A Discrete Hardy-Laptev-Weidl-Type Inequality and Associated Schrödinger-Type Operators

• Main Authors: W Desmond, Karl Michael Schmidt
• Other Authors: The Pennsylvania State University CiteSeerX Archives
• Format: Text
• Language: English
• Subjects: laptev
• Online Access: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.1041.1364; http://www.mat.ucm.es/serv/revista/vol22-1/vol22-1e.pdf

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 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.