Inverse spectral problems for Sturm Liouville operators with singular potentials

Abstract

Abstract. The inverse spectral problem is solved for the class of Sturm-Liouville operators with singular real-valued potentials from the space W −1 2 (0, 1). The potential is recovered via the eigenvalues and the corresponding norming constants. The reconstruction algorithm is presented and its stability proved. Also, the set of all possible spectral data is explicitly described and the isospectral sets are characterized.

Inverse Spectral Problems for Sturm–liouville Operators With Singular Potentials. Iv. Potentials in the Sobolev Space Scale

“…We note that under properties (B1) and (B2) with s = 0 the operator I + F 1 is uniformly positive in L 2 (0, 1), and the same is true of I + F 2 if (C1) and (C2) hold with s = 0 (see the details in [9]). Hence both GLM equations of interest possess unique solutions.…”

“…It was proved in [9] that any pair of sequences (µ n ) and (β n ) with µ 1 < µ 2 < · · · obeying (A2 ) and positive β n satisfying (1.7) form the spectral data of the operator T (q, h, ∞) for unique q ∈ L 2 (0, 1) and h ∈ R; moreover, the mapping between the space of the operators T (q, h, ∞) and the set of their spectral data becomes a real analytic isomorphism in suitable topologies. Similar statements hold for the case of Dirichlet boundary conditions, i.e.…”

“…In other words, the inverse spectral problem of recovering the potential from two spectra is uniquely soluble in the class of Sturm-Liouville operators with singular potentials from W −1 2 (0, 1). The papers [9] and [11] give the corresponding reconstruction algorithm and thus extend the classical inverse Sturm-Liouville theory.…”

“…Φ and Ψ are entire functions of order 1 2 and hence can be represented by their canonical Hadamard products [11], namely,…”

“…This method was suggested by Gel'fand and Levitan in [6] for the case of regular (i.e. locally integrable) potentials and was further developed in [9] to cover singular potentials from W −1 As in § 1, we denote by λ 1 < λ 2 < · · · and α 1 , α 2 , . .…”