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Maryland equation, renormalization formulas and mimimal meromorphic solutions to difference equations

Friday 27th March 2015 - 15:00 to 16:00
INI Seminar Room 1
Co-author: Fedor Sandomirskyi (Saint Petersburg State University)

Consider the difference Schrödinger equation $\psi_{k+1}+\psi_{k-1}+\lambda\ {cotan} (\pi\omega k+\theta)\psi_k=E\psi_k,\quad k\in{\mathbb Z}$,where $\lambda$, $\omega$, $\theta$ and $E$ are parameters. If $\omega$ is irrational, this equation is quasi-periodic. It was introduced by specialists in solid state physics from Maryland and is now called the Maryland equation. Computer calculations show that, for large $k$, its eigenfunctions have a multiscale, "mutltifractal" structure. We obtained renormalization formulas that express the solutions to the input Marryland equation for large $k$ in terms of solutions to the Marryland equation with new parameters for bounded $k$. The proof is based on the theory of meromorphic solutions of difference equations on the complex plane, and on ideas of the monodromization met hod -- the renormalization approach first suggested by V.S.Buslaev and A.A. Fedotov.

Our formulas are close to the renormalization formulas from the theory of the Gaussian exponential sums $S(N)=\sum_{n=0}^N\,e^{2\pi i (\omega n^2+\theta n)}$, where $\omega$ and $\theta$ 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 asymptotic effects (Fedotov-Klopp, 2012).

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons