Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter

Abstract

We study various direct and inverse spectral problems for the onedimensional Schrödinger equation with distributional potential and boundary conditions containing the eigenvalue parameter.

A Riesz basis criterion for Schrödinger operators with boundary conditions dependent on the eigenvalue parameter

“…In this paper we continue the study of one-dimensional Schrödinger operators with distributional potentials and boundary conditions containing rational Herglotz-Nevanlinna functions of the eigenvalue parameter initiated in [7]. These operators are generated by the differential equation − y [1] ′ (x) − s(x)y [1] (x) − s 2 (x)y(x) = λy(x) (1.1) and the boundary conditions y [1] (0) y(0) = −f (λ), y [1] (π) y(π) = F (λ), (1.2) where s ∈ L 2 (0, π) is real-valued, y [1] (x) := y ′ (x) − s(x)y(x) denotes the quasiderivative of y with respect to s, and…”

“…This notion allowed us in that paper (see also [6] and [7]) to formulate various direct and inverse spectral results for boundary value problems with boundary conditions of the form (1.2), (1.3) in a unified manner, i.e. without considering separate cases as it is usually done in the literature.…”

“…Since we have already discussed the case when only one of the boundary conditions depends on the eigenvalue parameter, we now assume that both of them contain the eigenvalue parameter (i.e., min{ind f, ind F } ≥ 1). In order to completely characterize those pairs Θ = {n 1 , n 2 } for which M Θ is invertible, we need some more definitions from [5], [7]. We assign to each rational Herglotz-Nevanlinna function f of the form (1.3) two polynomials f ↑ and f ↓ by writing this function as…”

“…where ϕ n and χ n are the (necessarily linearly dependent) eigenfunctions of (1.1)-(1.2) satisfying the conditions ϕ n (0) = f ↓ (λ n ), ϕ [1] n (0) = −f ↑ (λ n ), χ n (π) = F ↓ (λ n ), and χ [1] n (π) = F ↑ (λ n ). Using the results of [6] and [7] one can deduce that these numbers have alternating signs and asymptotically behave as…”

“…As one extreme example, we have β n = (−1) n for all n when the boundary value problem is symmetric (i.e., s(x) + s(π − x) = 0 and f = F ) and hence M Θ is never invertible for n 1 and n 2 of same parity. On the other hand, using the inverse spectral theory developed in [7], one can produce a boundary value problem of the form (1.1)-(1.2) for arbitrary sequence of distinct numbers β n , as long as they satisfy the requirements of the preceding paragraph. In the case of the latter boundary value problem, M Θ is invertible for all pairs of indices n 1 and n 2 .…”