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SUMMARY:Milivoje Lukic (Rice University)
DTSTART:20200501T185000Z
DTEND:20200501T195000Z
DTSTAMP:20260423T021528Z
UID:MPHA/3
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MPHA/3/">Sta
 hl--Totik regularity for continuum Schr\\"odinger operators</a>\nby Milivo
 je Lukic (Rice University) as part of TAMU: Mathematical Physics and Harmo
 nic Analysis Seminar\n\n\nAbstract\nThis talk describes joint work with Be
 njamin Eichinger: a\ntheory of regularity for one-dimensional continuum Sc
 hr\\"odinger\noperators\, based on the Martin compactification of the comp
 lement of\nthe essential spectrum. For a half-line Schr\\"odinger operator
 \n$-\\partial_x^2+V$ with a bounded potential $V$\, it was previously\nkno
 wn that the spectrum can have zero Lebesgue measure and even zero\nHausdor
 ff dimension\; however\, we obtain universal thickness statements\nin the 
 language of potential theory.\nNamely\, we prove that the essential spectr
 um is not polar\, it obeys\nthe Akhiezer--Levin condition\, and moreover\,
  the Martin function at\n$\\infty$ obeys the two-term asymptotic expansion
  $\\sqrt{-z} +\n\\frac{a}{2\\sqrt{-z}} + o(\\frac 1{\\sqrt{-z}})$ as $z \\
 to -\\infty$. The\nconstant $a$ in its asymptotic expansion plays the role
  of a\nrenormalized Robin constant suited for Schr\\"odinger operators and
 \nenters a universal inequality $a \\le \\liminf_{x\\to\\infty} \\frac 1x\
 n\\int_0^x V(t) dt$. This leads to a notion of regularity\, with\nconnecti
 ons to the exponential growth rate of Dirichlet solutions and\nthe zero co
 unting measures for finite restrictions of the operator. We\nalso present 
 applications to decaying and ergodic potentials.\n
LOCATION:https://researchseminars.org/talk/MPHA/3/
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