Geometrically induced spectral effects in tubes with a mixed Dirichlet-Neumann boundary

TL;DR

This paper studies how geometric features like bending and twisting in tubes with mixed boundary conditions affect the spectral properties of the Laplacian, revealing conditions for discrete spectrum presence and the impact of twists on the spectrum.

Contribution

It provides new insights into the spectral effects of bending and twisting in tubes with mixed Dirichlet-Neumann boundary conditions, including conditions for discrete spectrum and spectrum thresholds.

Findings

Bending direction influences discrete spectrum existence.

Constant twist raises the essential spectrum threshold.

Periodic twist elevates the spectral threshold.

Abstract

We investigate spectral properties of the Laplacian in $L^2(Q)$, where $Q$ is a tubular region in $\mathbb{R}^3$ of a fixed cross section, and the boundary conditions combined a Dirichlet and a Neumann part. We analyze two complementary situations, when the tube is bent but not twisted, and secondly, it is twisted but not bent. In the first case we derive sufficient conditions for the presence and absence of the discrete spectrum showing, roughly speaking, that they depend on the direction in which the tube is bent. In the second case we show that a constant twist raises the threshold of the essential spectrum and a local slowndown of it gives rise to isolated eigenvalues. Furthermore, we prove that the spectral threshold moves up also under a sufficiently gentle periodic twist.