On spectra of non-self-adjoint Sturm–Liouville operators

Abstract

The spectra of non-self-adjoint Sturm-Liouville operators with distributional potentials belonging to the space W −1 2 (0, 1) are studied. In particular, it is shown that any sequence of complex numbers obeying a specified asymptotics coincides with the spectrum of some non-self-adjoint SturmLiouville operator of the class under consideration. The inverse spectral problem of reconstructing an operator from two spectra or from one spectrum and suitably defined norming constants is also solved, and a complete description of the spectral data for the operators considered is given. (2000). Primary 34A55; Secondary 34B24, 47A56, 47E05. Mathematics Subject Classification

Inverse spectral theory for Sturm-Liouville operators with distributional potentials

“…Within the limited scope of this paper it is impossible to provide an exhaustive list of all pertinent references. Hence, we can only limit ourselves listing a few classical results, such as, [12], [13], [53], [54], [55,Chapter 3], [56], [59], [60,Section 3.4], and some of the more recent ones in connection with distributional potential coefficients, such as, [2], [39], [40], [42], [45], [46], [70]- [73] (some of these references also derive reconstruction algorithms for the potential term in Schrödinger operators); the interested reader will find many more references in these sources.…”

“…Their results include the reconstruction algorithm and its stability, and a description of isospectral sets. The case of the two spectra inverse problem (fixing the boundary condition at one end) is also discussed in [42] and continued in [2], [39], [45], [46]. In addition, Robin-type boundary conditions and a Hochstadt-Lieberman-type inverse spectral result, where the knowledge of the set of norming constants is replaced by knowledge of the potential over the interval (0, 1/2), and again a reconstruction algorithm is provided in [43]; transformation operators associated with Robin boundary conditions are treated in [44].…”

Local solvability and stability of the inverse problem for the non-self-adjoint Sturm–Liouville operator

“…Buterin [13] has proved the uniqueness theorem for this inverse problem and obtained a constructive algorithm for its solution based on the method of spectral mappings [8,14]. The question of GSD characterization for Sturm-Liouville operators with complex-valued potentials was investigated in [15,16]. However, necessary and sufficient conditions on GSD from [15,16] require solvability of some main equations.…”

An inverse problem for the non-self-adjoint matrix Sturm-Liouville operator

“…Buterin [6] has proved the uniqueness theorem for this inverse problem and obtained a constructive algorithm for its solution, based on the method of spectral mappings [4,7]. The question of GSD characterization for Sturm-Liouville operators with complex-valued potentials was investigated in [8,9]. However, necessary and sufficient conditions on GSD from [8,9] require solvability of some main equations.…”

“…The question of GSD characterization for Sturm-Liouville operators with complex-valued potentials was investigated in [8,9]. However, necessary and sufficient conditions on GSD from [8,9] require solvability of some main equations. Those requirements are difficult to verify.…”