10-143 V. Bach, W. de Siqueira Pedra and S.~Lakaev
Bounds on the Discrete Spectrum of Lattice Schr{\"o}dinger Operators (415K, pdf) Sep 14, 10
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Abstract. We discuss the the validity of the Weyl asymptotics -- in the sense of two-sided bounds -- for the size of the discrete spectrum of (discrete) Schr{\"o}diger operators on the \$d\$--dimensional, \$d\geq 1\$, cubic lattice \$\mathbb{Z}^d\$ at large couplings. We show that the Weyl asymptotics can be violated in any spatial dimension \$d \geq 1\$ -- even if the semi-classical number of bound states is finite. Furthermore, we prove for all dimensions \$d\geq 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 \geq 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 Schr{\"o}dinger operators if \$d\geq 3\$. We show here how to construct general upper bounds on the exact number of bound states of Schr{\"o}dinger operators on \$\mathbb{Z}^d\$ from semi-classical quantities in all space dimensions \$d\geq 1\$ and independently of the positivity-improving property of the free Hamiltonian.

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