By Stefan Teufel

Separation of scales performs a primary function within the figuring out of the dynamical behaviour of advanced platforms in physics and different ordinary sciences. A favorite instance is the Born-Oppenheimer approximation in molecular dynamics. This ebook specializes in a up to date method of adiabatic perturbation concept, which emphasizes the position of powerful equations of movement and the separation of the adiabatic restrict from the semiclassical limit.

A specified advent supplies an summary of the topic and makes the later chapters obtainable additionally to readers much less accustomed to the fabric. even if the final mathematical idea according to pseudodifferential calculus is gifted intimately, there's an emphasis on concrete and suitable examples from physics. purposes variety from molecular dynamics to the dynamics of electrons in a crystal and from the quantum mechanics of partly restrained structures to Dirac debris and nonrelativistic QED.

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**Example text**

Up to Chapter 6 we always assume a gap condition, at least locally in the conﬁguration space of the slow degrees of freedom. In the presence of eigenvalue crossings our results permit to derive an eﬀective Hamiltonian for a group of bands, inside of which the crossing occurs. However, the study of such an eﬀective Hamiltonian is in itself a veritable task. Much work has been spent in order to study model Hamiltonians displaying diﬀerent types of eigenvalue crossings, among which we mention only some recent work [Ha3 , HaJo1 , CLP, FeGe, FeLa, Col1 , Col2 , LaTe].

4, the leading orders of this expansion provide very relevant information about the dynamics of the slow degrees of freedom. In many cases the eﬀective Hamiltonian is the quantization of a matrixvalued symbol with scalar principal symbol. 4 some results on semiclassics for matrix-valued Hamiltonians. Moreover the geometrical interpretation of the subprincipal symbol of the eﬀective Hamiltonian as coming partly from the famous Berry connection is most conveniently discussed in the context of the semiclassical limit.

7) suggests that this diﬀerence might add up to a diﬀerence of order 1 on the macroscopic time-scale. 7) oscillates and can be written as the time-derivative of a function which is of order ε. In principle the situation is the same as for the function eit/ε , which is O(1), but integrated yields t t0 dt eit /ε = −iε eit0 /ε − eit/ε = O(ε) . 1 The classical time-adiabatic result 37 In order to motivate the following deﬁnition, consider for the moment the special case when σ∗ (t) = {E(t)} consists of a single eigenvalue E(t).