Schroedinger operators with singular interactions: a model of tunneling resonances

Abstract

We discuss a generalized Schrödinger operator in $L^2({\mathbb{R}}^d),d=2,3$, with an attractive singular interaction supported by a $(d-1)$-dimensional hyperplane and a finite family of points. It can be regarded as a model of a leaky quantum wire and a family of quantum dots if $d=2$, or surface waves in presence of a finite number of impurities if $d=3$. We analyze the discrete spectrum, and furthermore, we show that the resonance problem in this setting can be explicitly solved; by Birman-Schwinger method it is cast into a form similar to the Friedrichs model.

LaTeX2e, 34 pages

References in corpus (7)

• Asymptotics of eigenvalues of the Schroedinger operator with a strong delta-interaction on a loop
• Leaky quantum graphs: approximations by point interaction Hamiltonians
• Boundary triples and Weyl functions for singular perturbations of self-adjoint operators
• Curvature-induced bound states for a $\delta$ interaction supported by a curve in ${\mathbb{R}}^3$
• Spectra of soft ring graphs
• Eigenvalue asymptotics for the Schrödinger operator with a $\delta$-interaction on a punctured surface
• On Eigenvalues Problem for Self-adjoint Operators with Singular Perturbations

Cited by in corpus (7)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Leaky Quantum Graphs: A Review
• Scattering by local deformations of a straight leaky wire
• A straight waveguide with a wire inducing resonances
• Fermi's golden rule in tunneling models with quantum waveguides perturbed by Kato class measures
• Straight quantum layer with impurities inducing resonances
• Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators