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SUMMARY:The isospectral torus of quasi-periodic Schrodinger operators via 
 periodic approximations - Lukic\, M (University of Toronto)
DTSTART:20150410T140000Z
DTEND:20150410T142500Z
UID:TALK58848@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Co-authors: David Damanik (Rice University)\, Michael Goldstei
 n (University of Toronto) \n\nThis talk describes joint work with D. Daman
 ik and M. Goldstein.\n\nWe study quasi-periodic Schr"odinger operators $H 
 = -rac{d^2}{dx^2} +V$ in the regime of analytic sampling function and sma
 ll coupling. More precisely\, the potential is\n[\nV(x)= um_{min mathbb{Z}
 ^\nu} c(m) xp(2pi i m omega x)\n]\n with $|c(m)|le psilon xp(-kappa |m|
 )$. Our main result is that any reflectionless potential $Q$ isospectral w
 ith $V$ is also quasi-periodic and in the same regime\, with the same Diop
 hantine frequency $omega$\, i.e.\n[\nQ(x)= um_{min mathbb{Z}^\nu} d(m) xp
 (2pi i m omega x)\n]\n with $|d(m)|le  qrt{2psilon} xp(-rac{kappa}2 |m|
 )$.\n\nThe proof relies on approximation by periodic potentials $	ilde V$\
 , which are obtained by replacing the frequency $omega$ by rational approx
 imants $	ilde omega$. We adapt the multiscale analysis\, developed by Dama
 nik--Goldstein for $V$\, so that it applies to the periodic approximants $
 	ilde V$. This allows us to establish estimates for gap lengths and Fourie
 r coefficients of $	ilde V$ which are independent of period\, unlike the s
 tandard estimates known in the theory of periodic Schr"odinger operators. 
 Starting from these estimates\, we obtain the main result by comparing the
  isospectral tori and translation flows of $	ilde V$ and $V$.\n\n
LOCATION:Seminar Room 1\, Newton Institute
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