Existence of Bound States for Layers Built Over Hypersurfaces of Euclidean Space

TL;DR

This paper investigates the existence of bound states in quantum layers constructed over hypersurfaces in Euclidean space, establishing conditions related to curvature decay and convexity that guarantee bound states.

Contribution

It provides new criteria for the existence of bound states in quantum layers over certain hypersurfaces, extending previous results to non-totally geodesic and convex cases.

Findings

Bound states exist over parabolic manifolds with fast-decaying second fundamental form.

Quantum layers over convex surfaces with vanishing second fundamental form at infinity have bound states.

Abstract

In this paper, we study the bound states of quantum layers. We prove that for the quantum layer built over a parabolic manifold which is not totally geodesic, if the second fundamantal form decays sufficiently fast, then the bound states exist. In the 2d case, we prove that the quantum layer over a convex surface whose second fundamental form tends to zero at infinity must have bound states.