Absolute continuity in periodic thin tubes and strongly coupled leaky wires

Abstract

Using a perturbative argument, we show that in any finite region containing the lowest transverse eigenmode, the spectrum of a periodically curved smooth Dirichlet tube in two or three dimensions is absolutely continuous provided the tube is sufficiently thin. In a similar way we demonstrate absolute continuity at the bottom of the spectrum for generalized Schrödinger operators with a sufficiently strongly attractive $\delta$ interaction supported by a periodic curve in ${\mathbb{R}}^d,d=2,3$.

LaTeX 2e, 10 pages

References in corpus (4)

• Curvature-induced bound states for a $\delta$ interaction supported by a curve in ${\mathbb{R}}^3$
• Band gap of the Schroedinger operator with a strong delta-interaction on a periodic curve
• Persistent currents for 2D Schrödinger operator with a strong $\delta$-interaction on a loop
• Magnetic transport in a straight parabolic channel

Cited by in corpus (9)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• On the spectrum of curved quantum waveguides
• Leaky Quantum Graphs: A Review
• Quantum waveguides with small periodic perturbations: gaps and edges of Brillouin zones
• Absolute continuity of the spectrum for periodically modulated leaky wires in ${\mathbb{R}}^3$
• Spectral theory for Schrödinger operators with $\delta$-interactions supported on curves in ${\mathbb{R}}^3$
• A note on the spectrum of the Neumann Laplacian in periodic waveguides
• Influence of the bound states in the Neumann Laplacian in a thin waveguide