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SUMMARY:Maryland equation\, renormalization formulas and mimimal meromorph
 ic solutions to difference equations - Fedotov\, A (Saint Petersburg State
  University)
DTSTART:20150327T150000Z
DTEND:20150327T160000Z
UID:TALK58626@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Co-author: Fedor Sandomirskyi (Saint Petersburg State Universi
 ty) \n\nConsider the difference Schr&ouml\;dinger equation $psi_{k+1}+psi_
 {k-1}+lambda {cotan} (piomega k+	heta)psi_k=Epsi_k\,quad kin{mathbb Z}$\,w
 here $lambda$\, $omega$\, $	heta$ and $E$ are parameters. If $omega$ is ir
 rational\, this equation is quasi-periodic. It was introduced by specialis
 ts in solid state physics from Maryland and is now called the Maryland equ
 ation. Computer calculations show that\, for large  $k$\, its eigenfunctio
 ns have a multiscale\, "mutltifractal" structure. We obtained renormalizat
 ion formulas that express the solutions to the input Marryland equation fo
 r large $k$ in terms of solutions to the Marryland equation with new param
 eters for bounded $k$. The proof is based on the theory of meromorphic sol
 utions of difference equations on the complex plane\, and on ideas of the 
 monodromization met\nhod -- the renormalization approach first suggested b
 y V.S.Buslaev and A.A. Fedotov.\n\n\n\nOur formulas are close to the renor
 malization formulas from the theory\nof the Gaussian exponential sums $S(N
 )= um_{n=0}^N\,e^{2pi i (omega n^2+	heta n)}$\, where  $omega$ and $	heta$
  are parametrs. For large $N$\, these sums also have a multiscale behavior
 . The renormalization formulas lead to a natural explanation of the famous
  mutiscale structure that appears to reflect certain quasi-classical asymp
 totic effects (Fedotov-Klopp\, 2012).\n
LOCATION:Seminar Room 1\, Newton Institute
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