Spectral analysis on ruled surfaces with combined Dirichlet and Neumann boundary conditions

TL;DR

This paper investigates the spectral properties of the Laplacian on unbounded ruled surfaces with mixed boundary conditions, revealing how geometry influences the existence of discrete spectra and eigenvalue asymptotics.

Contribution

It provides new insights into the spectral behavior of Laplacians with combined boundary conditions on ruled surfaces, including eigenvalue asymptotics and effects of geometric twisting and bending.

Findings

Existence and absence conditions for discrete spectrum influenced by geometry.

Asymptotic behavior of eigenvalues for thin ruled surfaces.

Comparison of spectral properties under different geometric transformations.

Abstract

Let $\Omega$ be an unbounded two dimensional strip on a ruled surface in $\mathbb{R}^d$, $d\geq2$. Consider the Laplacian operator in $\Omega$ with Dirichlet and Neumann boundary conditions on opposite sides of $\Omega$. We prove some results on the existence and absence of the discrete spectrum of the operator; which are influenced by the twisted and bent effects of $\Omega$. Provided that $\Omega$ is thin enough, we show an asymptotic behavior of the eigenvalues. The interest in those considerations lies on the difference from the purely Dirichlet case. Finally, we perform an appropriate dilatation in $\Omega$ and we compare the results.