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(Reference retrieved automatically from Web of Science through information on FAPESP grant and its corresponding number as mentioned in the publication by the authors.)

Bounds on the discrete spectrum of lattice Schrodinger operators

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Bach, V. [1] ; de Siqueira Pedra, W. [2] ; Lakaev, S. N. [3]
Total Authors: 3
[1] TU Braunschweig, Inst Anal & Algebra, D-38106 Braunschweig - Germany
[2] Univ Sao Paulo, IF Inst Fis, Sao Paulo - Brazil
[3] Samarkand State Univ, Samarkand - Uzbekistan
Total Affiliations: 3
Document type: Journal article
Source: Journal of Mathematical Physics; v. 59, n. 2 FEB 2018.
Web of Science Citations: 2

We discuss the validity of the Weyl asymptotics-in the sense of two-sided bounds-for the size of the discrete spectrum of (discrete) Schrodinger operators on the d-dimensional, d >= 1, cubic lattice Z(d) at large couplings. We show that the Weyl asymptotics can be violated in any spatial dimension d >= 1-even if the semi-classical number of bound states is finite. Furthermore, we prove for all dimensions d >= 1 that, for potentials well behaved at infinity and fulfilling suitable decay conditions, the Weyl asymptotics always hold. These decay conditions are mild in the case d >= 3 while stronger for d = 1, 2. It is well known that the semi-classical number of bound states is-up to a constant-always an upper bound on the size of the discrete spectrum of Schrodinger operators if d >= 3. We show here how to construct general upper bounds on the number of bound states of Schrodinger operators on Z(d) from semi-classical quantities in all space dimensions d >= 1 and independently of the positivity-improving property of the free Hamiltonian. Published by AIP Publishing. (AU)

FAPESP's process: 16/02503-8 - Macroscopic behavior of non-relativistic interacting many fermion systems
Grantee:Walter Alberto de Siqueira Pedra
Support type: Scholarships abroad - Research