# Quantum strips in higher dimensions

## TL;DR

This paper investigates the spectral properties of the Dirichlet Laplacian on unbounded strips in higher-dimensional ruled surfaces, establishing conditions for the essential and discrete spectrum, and deriving effective one-dimensional operators.

## Contribution

It introduces new spectral analysis results for Laplacians on higher-dimensional strips, including conditions for essential and discrete spectra, and develops an effective operator approach.

## Key findings

- Essential spectrum location under asymptotic flatness.
- Existence of discrete spectrum when the curve is not a geodesic and curvature conditions.
- Derivation of an effective one-dimensional Schrödinger operator.

## Abstract

We consider the Dirichlet Laplacian in unbounded strips on ruled surfaces in any space dimension. We locate the essential spectrum under the condition that the strip is asymptotically flat. If the Gauss curvature of the strip equals zero, we establish the existence of discrete spectrum under the condition that the curve along which the strip is built is not a geodesic. On the other hand, if it is a geodesic and the Gauss curvature is not identically equal to zero, we prove the existence of Hardy-type inequalities. We also derive an effective operator for thin strips, which enables one to replace the spectral problem for the Laplace-Beltrami operator on the two-dimensional surface by a one-dimensional Schroedinger operator whose potential is expressed in terms of curvatures. In the appendix, we establish a purely geometric fact about the existence of relatively parallel adapted frames for any curve under minimal regularity hypotheses.