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Texas A&M University

Events for 05/22/2020 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

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Time: 1:50PM - 2:50PM

Location: Zoom Seminar

Speaker: Ilya Kachkovskiy, MSU

Title: On spectral band edges of discrete periodic Schrodinger operators

Abstract: We consider discrete Schrodinger operators on $\ell^2(\mathbb Z^d)$, periodic with respect to some lattice $\Gamma$ in $\mathbb Z^d$ of full rank. Our main goal is to study dimensions of level sets of spectral band functions at the energies corresponding to their extremal values (the edges of the bands). Suppose that $d\ge 3$ and the dual lattice $\Gamma’$ does not contain the vector $(1/2,…,1/2)$. Then the above mentioned level sets have dimension at most $d-2$. Suppose that $d=2$ and the dual lattice does not contain vectors of the form $(1/p,1/p)$ and $(1/p,-1/p)$ for all $p\ge 2$. Then the same statement holds (in other words, the corresponding level sets are finite modulo $\mathbb Z^d$). For all lattices that do not satisfy the above assumptions, there are known counterexamples of level sets of dimensions $d-1$. Part of the argument also implies a discrete Bethe-Sommerfeld property: if $d\ge 2$ and the dual lattice does not contain the vector $(1/2,…,1/2)$, then, for sufficiently small potentials (depending on the lattice), the spectrum of the periodic Schrodinger operator is an interval. Previously, this property was studied by Kruger, Embree-Fillman, Jitomirskaya-Han, and Fillman-Han. Our proof is different and implies some new cases. The talk is based on joint work with in progress with N. Filonov.