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

Evans, William Desmond and Schmidt, Karl Michael ORCID: https://orcid.org/0000-0002-0227-3024 2009. A discrete Hardy-Laptev-Weidl type inequality and associated Schrödinger-type operators. Revista Matemática Complutense 22 (1) , pp. 75-90.

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Abstract

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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Uncontrolled Keywords:	discrete Schrödinger operator, Aharonov-Bohm magnetic potential
Publisher:	Springer Verlag
ISSN:	1139-1138
Last Modified:	18 Oct 2022 13:25
URI:	https://orca.cardiff.ac.uk/id/eprint/13823

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