Eigenvalues of Schrödinger operators on finite and infinite intervals

Abstract

We consider a Sturm-Liouville operator with an integrable potential 𝑞 on the unit interval 𝐼 = [0, 1]. We consider a Schrödinger operator with a real compactly supported potential on the half line and on the line, where this potential coincides with 𝑞 on the unit interval and vanishes outside 𝐼. We determine the relationships between eigenvalues of such operators and obtain estimates of eigenvalues in terms of potentials.

Inverse problems for Jacobi operators with finitely supported perturbations

“…Remark. In the case of Schrodinger operators on the half-line it is also possible to evaluate the number and location of eigenvalues of the operator on the half-line using the eigenvalues of the operator on the finite interval, see [32]. The relations from [32] are similar to (1.20-1.23).…”

Discretization of inverse scattering on a half line

“…In our consideration we use the following crucial fact: q ∈ L 2 + iff the first eigenvalue τ 1 0 of (1.11), i.e., q ∈ L 2 + ⇔ τ 1 0, (1.13) see e.g., [16]. We fix model (unperturbed) sequences (corresponding to potential q = 0) by…”

“…Recall that µ n = µ n (q), n 1 are the Dirichlet eigenvalues of the problem (1.10) and τ n = τ n (q), n 1 are the mixed eigenvalues of the problem (1.11) on the unite interval corresponding to q and they satisfy τ 1 < µ 1 < τ 2 < µ 2 < .... We need to recall results from [16], which will be crucial for us:…”

“…1 can be any odd number for a Jost function ψ(k, q) with a specific potential q, see [16]. We describe potentials q * corresponding the Jost function ψ * (k) = ψ(k) k−k * k+k * .…”

“…We show (4. We need the following fact from [16]: if k * ∈ (k j , k j+1 ), then k 2 * ∈ (µ j , µ j+1 ), which yields (−1) j ϕ(1, k * ) > 0. Here (µ n ) n∈N is an increasing sequence of eigenvalues of the problem −y ′′ + qy = λy, y(0) = y(1) = 0 on the unit interval [0, 1].…”