Inverse Spectral Problems for Sturm-Liouville Operators with Singular Potentials, II. Reconstruction by Two Spectra*
Introduction
Suppose that q is a real-valued distribution from the space . A Sturm-Liouville operator T in a Hilbert space corresponding formally to the differential expressioncan be defined as follows [18]. We take a distributional primitive of q, putfor absolutely continuous functions u whose quasi-derivative is again absolutely continuous, and then setfor suitable u (see Section 2 for rigorous definitions). When considered on this natural domain subject to the boundary conditionsthe 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, 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 , 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 and 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 is the spectrum of the operator T(σ, H, h1) and is the spectrum of the operator T(σ, H, h2) with . 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 and so that they could serve as spectra of Sturm-Liouville operators on (0,1) with some potential 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 for any , so that it is impossible to recover σ, H, h1, and h2 uniquely; however, we shall show that the potential 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 and is any of its real-valued distributional primitives. For h ∈ ℝ we denote by the operator in ℋ
Reduction to the inverse spectral problem by one spectrum and norming constants
Suppose that σ ∈ ℋ is real-valued and that and 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 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 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 ), i.e., the boundary conditions under considerations become Dirichlet and Dirichlet-Neumann ones.
Suppose therefore that is real valued and that and 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 in a Hilbert space ℋ is a Riesz basis if and only if any element e ∈ ℋ has a unique expansion with . If (en) is a Riesz basis, then in the above expansion the Fourier coefficients cn are given by , where 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.