Inverse Spectral Problems for Sturm-Liouville Operators with Singular Potentials, II. Reconstruction by Two Spectra*

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Abstract

We solve the inverse spectral problem of recovering the singular po of Sturm-Liouville operators by two spectra. The reconstruction algorithm is presented and necessary and sufficient conditions on two sequences to be spectral data for Sturm-Liouville operators under consideration are given.

Introduction

Suppose that q is a real-valued distribution from the space W21(0,1). A Sturm-Liouville operator T in a Hilbert space :=L2(0,1) corresponding formally to the differential expressiond2dx2+qcan be defined as follows [18]. We take a distributional primitive σ of q, putlσ(u):=(uσ)σufor absolutely continuous functions u whose quasi-derivative u[1]:=uσu is again absolutely continuous, and then setTu:=lσ(u)for suitable u (see Section 2 for rigorous definitions). When considered on this natural domain subject to the boundary conditionsu[1](0)Hu(0)=0,u[1](1)+hu(1)=0,H,h{},the operator T = T(σ, H, h) becomes 17., 18. selfadjoint, bounded below, and possesses a simple discrete spectrum accumulating at +∞. (If H = ∞ or h = ∞, then the corresponding boundary condition is understood as a Dirichlet one.)

We observe that among the singular potentials included in this scheme there are, e.g., the Dirac δ-potentials and Coulomb 1/x-like potentials that have been widely used in quantum mechanics to model particle interactions of various types (see, e.g., 1., 2.). It is also well known that singular potentials of this kind usually do not produce a single Sturm-Liouville operator. Still for many reasons T(σ, H, h) can be regarded as a natural operator associated with differential expression (1.1) and boundary conditions (1.2). For instance, lσ(u)=u+qu in the sense of distributions and hence for regular (i.e., locally integrable) potentials q the above definition of T(σ, H, h) coincides with the classical one. Also T(σ, H, h) depends continuously in the uniform resolvent sense on σ, which allows us to regard T(σ, H, h) for singular q = σ′ as a limit of regular Sturm-Liouville operators.

Sturm-Liouville and Schrödinger operators with distributional potentials from the space W21(0,1) and W2,unif1() respectively have been shown to possess many properties similar to those for operators with regular potentials, see, e.g., [9,10,17,18]. In particular, just as in the regular case, the potential q = σ′ and boundary conditions (1.2) of the Sturm-Liouville operator T(σ, H, h) can be recovered via the corresponding spectral data, the sequences of eigenvalues and so-called norming constants [11].

The main aim of the present paper is to treat the following inverse spectral problem. Suppose that {λn2} is the spectrum of the operator T(σ, H, h1) and {μn2} is the spectrum of the operator T(σ, H, h2) with h2h1. Is it possible to recover σ, H, h1, and h2 from the two given spectra? More generally, the task is to find necessary and sufficient conditions for two sequences (λn2) and (μn2) so that they could serve as spectra of Sturm-Liouville operators on (0,1) with some potential q=σW21(0,1) and boundary conditions (1.2) for two different values of h and, then, to present an algorithm recovering these two operators (i.e., an algorithm determining σ and the boundary conditions).

For the case of a regular (i.e., locally integrable) potential q the above problem was treated by Levitan and Gasymov 14., 15. when h1 and h2 are finite and by Marchenko [16, Ch. 3.4] when one of h1, h2 is infinite. Earlier, BORG [4] proved that two such spectra determine the regular potential uniquely.

Observe that T(σ+h,Hh,h+h)=T(σ,H,h) for any h, so that it is impossible to recover σ, H, h1, and h2 uniquely; however, we shall show that the potential q=σ and the very operators T(σ, H, h1) and T(σ, H, h2) are determined uniquely by the two spectra. The reconstruction procedure consists in reduction of the problem to recovering the potential q = σ′ and the boundary conditions based on the spectrum and the sequence of so-called norming constants. The latter problem has been completely solved in our paper [11], and this allows us to give a complete description of the set of spectral data and to develop a reconstruction algorithm for the inverse spectral problem under consideration.

The organization of the paper is as follows. In the next two sections we restrict ourselves to the case where H = ∞ and h1, h2 ∈ ℝ. In Section 2 refined eigenvalue asymptotics is found and some other necessary conditions on spectral data are established. These results are then used in Section 3 to determine the set of norming constants and completely solve the inverse spectral problem. The case where one of h1 and h2 is infinite is treated in Section 4, and in Section 5 we comment on the changes to be made if H is finite. Finally, Appendix A contains some facts about Riesz bases of sines and cosines in L2(0, 1) that are frequently used throughout the paper.

Section snippets

Spectral asymptotics

In this section (and until Section 4) we shall consider the case of the Dirichlet boundary condition at the point x = 0 and the boundary conditions of the third type at the point x = 1 (i. e., the case where H = ∞ and h1, h2 ∈ ℝ). We start with the precise definition of the Sturm-Liouville operators under consideration.

Suppose that q is a real-valued distribution from the class W21(0,1) and σ is any of its real-valued distributional primitives. For h ∈ ℝ we denote by Tσ,h=T(σ,,h) the operator in

Reduction to the inverse spectral problem by one spectrum and norming constants

Suppose that σ is real-valued and that {λn2} and {μn2} are eigenvalues of the operators Tσ, h1 and Tσ, h2 respectively introduced in the previous section. We shall show how the problem of recovering σ, h1, and h2 based on these spectra can be reduced to the problem of recovering σ and h1 based on the spectrum {λn2} of Tσ, h1 and the so-called norming constants {αn} defined below. This latter problem for the class of Sturm-Liouville operators with singular potentials from W21(0,1) is

Reconstruction by Dirichlet and Dirichlet-Neumann spectra

The analysis of the previous two sections does not cover the case where one of h1, h2 is infinite. In this case the other number may be taken 0 without loss of generality (recall that Tσ,h=Tσ+h,0), i.e., the boundary conditions under considerations become Dirichlet and Dirichlet-Neumann ones.

Suppose therefore that σH is real valued and that (λn2) and (μn2) are spectra of the operators Tσ,0 and Tσ,∞ respectively; without loss of generality we assume that λn and μn are positive and strictly

The case of the Neumann boundary condition at x=0

The analysis of the previous sections can easily be modified to cover the case H = 0, i.e., the Neumann boundary condition u[1](0) = 0. As before, we use the two spectra to determine the sequence of norming constants and then apply the reconstruction procedure of [11] to find the corresponding Sturm-Liouville operators. Also the necessary and sufficient conditions on the two spectra can be established. We formulate the corresponding results in the following two theorems.

Appendix Appendix A. Riesz bases

In this appendix we gather some well known facts about Riesz bases of sines and cosines (see, e.g., 6., 7., 8. and the references therein for a detailed exposition of this topic).

Recall that a sequence (en)1 in a Hilbert space is a Riesz basis if and only if any element e has a unique expansion e=n=1cnen with (cn)2. If (en) is a Riesz basis, then in the above expansion the Fourier coefficients cn are given by (cn)=(e,en), where (en)1 is a system biorthogonal to (en), i.e., a

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The work was partially supported by Ukrainian Foundation for Basic Research DFFD under grant No. 01.07/00172. R. H. acknowledges support of the Alexander von Humboldt Foundation.

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