On spectral theory for Schrödinger operators with strongly singular potentials

Abstract

Abstract. We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrödinger operators on [a, ∞), a ∈ R, with a regular finite end point a and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2 × 2 matrix-valued Herglotz functions representing the associated Weyl-Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schrödinger operators on (a, ∞) with a potential strongly singular at the end point a. We focus on situations where the potential is so singular that the associated maximally defined Schrödinger operator is selfadjoint (equivalently, the associated minimally defined Schrödinger operator is essentially self-adjoint) and hence no boundary condition is required at the finite end point a. For this case we show that the Weyl-Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schrödinger operators.

Equality of the Spectral and Dynamical Definitions of Reflection

“…By the resolvent formula for J + 0 and the analog of (1.28) (with the normalization (1.25)), 18) and similarly,…”

“…2.5 of Teschl [46], it is as easy to establish it from first principles as to manipulate the results of [46] to the form we need. Our use of Stone's formula is similar to that of Gesztesy-Zinchenko [18].…”

“…The one difference is that since δ(x) is not in L 2 , we do not have the analog of (2.23). However, 18) which implies that 19) and that suffices to define an inverse transform on L 2 (…”

Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

“…Most of the material summarized here can be found in [GZ06], [Tes09]. Let A = − d 2 dx 2 + V (x) be a self-adjoint Schrödinger operator, and let u and y be the solutions of the differential equation Aφ = zφ corresponding to Neumann and Dirichlet boundary conditions respectively.…”

“…We use the well known formula relating the ρ(ξ) and G, where G is the Green's function. Gesztesy-Zinchenko ((2.18) of [GZ06]) gives the Green's function explicitly, so we compute:…”