Equivalence of inverse Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter

Abstract

Three inverse problems for a Sturm-Liouville boundary value problem −y + qy = λy, y(0) cos α = y (0) sin α and y (1) = f (λ)y(1) are considered for rational f . It is shown that the Weyl m-function uniquely determines α, f , and q, and is in turn uniquely determined by either two spectra from different values of α or by the Prüfer angle. For this it is necessary to produce direct results, of independent interest, on asymptotics and oscillation.

Transformation of Sturm–liouville Problems With Decreasing Affine Boundary Conditions

“…The first sentence follows from [16], as does the fact that all but finitely many eigenvalues are real and simple. The asymptotic development in (i) is derived in [9]. …”

Recovering singular Sturm-Liouville differential pencils from spectral data

“…Let M(λ) be the Weyl function for L . We find P 00 using (14), and construct P k (λ) by (3). We choose a model pairL = L(q, U ) such that P k (λ) =P k (λ), k = 0, 1, and arbitrary in the rest.…”

“…Inverse problems for differential operators with boundary conditions dependent on the spectral parameter are more difficult to investigate, and nowadays there are only a number of papers in this direction (see [11][12][13][14][15][16][17]). In particular, [11][12][13][14][15][16] study such problems on a finite interval. Inverse spectral problems for the non-self-adjoint Sturm-Liouville pencil (1) and (2) on the half-line were considered in [17], where recovering L from the Weyl function was studied.…”

Inverse spectral problems for Dirac operator with eigenvalue dependent boundary and jump conditions

“…[1], [3], [8], [9], [12] and [19] are examples of works with boundary conditions that depend linearly on eigenvalue parameters. Boundary conditions depending nonlinearly on spectral parameter were also considered in [4][5][6], [18], [21][22][23]. These works are relevant to Hilbert and Pontryagin space formulations, expansion theory, direct and inverse spectral theory.…”