A class of inverse problems for a quadratic pencil of Sturm-Liouville operators

An inverse nodal problem for differential pencils with complex spectral parameter dependent boundary conditions

“…Some aspects of this type problems were studied by many authors (see [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38]). Such problems play an important role in mathematics and have many applications in natural sciences and engineering.…”

Integral Representations for the Solutions of the Generalized Schroedinger Equation in a Finite Interval

“…Various inverse scattering problems for the Schroedinger equation with an energy dependent potential on the half line and full line were investigated in [24]- [30]. Direct and inverse spectral problems in a finite interval for the Equation (1) in the case ( ) 1 x ρ = were first investigated in [17] [18].…”

Reconstruction of energy-dependent Sturm-Liouville equations from two spectra. II

“…q = r ′ with a real-valued r ∈ L 2 (0, 1). We consider this equation under two types of boundary conditions: the Dirichlet ones y(0) = y(1) = 0 (2) and the so-called mixed conditions y(0) = y [1] (1) + Hy(1) = 0, where H ∈ R is some constant and y [1] := y ′ − ry is a quasi-derivative of the function y used in the regularization procedure due to Savchuk and Shkalikov (see [19,20] and the next section for details). Since primitive of q is defined only up to an additive constant, by replacing r with r − H we reduce the above mixed boundary conditions to the following ones:…”

“…In what follows, we shall denote by L D (p, r) and L M (p, r) the spectral problems (1), (2) and (1), (3) respectively. Our main aim in this paper is to solve the inverse problem of reconstructing the potentials p and r given the spectra of L D (p, r) and L M (p, r).…”

“…The inverse spectral problems for (1) with p ∈ W 1 2 (0, 1) and q ∈ L 2 (0, 1) and with Robin boundary conditions were discussed by M. Gasymov and G. Guseinov in their short paper [3] of 1981 containing no proofs. Such problems were also considered in [2,4,13,14,22], but only Borg-type uniqueness results were obtained therein.…”