A discrete Hardy-Laptev-Weidl type inequality and associated Schrödinger-type operators
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 plan...
| Main Authors: | , |
|---|---|
| Format: | Article in Journal/Newspaper |
| Language: | unknown |
| Published: |
Springer Verlag
2009
|
| Subjects: | |
| Online Access: | https://orca.cardiff.ac.uk/id/eprint/13823/ http://www.mat.ucm.es/serv/revmat/vol22-1e.html |
| Summary: | 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. |
|---|