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SUMMARY:Ilya Kachkovskiy (MSU)
DTSTART:20200522T185000Z
DTEND:20200522T195000Z
DTSTAMP:20260423T021603Z
UID:MPHA/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MPHA/6/">On 
 spectral band edges of discrete periodic Schrodinger operators</a>\nby Ily
 a Kachkovskiy (MSU) as part of TAMU: Mathematical Physics and Harmonic Ana
 lysis Seminar\n\n\nAbstract\nWe 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 l
 evel sets of spectral band functions at the energies corresponding to thei
 r extremal values (the edges of the bands).Suppose that $d\\ge 3$ and the 
 dual lattice $\\Gamma’$ does not contain the vector $(1/2\,…\,1/2)$. T
 hen the above mentioned level sets have dimension at most $d-2$.\n\nSuppos
 e 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 hol
 ds (in other words\, the corresponding level sets are finite modulo $\\mat
 hbb Z^d$).For all lattices that do not satisfy the above assumptions\, the
 re are known counterexamples of level sets of dimensions $d-1$.\n\nPart of
  the argument also implies a discrete Bethe-Sommerfeld property: if $d\\ge
  2$ and the dual lattice does not contain the vector $(1/2\,…\,1/2)$\, t
 hen\, for sufficiently small potentials (depending on the lattice)\, the s
 pectrum 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.\n\nThe
  talk is based on joint work with in progress with N. Filonov.\n
LOCATION:https://researchseminars.org/talk/MPHA/6/
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