On the guided states of 3D biperiodic Schr\"odinger operators

TL;DR

This paper studies guided states of a 3D Schrödinger operator with a periodic potential, showing that under certain conditions, these states are characterized by specific quasi-momenta and decay rapidly in the non-periodic direction.

Contribution

It provides a characterization of guided states for a 3D periodic Schrödinger operator, including their quasi-momenta and decay properties, under small and fast-decaying potentials.

Findings

Guided states are associated with a one-dimensional real analytic submanifold of the Brillouin zone.

These states decay faster than any polynomial in the non-periodic direction.

The characterization holds for sufficiently small and rapidly decreasing potentials.

Abstract

We consider the Laplacian operator H_0 perturbed by a non-positive potential $V$, which is periodic in two directions, and decays in the remaining one. We are interested in the characterization and decay properties of the guided states, defined as the eigenfunctions of the reduced operators in the Bloch-Floquet-Gelfand transform of H_0+V in the periodic variables. If V is sufficiently small and decreases fast enough in the infinite direction, we prove that, generically, these guided states are characterized by quasi-momenta belonging to some one-dimensional compact real analytic submanifold of the Brillouin zone. Moreover they decay faster than any polynomial function in the infinite direction.