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By Stefan Teufel

Separation of scales performs a primary function within the realizing of the dynamical behaviour of complicated structures in physics and different typical sciences. A in demand instance is the Born-Oppenheimer approximation in molecular dynamics. This booklet specializes in a up to date method of adiabatic perturbation concept, which emphasizes the position of potent equations of movement and the separation of the adiabatic restrict from the semiclassical restrict. a close creation offers an summary of the topic and makes the later chapters available additionally to readers much less accustomed to the fabric. even though the overall mathematical thought according to pseudodifferential calculus is gifted intimately, there's an emphasis on concrete and appropriate examples from physics. functions diversity from molecular dynamics to the dynamics of electrons in a crystal and from the quantum mechanics of in part restricted structures to Dirac debris and nonrelativistic QED.

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5 we study the effective Born-Oppenheimer Hamiltonian near a conical eigenvalue crossing. Chapter 5: Dynamics in periodic structures As a not so obvious application of space-adiabatic perturbation theory we discuss the dynamics of an electron in a periodic potential based on Panati, Spohn, Teufel [PST3 ]. Indeed it requires considerable insight into the problem and some analysis to even formulate this question as a space-adiabatic problem. 1. 2 the general scheme of Chapter 3 is applied, however, with several technical innovations.

While this is the topic of Chapter 3, let us end this section with a short example for the physical relevance of effective Hamiltonians. The so called T-BMT equation was derived by Thomas [Tho] and, in a more general form, by Bargmann, Michel and Telegdi [BMT] on purely classical grounds as the simplest Lorentz invariant equation for the spin dynamics of a classical relativistic particle. It is of great physical importance, since it was used to compute the anomalous magnetic moment of the electron from experimental data before the invention of particle traps.

The case of regular eigenvalue crossings is included as a corollary. The presentation is such that the generalization to the first order space-adiabatic theorem is straightforward. 2 the first order space-adiabatic theorem is formulated for a certain class of adiabatic problems, namely for perturbations of fibered Hamiltonians. This is done under simplifying assumptions in order to provide a pedagogical introduction. The idea to translate the time-adiabatic theorem to a space-adiabatic version was first used by H¨ overmann, Spohn and Teufel [HST] in the context of the semiclassical limit in periodic structures and subsequently modified and applied to the semiclassical limit of dressed electrons by Teufel and Spohn [TeSp] and to the derivation of the time-dependent Born-Oppenheimer approximation in molecular dynamics by Spohn and Teufel [SpTe].

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