The Schrödinger equation with a quasi-periodic potential

Abstract:

We consider the Schràdinger equation \[ - \frac {{{d^2}}} {{d{x^2}}}\psi + \varepsilon (\cos x + \cos (\alpha x + \vartheta ))\psi = E\psi \] where $\varepsilon$ is small and $\sigma$ satisfies the Diophantine inequality \[ |p + q\alpha | \geq C/{q^2}{\text {for}}p{\text {,}}q \in {\mathbf {Z}},q \ne 0.\]. We look for solutions of the form \[ \psi (x) = {e^{iKx}}q(x) = {e^{iKx}}\sum {{\psi _{mn}}{e^{inx}}} {e^{im(\alpha x + \vartheta )}}\]. If we try to solve for $\psi = {\psi _{mn}}$ we are led to the Schràdinger equation on the lattice ${{\mathbf {Z}}^2}$ \[ H(K)\psi = (\varepsilon \Delta + V(K))\psi = E\psi \] where $\Delta$ is the discrete Laplacian (without diagonal terms) and $V(K)$ is some potential on ${{\mathbf {Z}}^2}$ . We have two main results: (1) For $\varepsilon$ sufficiently small, $H(K)$ has pure point spectrum for almost every $K$. (2) For $\varepsilon$ sufficiently small, the operator \[ - {d^2}/d{x^2} + \varepsilon (\cos x + \cos (\alpha x + \vartheta ))\] has no point spectrum. To prove our results, we must get decay estimates on the Green’s function ${(E - H)^{ - 1}}$. The decay of the eigenfunction follows from this. In general, we must keep track of small divisors which can make the Green’s function large. This is accomplished by a KAM (Kolmogorov, Arnold, Moser) type of multiscale perturbation analysis.