V. Bach, W. de Siqueira Pedra and S.~Lakaev
Bounds on the Discrete Spectrum of Lattice Schr{\"o}dinger Operators
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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.