Semiclassics, adiabatic decoupling and perturbed periodic structures

Abstract

This thesis deals with the consequences of periodic structures in quantum mechanics in different semiclassical regimes. The first chapter introduces the Floquet theory. It is applied on the Hill equation. As a special case, the solutions to the Mathieu equation are studied exemplifying the properties of the spectrum of periodic Schrödinger operators. Special attention is brought to the properties of the spectrum and the corresponding Floquet exponent of solutions. We present new results concerning the relation between the eigenvalues and the Floquet exponent. We introduce the Weyl quantization in the second chapter. In particular, we introduce semiclassical propagation results which, up to some known errors, describes the propagation of an initial Gaußian under quantum evolution (Combescure, Robert 1997). This approximation breaks down at what is commonly known as the Ehrenfest time. We use these results to investigate the localization properties of an initial Gaußian. We can show localization of the state up to the Ehrenfest time under the assumption that the flow differential of the corresponding classical motion admits Floquet solutions with purely imaginary Floquet exponents. In this case, we also show the existence of what we call classical revivals. The last part is a study of the spectral properties of perturbed periodic Schrödinger operators. This is done in the limit of weak perturbation epsilon . As the structure of the band spectrum of the unperturbed operator remains unchanged under small enough perturbations, the effective dynamics can be described adiabatically under certain prerequisites. We consider the construction of WKB ansätze (Littlejohn, Flynn, 1991, Emmrich, Weinstein 1996). Our results include Bohr-Sommerfeld quantization conditions modulo O(epsilon²). This is made possible by including geometric terms. The Wannier-Stark system is studied as an example.