The Schrödinger equation with a quasi-periodic potential

Abstract

We consider the Schràdinger equation \[ − d 2 d x 2 ψ + ε ( cos ⁡ x + cos ⁡ ( α x + ϑ ) ) ψ = E ψ - \frac {{{d^2}}} {{d{x^2}}}\psi + \varepsilon (\cos x + \cos (\alpha x + \vartheta ))\psi = E\psi \] where ε \varepsilon is small and σ \sigma satisfies the Diophantine inequality \[ | p + q α | ≥ C / q 2 for p , q ∈ Z , q ≠ 0. |p + q\alpha | \geq C/{q^2}{\text {for}}p{\text {,}}q \in {\mathbf {Z}},q \ne 0. \] . We look for solutions of the form \[ ψ ( x ) = e i K x q ( x ) = e i K x ∑ ψ m n e i n x e i m ( α x + ϑ ) \psi (x) = {e^{iKx}}q(x) = {e^{iKx}}\sum {{\psi _{mn}}{e^{inx}}} {e^{im(\alpha x + \vartheta )}} \] . If we try to solve for ψ = ψ m n \psi = {\psi _{mn}} we are led to the Schràdinger equation on the lattice Z 2 {{\mathbf {Z}}^2} \[ H ( K ) ψ = ( ε Δ + V ( K ) ) ψ = E ψ H(K)\psi = (\varepsilon \Delta + V(K))\psi = E\psi \] where Δ \Delta is the discrete Laplacian (without diagonal terms) and V ( K ) V(K) is some potential on Z 2 {{\mathbf {Z}}^2} . We have two main results: (1) For ε \varepsilon sufficiently small, H ( K ) H(K) has pure point spectrum for almost every K K . (2) For ε \varepsilon sufficiently small, the operator \[ − d 2 / d x 2 + ε ( cos ⁡ x + cos ⁡ ( α x + ϑ ) ) - {d^2}/d{x^2} + \varepsilon (\cos x + \cos (\alpha x + \vartheta )) \] has no point spectrum. To prove our results, we must get decay estimates on the Green’s function ( E − H ) − 1 {(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.