A discrete Hardy-Laptev-Weidl type inequality and associated Schrödinger-type operators

• Main Authors: Evans, William Desmond, Schmidt, Karl Michael
• Format: Article in Journal/Newspaper
• Language: unknown
• Published: Springer Verlag 2009
• Subjects: QA Mathematics; Satis; laptev
• Online Access: https://orca.cardiff.ac.uk/id/eprint/13823/; http://www.mat.ucm.es/serv/revmat/vol22-1e.html

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 Schrödingertype operator in the Aharonov-Bohm feld has essential spectrum concentrated at 0, and the multiplicity of its lower spectrum satis�es a CLR-type inequality.