On the discrete spectrum of quantum layers

TL;DR

This paper investigates the conditions under which a quantum particle in a curved, non-compact surface layer has bound states, linking geometric properties like curvature and parabolicity to the existence of discrete spectrum.

Contribution

It establishes new criteria connecting surface curvature, parabolicity, and layer thickness to the presence of discrete spectrum in quantum layers.

Findings

Discrete spectrum exists if Gauss curvature is integrable and surface is weakly κ-parabolic.

Non-positive total Gauss curvature guarantees discrete spectrum.

Sufficiently thin layers ensure bound states for parabolic surfaces.

Abstract

Consider a quantum particle trapped between a curved layer of constant width built over a complete, non-compact, $\mathcal C^2$ smooth surface embedded in $\mathbb{R}^3$. We assume that the surface is asymptotically flat in the sense that the second fundamental form vanishes at infinity, and that the surface is not totally geodesic. This geometric setting is known as a quantum layer. We consider the quantum particle to be governed by the Dirichlet Laplacian as Hamiltonian. Our work concerns the existence of bound states with energy beneath the essential spectrum, which implies the existence of discrete spectrum. We first prove that if the Gauss curvature is integrable, and the surface is weakly $\kappa$-parabolic, then the discrete spectrum is non-empty. This result implies that if the total Gauss curvature is non-positive, then the discrete spectrum is non-empty. We next prove that if the Gauss curvature is non-negative, then the discrete spectrum is non-empty. Finally, we prove that if the surface is parabolic, then the discrete spectrum is non-empty if the layer is sufficiently thin.