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SUMMARY:Conductance and absolutely continuous spectrum of 1D samples - Jak
 sic\, V (McGill University)
DTSTART:20150513T130000Z
DTEND:20150513T140000Z
UID:TALK59449@talks.cam.ac.uk
CONTACT:42080
DESCRIPTION:In this talk I shall describe the characterization of the abso
 lutely continuous spectrum of the one-dimensional Schr ̈odinger operators
  h = &minus\;∆ + v acting on 2 (Z + ) in terms of\nthe limiting behavior
  of the Landauer-B ̈\nuttiker and Thouless conductances of the associated
 \nfinite samples. The finite samples are defined by restricting h to a fin
 ite interval [1\, L] &cap\; Z +\nand the conductance refers to the charge 
 current across the sample in the open quantum\nsystem obtained by attachin
 g independent electronic reservoirs to the sample ends. Our\nmain result i
 s that the conductances associated to an energy interval I are non-vanishi
 ng\nin the limit L &rarr\; &infin\; (physical characterization of the meta
 llic regime) iff sp ac (h) &cap\; I = &empty\;\n(mathematical characteriza
 tion of the metallic regime). This result is of importance for the\nfounda
 tions of quantum mechanics since it provides the first complete dynamical 
 character-\nization of the absolutely continuous spectrum of Schr ̈odinge
 r operators. I shall also discuss\nits relation with Avila&rsquo\;s counte
 rexample to the Schr ̈odinger Conjecture.\nThis talk is based on a joint 
 work with L. Bruneau\, Y. Last\, and C-A. Pillet.\n\n
LOCATION:Seminar Room 2\, Newton Institute Gatehouse
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