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

• Published in: Revista Matemática Complutense
• Main Authors: Evans, W. Desmond, Schmidt, Karl Michael
• Format: Article in Journal/Newspaper
• Language: Spanish
• Published: Springer 2009
• Subjects: Discrete Schrödinger operator; Aharonov-Bohm magnetic potential; laptev
• Online Access: https://revistas.ucm.es/index.php/REMA/article/view/REMA0909120075A; https://doi.org/10.5209/rev_REMA.2009.v22.n1.16316

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. 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.