Resolvent expansions for the Schr\"odinger operator on the discrete half-line

TL;DR

This paper provides a comprehensive analysis of the resolvent expansion at thresholds for the Schrödinger operator on the discrete half-line, crucial for understanding transport in mesoscopic systems with boundary conditions.

Contribution

It offers a complete description of threshold resonance states and resolvent expansions for the discrete half-line Schrödinger operator with general boundary conditions.

Findings

Explicit resolvent expansion coefficients linked to generalized eigenspaces

Detailed characterization of threshold resonance states

Implementation of Jensen-Nenciu expansion scheme in full generality

Abstract

Simplified models of transport in mesoscopic systems are often based on a small sample connected to a finite number of leads. The leads are often modelled using the Laplacian on the discrete half-line $\mathbb N$. Detailed studies of the transport near thresholds require detailed information on the resolvent of the Laplacian on the discrete half-line. This paper presents a complete study of threshold resonance states and resolvent expansions at a threshold for the Schr\"odinger operator on the discrete half-line $\mathbb N$ with a general boundary condition. A precise description of the expansion coefficients reveals their exact correspondence to the generalized eigenspaces, or the threshold types. The presentation of the paper is adapted from that of Ito-Jensen [Rev.\ Math.\ Phys.\ {\bf 27} (2015), 1550002 (45 pages)], implementing the expansion scheme of Jensen-Nenciu [Rev.\ Math.\ Phys.\ \textbf{13} (2001), 717--754, \textbf{16} (2004), 675--677] in its full generality.