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The isospectral torus of quasi-periodic Schrodinger operators via periodic approximations

Friday 10th April 2015 - 15:00 to 15:25
INI Seminar Room 1
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 $H = -\frac{d^2}{dx^2} +V$ in the regime of analytic sampling function and small coupling. More precisely, the potential is \[ V(x)=\sum_{m\in \mathbb{Z}^\nu} c(m) \exp(2\pi i m \omega x) \] with $|c(m)|\le \epsilon \exp(-\kappa |m|)$. 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. \[ Q(x)=\sum_{m\in \mathbb{Z}^\nu} d(m) \exp(2\pi i m \omega x) \] with $|d(m)|\le \sqrt{2\epsilon} \exp(-\frac{\kappa}2 |m|)$. The proof relies on approximation by periodic potentials $\tilde V$, which are obtained by replacing the frequency $\omega$ by rational approximants $\tilde \omega$. We adapt the multiscale analysis, developed by Damanik--Goldstein for $V$, so that it applies to the periodic approximants $\tilde V$. This allows us to establish estimates for gap lengths and Fourier coefficients of $\tilde V$ 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 $\tilde V$ and $V$.

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