Spectral theory of semibounded Schr\"odinger operators with $\delta'$-interactions

TL;DR

This paper investigates the spectral properties of Schrödinger operators with delta-prime interactions on discrete sets, establishing classical spectral results and revealing their connection to Neumann realizations of related operators.

Contribution

It extends classical spectral analysis results to Schrödinger operators with delta-prime interactions, linking their spectral properties to Neumann realizations on subintervals.

Findings

Established analogs of classical spectral theorems for delta-prime interaction operators.

Connected spectral properties of these operators to Neumann realizations on subintervals.

Provided criteria for discreteness and stability of the essential spectrum.

Abstract

We study spectral properties of Hamiltonians $\rH_{X,\gB,q}$ with $\delta'$-point interactions on a discrete set $X={x_k}_{k=1}^\infty\subset\R_+$. %at the centers $x_n$ on the positive half line in terms of energy forms. Using the form approach, we establish analogs of some classical results on operators $\rH_q=-d^2/dx^2+q$ with locally integrable potentials $q\in L^1_{\loc}(\R_+)$. In particular, we establish analogues of the Glazman-Povzner-Wienholtz theorem, the Molchanov discreteness criterion, and the Birman theorem on stability of an essential spectrum. It turns out that in contrast to the case of Hamiltonians with $\delta$-interactions, spectral properties of operators $\rH_{X,\gB,q}$ are closely connected with those of $\rH_{X,q}^N=\oplus_{k}\rH_{q,k}^N$, where $\rH_{q,k}^N$ is the Neumann realization of $-d^2/dx^2+q$ in $L^2(x_{k-1},x_k)$.