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

• Autores: W. Desmond Evans, Karl Michael Schmidt
• Localización: Revista matemática complutense, ISSN-e 1988-2807, ISSN 1139-1138, Vol. 22, Nº 1, 2009, págs. 75-90
• Idioma: inglés
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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 field has essential spectrum concentrated at 0, and the multiplicity of its lower spectrum satisfies a CLR-type inequality.