Some properties of threshold eigenstates and resonant states of discrete Schr\"odinger operators

TL;DR

This paper investigates the characteristics of threshold eigenstates and resonant states in discrete Schrödinger operators, establishing their asymptotic behavior at elliptic thresholds and proving the absence of resonances at hyperbolic thresholds.

Contribution

It provides new theoretical results on the asymptotic properties and existence of resonant states in discrete Schrödinger operators, extending understanding from continuous to discrete cases.

Findings

Resonant states at elliptic thresholds share asymptotic expansions with continuous operators.

No resonant states exist at hyperbolic thresholds.

Theoretical advancement in spectral analysis of discrete Schrödinger operators.

Abstract

In this note, we study some properties of threshold resonant states or threshold eigenfunctions for discrete Schr\"odinger operators. We mainly prove two theorems. First, we prove that resonant states at the elliptic threshold have the same asymptotic expansion as the continuous Schr\"odinger operator. Second, we prove absence of resonant states at hyperbolic thresholds