Spectral theory of semibounded Schrödinger operators with ${\delta }^{\prime }$-interactions

Abstract

We study spectral properties of Hamiltonians $\rH_{X,\gB,q}$ with ${\delta }^{\prime }$-point interactions on a discrete set $X={x_k}_{k=1}^{\infty }\subset {\mathbb{R}}_{+}$. %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 $-d^2/d{x}^2+q$ in $L^2(x_{k-1},x_k)$.

33 pages

References in corpus (2)

• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• On norm resolvent convergence of Schrödinger operators with ${\delta }^{\prime }$-like potentials

Cited by in corpus (5)

• One-dimensional Schroedinger operators with delta-prime-interactions on Cantor-type sets
• On the absolutely continuous spectrum of generalized indefinite strings
• On absence of bound states for weakly attractive ${\delta }^{\prime }$-interactions supported on non-closed curves in ${\mathbb{R}}^2$
• Stability of ground states for logarithmic Schrödinger equation with a ${\delta }^{\prime }$-interaction
• Generalized indefinite strings with purely discrete spectrum