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Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case. (English) Zbl 1004.81008

Summary: This work is devoted to the study of a family of almost periodic one-dimensional Schrödinger equations. Using results on the asymptotic behavior of a corresponding monodromy matrix in the adiabatic limit, we prove the existence of an asymptotically sharp Anderson transition in the low energy region. More explicitly, we prove the existence of energy intervals containing only singular spectrum, and of other energy intervals containing absolutely continuous spectrum; the zones containing singular spectrum and those containing absolutely continuous are separated by asymptotically sharp transitions. The analysis may be viewed as utilizing a complex WKB method for adiabatic perturbations of periodic Schrödinger equations. The transition energies are interpreted in terms of phase space tunneling.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34M30 Asymptotics and summation methods for ordinary differential equations in the complex domain
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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