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$%W. D. Evans and K. M. Schmidt, A discrete Hardy-Laptev-Weidl-type inequality and associated Schrödinger-type operators, Rev. Mat. Complut. 22 (2009), no. 1, 75–90.%$
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.
Key words: discrete Schrödinger operator, Aharonov-Bohm magnetic potential.
2000 Mathematics Subject Classification: 47B37, 35P05, 39A70, 47A30, 81Q10.