Resolvent expansions for the Schrödinger operator on the discrete half-line

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 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 Schrodinger operator on the discrete half-line 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. 27, 1550002 (2015)], implementing the expansion scheme of Jensen-Nenciu [Rev. Math. Phys. 13(6), 717–754 (2001) and Rev. Math. Phys. 16(5), 675–677 (2004)] in its full generality.