N I Karachalios 2008 J. Phys. A: Math. Theor. 41 455201 doi:10.1088/1751-8113/41/45/455201
The number of bound states for a discrete Schrödinger operator on 
, lattices
N I Karachalios
Show affiliationsWe consider the discrete Schrödinger operator 
in 
in the case of a potential with negative part in an appropriate ℓσ-space (decays with an appropriate rate). We present a discrete analog of the method of Li and Yau (1983 Commun. Math. Phys. 88 309–18), proving an explicit upper estimate on the number of bound states 
, which is independent of the dimension of the lattice. This is a major difference with the continuous counterpart estimate, which is not valid when N = 1, 2. As a consequence, a dimension-independent smallness criterion for the existence of bound states is derived in contrast to the continuous case as well as to the discrete case of vanishing potential. A short comment is made on possible applications of the results to the study of the dynamics of some particular spatially discrete nonlinear systems.
- PACS
-
03.65.Ge Solutions of wave equations: bound states
- MSC
-
81Rxx Groups and algebras in quantum theory
- Subjects
- Dates
-
Issue 45 (14 November 2008)
Received 11 June 2008, in final form 4 September 2008
Published 8 October 2008
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The number of bound states for a discrete Schrödinger operator on , lattices
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