Inverse indefinite Sturm–Liouville problems with three spectra

Abstract

We prove that the potential q(x) of an indefinite Sturm-Liouville problem on the closed interval [a, b] with the indefinite weight function w(x) can be determined uniquely by three spectra, which are generated by the indefinite problem defined on [a, b] and two right-definite problems defined on [a, 0] and [0, b], where point 0 lies in (a, b) and is the turning point of the weight function w(x).

On solvability of three spectra problem

“…While uniqueness of solution for the so‐called three spectra problem was considered in many publications (see ), the problem of existence of solution has been treated less intensively. Here we should mention and .…”

On a generalization of the three spectral inverse problem

“…Uniqueness of solution for the three spectral problem was considered in many publications (see [8], [2], [3], [4], [7], [23], [5], [6]), the problem of existence of its solution has been treated in [21], [18] and [11]. Let us describe the three spectral problem.…”