Full paper in PDF format:
$%W. D. Evans and K. M. Schmidt, A discrete Hardy-Laptev-Weidl-type inequality and associated Schrödinger-type operators, Rev. Mat. Complut. 22 (2009), no. 1, 75–90.%$

A Discrete Hardy-Laptev-Weidl-Type Inequality and Associated Schrödinger-Type Operators
W. Desmond EVANS and Karl Michael SCHMIDT
School of Mathematics
Cardiff University
Senghennydd Road
Cardiff CF24 4AG — Wales

EvansWD@cardiff.ac.uk  schmidtkm@cardiff.ac.uk

Received: October 8, 2007
Accepted: April 7, 2008


Although the classical Hardy inequality is valid only in the three- and higher dimensional case, Laptev and Weidl established a two-dimensional Hardy-type inequality for the magnetic gradient with an Aharonov-Bohm magnetic potential. Here we consider a discrete analogue, replacing the punctured plane with a radially exponential lattice. In addition to discrete Hardy and Sobolev inequalities, we study the spectral properties of two associated self-adjoint operators. In particular, it is shown that, for suitable potentials, the discrete Schrödinger-type operator in the Aharonov-Bohm field has essential spectrum concentrated at 0, and the multiplicity of its lower spectrum satisfies a CLR-type inequality.

Key words: discrete Schrödinger operator, Aharonov-Bohm magnetic potential.
2000 Mathematics Subject Classification:
47B37, 35P05, 39A70, 47A30, 81Q10.