Abstract

Co-authors: David Damanik (Rice University), Michael Goldstein (University of Toronto)

This talk describes joint work with D. Damanik and M. Goldstein.

We study quasi-periodic Schr”odinger operators $H = – rac{d^2}{dx2} +V$ in the regime of analytic sampling function and small coupling. More precisely, the potential is [ V(x)= um_{min mathbb{Z} ^{u} c(m) xp(2pi i m omega x) ] with $|c(m)|le psilon xp(-kappa |m|)$. Our main result is that any reflectionless potential $Q$ isospectral with $V$ is also quasi-periodic and in the same regime, with the same Diophantine frequency $omega$, i.e. [ Q(x)= um_{min mathbb{Z}} u} d(m) xp(2pi i m omega x) ] with $|d(m)|le qrt{2psilon} xp(-rac{kappa}2 |m|)$.

The proof relies on approximation by periodic potentials $ ilde V$, which are obtained by replacing the frequency $omega$ by rational approximants $ ilde omega$. We adapt the multiscale analysis, developed by Damanik—Goldstein for $V$, so that it applies to the periodic approximants $ ilde V$. This allows us to establish estimates for gap lengths and Fourier coefficients of $ ilde V$ which are independent of period, unlike the standard 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 $V$.