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SUMMARY:Simon Larson (Caltech)
DTSTART:20210506T170000Z
DTEND:20210506T180000Z
DTSTAMP:20260423T040039Z
UID:Thouless/14
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/Thouless/14/
 ">On the spectrum of the Kronig-Penney model in a constant electric field<
 /a>\nby Simon Larson (Caltech) as part of UCI Mathematical Physics\n\n\nAb
 stract\nWe are interested in the nature of the spectrum of the one-dimensi
 onal Schr\\"odinger operator\n$$\n  - \\frac{d^2}{dx^2}-Fx + \\sum_{n \\in
  \\mathbb{Z}}g_n \\delta(x-n)\n$$\nwith $F>0$ and two different choices of
  the coupling constants $\\{g_n\\}_{n\\in \\mathbb{Z}}$. In the first mode
 l $g_n \\equiv \\lambda$ and we prove that if $F\\in \\pi^2 \\mathbb{Q}$ t
 hen the spectrum is $\\mathbb{R}$ and is furthermore absolutely continuous
  away from an explicit discrete set of points. In the second model $g_n$ a
 re independent random variables with mean zero and variance $\\lambda^2$. 
 Under certain assumptions on the distribution of these random variables we
  prove that almost surely the spectrum is dense pure point if $F < \\lambd
 a^2/2$ and purely singular continuous if $F> \\lambda^2/2$. Based on joint
  work with Rupert Frank.\n
LOCATION:https://researchseminars.org/talk/Thouless/14/
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