there is an open disc Kζ ⊂ G centered at ζ and an interval Jτ ⊂ J centeredat τ such that

∀λ ∈ Gζ ∀t ∈ Jτ∂ϕ

∂w(λ, t) 6= 0.

By the assumption of induction the sets

Γζ,τ := Γ(Kζ , Jτ ), ζ ∈ G1, τ ∈ Jhave Hausdorff dimension ≤ 1. Since it is always possible to choose an at most count-able sequence of points (ζi, τi) ∈ G1 × J such that⋃

(ζ,τ)∈G1×J

Kζ × Jτ =⋃n

Kζn × Jτn ,

it follows that the set

Γ(G, J) = Γ1

⋃(⋃n

Γ(Kζn , Jτn)

)is of Hausdorff dimension ≤ 1.

It therefore suffices to prove the lemma under condition (C1). Without loss of gen-erality we shall assume that G is a sufficiently small circle centered at w0 and J is a


sufficiently small interval with centre at t0 so that for all (w, t) ∈ G× J the followinginequality holds:

(A.1) |ϕ(w, t)− ϕ(w0, t0)| ≤1

2|ϕ(w0, t0)|,

where ϕ := ∂ϕ/∂w. Moreover, in view of the implicit function theorem we can assumethat G and J are such that

Γ(G, J) = g(J)

for some continuous function g : J → G. We shall show below that the functiong is absolutely continuous. Since the range g(J) is a rectifiable curve in C, by [6,Lemma 3.5] the Hausdorff dimension of g(J) is one.

It remains to prove absolute continuity of g. To this end we note that (A.1) yieldsthe inequality

(A.2) |ϕ(w1, t)− ϕ(w2, t)| =∣∣∣∫ w2

w1

ϕ(w, t) dw∣∣∣ ≥ 1

2|ϕ(w0, t0)||w1 − w2|

for arbitrary w1, w2 ∈ G and t ∈ J . On the other hand, there exists a constant C suchthat, for all w ∈ G and all t1, t2 ∈ J ,

(A.3) |ϕ(w, t1)− ϕ(w, t2)| ≤ Cn∑

k=0

|ϕk(t1)− ϕk(t2)|.

Since, by definition,ϕ(g(t1), t1) = ϕ(g(t2), t2) = 0,

relations (A.2) and (A.3) yield

1

2|ϕ(w0, t0)||g(t1)− g(t2)| ≤ |ϕ(g(t1), t1)− ϕ(g(t2), t1)|

= |ϕ(g(t2), t1)− ϕ(g(t2), t2)| ≤ Cn∑

k=0

|ϕk(t1)− ϕk(t2)|,

and hence g is absolutely continuous on J . The proof is complete.

References

[1] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Me-chanics, 2nd ed. (with an appendix by P. Exner), AMS Chelsea Publishing, Providence, RI, 2005.

[2] S. Albeverio, R. Hryniv, and Ya. Mykytyuk Inverse spectral problems for Sturm–Liouville oper-ators in impedance form, J. Funct. Anal. 222(2005), no. 1, 143–177.

[3] S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators. Solvable Schro-dinger Type Operators, Cambridge University Press, Cambridge, 2000.

[4] C. F. Coleman and J. R. McLaughlin, Solution of the Inverse Spectral Problem for an Impedancewith Integrable Derivative, I, Comm. Pure Appl. Math. 46(1993), 145–184; II, Comm. Pure Appl.Math. 46(1993), 185–212.

[5] D. E. Dahlberg and E. Trubowitz, The inverse Sturm–Liouville problem. III, Comm. Pure Appl.Math. 37 (1984), no. 2, 255–267.

[6] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, vol. 85, Cam-bridge University Press, Cambridge, 1986.

[7] I. M. Gelfand and B. M. Levitan, On determination of a differential equation by its spectralfunction, Izv. AN USSR, Ser. Mat. 15(1951), no. 4, 309–360.

[8] I. Gohberg, S. Goldberg, and M. Kaashoek, Classes of linear operators. Vol. II, Oper. TheoryAdv. Appl. 63, Birkhauser Verlag, Basel, 1993.

[9] I. Gohberg and M. Krein, Theory of Volterra Operators in Hilbert Space and its Applications,Nauka Publ., Moscow, 1967 (Russian); Engl. transl.: Amer. Math. Soc. Transl. Math. Mono-graphs 24, Amer. Math. Soc., Providence, RI, 1970.


[10] X. He and H. Volkmer, Riesz bases of solutions of Sturm-Liouville equations, J. Fourier Anal.Appl. 7 (2001), no. 3, 297–307.

[11] R. Hryniv and Ya. Mykytyuk, Inverse spectral problems for Sturm–Liouville operators withsingular potentials, Inverse Problems 19 (2003), 665–684.

[12] R. O. Hryniv and Ya. V. Mykytyuk, Inverse spectral problems for Sturm–Liouville operatorswith singular potentials, II. Reconstruction by two spectra, in Functional Analysis and its Ap-plications, V. Kadets and W. Zelazko, eds., North-Holland Mathematics Studies, 197, 97–114,North-Holland Publishing Co., Amsterdam, 2004.

[13] R. O. Hryniv and Ya. V. Mykytyuk, Transformation operators for Sturm–Liouville operators withsingular potentials, Math. Phys. Anal. Geom. 7 (2004), 119–149.

[14] R. Hryniv and Ya. Mykytyuk, On zeros of some entire functions, arXiv preprinthttp://arxiv.org/abs/math.CV/0410420.

[15] E. L. Isaacson, H. P. McKean, and E Trubowitz, The inverse Sturm–Liouville problem. II, Comm.Pure Appl. Math. 37 (1984), no. 1, 1–11.

[16] E. L. Isaacson and E. Trubowitz, The inverse Sturm–Liouville problem, Comm. Pure Appl. Math.36 (1983), no. 6, 767–783.

[17] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Grundlehren der MathematischenWissenschaften, vol. 132, Springer-Verlag, Berlin-New York, 1976.

[18] M. G. Krein, On the inverse problem for a non-homogeneous cord, Dokl. Akad. Nauk SSSR82(1951), 669–672.

[19] B. M. Levitan, Inverse Sturm-Liouville Problems, Nauka Publ., Moscow, 1984 (in Russian); Engl.transl.: VNU Science Press, Utrecht, 1987.

[20] V. A. Marchenko, Some questions of the theory of second order differential operators, USSRDoklady 72(1950), no. 3, 457–460.

[21] V. A. Marchenko, Sturm-Liouville Operators and Their Applications, Naukova Dumka Publ.,Kiev, 1977 (in Russian); Engl. transl.: Birkhauser Verlag, Basel, 1986.

[22] Ya. V. Mykytyuk, Factorization of Fredholm operators, Mat. Stud. 20 (2003), no. 2, 185–199(Ukrainian).

[23] Ya. V. Mykytyuk, Factorization of Fredholm operators in operator algebras, Mat. Stud. 21 (2004),no. 1, 87–97 (Ukrainian).

[24] J. Poschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Orlando, Florida, 1987(Pure and Applied Math., Vol. 130).

[25] W. Rundell and P. E. Sacks, The reconstruction of Sturm-Liouville operators, Inverse Probl. 8(1992), 457–482.

[26] J.-J. Sansuc, V. Tkachenko, Spectral parametrization of non-selfadjoint Hill’s operators, J. Dif-ferential Equations 125 (1996), no. 2, 366–384.

[27] A. M. Savchuk and A. A. Shkalikov, Sturm–Liouville operators with distributional potentials,Trudy Mosk. Matem Ob-va, 64 (2003), 159–212 (in Russian); Engl. transl.: Trans. MoscowMath. Soc. 2003, 143–192.

[28] A. M. Savchuk and A. A. Shkalikov, Inverse problem for Sturm–Liouville operators with distribu-tional potentials: Reconstruction from two spectra, Russian J. Math. Phys. 12 (2005), 507–514.

[29] V. Tkachenko, Non-selfadjoint Sturm-Liouville operators with multiple spectra, in Interpolationtheory, systems theory and related topics, (Tel Aviv/Rehovot, 1999), 403–414, Oper. Theory Adv.Appl., 134, Birkhauser, Basel, 2002.

Bestellungen nimmt entgegen: Institut für Angewandte Mathematik der Universität Bonn Sonderforschungsbereich 611 Wegelerstr. 6

D - 53115 Bonn Telefon: 0228/73 3411 Telefax: 0228/73 7864 E-mail: [email protected] Homepage: http://www.iam.uni-bonn.de/sfb611/

Verzeichnis der erschienenen Preprints ab No. 235

235. Grunewald, Natalie; Otto, Felix; Reznikoff, Maria G.; Villani, Cédric: A Two-Scale Proof of a Logarithmic Sobolev Inequality 236. Albeverio, Sergio; Hryniv, Rostyslav; Mykytyuk, Yaroslav: Inverse Spectral Problems for Coupled Oscillating Systems: Reconstruction by Three Spectra

237. Albeverio, Sergio; Cebulla, Christof: Müntz Formula and Zero Free Regions for the Riemann Zeta Function 238. Marinelli, Carlo: The Stochastic Goodwill Problem; erscheint in: European Journal of Operational Research 239. Albeverio, Sergio; Lütkebohmert, Eva: The Price of a European Call Option in a Black- Scholes-Merton Model is given by an Explicit Summable Asymptotic Series 240. Albeverio, Sergio; Lütkebohmert, Eva: Asian Option Pricing in a Lévy Black-Scholes Setting 241. Albeverio, Sergio; Yoshida, Minoru W.: Multiple Stochastic Integrals Construction of non-Gaussian Reflection Positive Generalized Random Fields 242. Albeverio, Sergio; Bernabei, M. Simonetta; Röckner, Michael; Yoshida, Minoru W.:

Homogenization of Diffusions on the Lattice Zd with Periodic Drift Coefficients;

Application to Logarithmic Sobolev Inequality 243. Albeverio, Sergio; Konstantinov, Alexei Yu.: On the Absolutely Continuous Spectrum of Block Operator Matrices 244. Albeverio, Sergio; Liang, Song: A Remark on the Nonequivalence of the Time-Zero \phi_3^4-Measure with the Free Field Measure 245. Albeverio, Sergio; Liang, Song; Zegarlinski, Boguslav: A Remark on the Integration by Parts Formula for the \phi_3^4-Quantum Field Model 246. Grün, Günther; Mecke, Klaus; Rauscher, Markus: Thin-Film Flow Influenced by Thermal Noise 247. Albeverio, Sergio; Liang, Song: A Note on the Renormalized Square of Free Quantum Fields in Space-Time Dimension d > 4

248. Griebel, Michael: Sparse Grids and Related Approximation Schemes for Higher Dimensional Problems; erscheint in: Proc. Foundations of Computational Mathematics 2005

249. Albeverio, Sergio; Jushenko, Ekaterina; Proskurin, Daniil; Samoilenko, Yurii: *-Wildness of

Some Classes of C*-Algebras

250. Albeverio, Sergio; Ostrovskyi, Vasyl; Samoilenko, Yurii: On *-Representations of a Certain

Class of Algebras Related to Extended Dynkin Graphs

251. Albeverio, Sergio; Ostrovskyi, Vasyl; Samoilenko, Yurii: On Functions on Graphs and

Representations of a Certain Class of *-Algebras

252. Holtz, Markus; Kunoth, Angela: B-Spline-Based Monotone Multigrid Methods 253. Albeverio, Sergio; Kuzhel, Sergej; Nizhnik, Leonid: Singularly Perturbed Self-Adjoint Operators in Scales of Hilbert Spaces

254. Albeverio, Sergio; Hryniv, Rostyslav; Mykytyuk, Yaroslav: On Spectra of Non-Self-Adjoint Sturm-Liouville Operators

On Spectra of Non-Self-Adjoint Sturm-Liouville Operators ...

Documents

Transcript of On Spectra of Non-Self-Adjoint Sturm-Liouville Operators ...