Asymptotic eigenfunctions for a class of difference operators

Article type: Research Article

Authors: Klein, Markus | Rosenberger, Elke

Affiliations: Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany. E-mails: mklein@math.uni-potsdam.de; erosen@uni-potsdam.de

Abstract: We analyze a general class of difference operators Hε=Tε+Vε on ℓ2((εZ)d), where Vε is a one-well potential and ε is a small parameter. We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of Hε. These are obtained from eigenfunctions or quasimodes for the operator Hε, acting on L2(Rd), via restriction to the lattice (εZ)d.

Keywords: difference operator, tunneling, WKB-expansion, quasimodes

DOI: 10.3233/ASY-2010-1025

Journal: Asymptotic Analysis, vol. 73, no. 1-2, pp. 1-36, 2011