Abstract

Co-author: Fedor Sandomirskyi (Saint Petersburg State University)

Consider the difference Schrödinger equation $psi_{k+1}psi_{k-1}+lambda {cotan} (piomega k heta)psi_k=Epsi_k,quad kin{mathbb Z}$,where $lambda$, $omega$, $ heta$ 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 $S(N)= um_{n=0}^N,e{2pi i (omega n^2+ heta n)}$, 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 asymptotic effects (Fedotov-Klopp, 2012).