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arXiv · 1709.09200

Bounds on the Pure Point Spectrum of Lattice Schrödinger Operators

Volker Bach·Walter de Siqueira Pedra·Saidakhmat Lakaev
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Abstract

In dimension $d\geq 3$, a variational principle for the size of the pure point spectrum of (discrete) Schrödinger operators $H(\mathfrak{e},V)$ on the hypercubic lattice $\mathbb{Z}^{d}$, with dispersion relation $\mathfrak{e}$ and potential $V$, is established. The dispersion relation $\mathfrak{e}$ is assumed to be a Morse function and the potential $V(x)$ to decay faster than $|x|^{-2(d+3)}$, but not necessarily to be of definite sign. Our estimate on the size of the pure-point spectrum yields the absence of embedded and threshold eigenvalues of $H(\mathfrak{e},V)$ for a class ot potentials of this kind. The proof of the variational principle is based on a limiting absorption principle combined with a positive commutator (Mourre) estimate, and a Virial theorem. A further observation of crucial importance for our argument is that, for any selfadjoint operator $B$ and positive number $λ>0$, the number of negative eigenvalues of $λB$ is independent of $λ$.

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Volker Bach, Walter de Siqueira Pedra, Saidakhmat Lakaev. 2017-09-26. Bounds on the Pure Point Spectrum of Lattice Schrödinger Operators. https://arxiv.org/abs/1709.09200

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