By Stefan Teufel

Separation of scales performs a basic function within the knowing of the dynamical behaviour of complicated structures in physics and different common sciences. A favorite instance is the Born-Oppenheimer approximation in molecular dynamics. This booklet makes a speciality of a up to date method of adiabatic perturbation idea, which emphasizes the function of powerful equations of movement and the separation of the adiabatic restrict from the semiclassical restrict. an in depth advent supplies an summary of the topic and makes the later chapters available additionally to readers much less acquainted with the fabric. even though the overall mathematical concept in line with pseudodifferential calculus is gifted intimately, there's an emphasis on concrete and correct examples from physics. purposes variety from molecular dynamics to the dynamics of electrons in a crystal and from the quantum mechanics of partly restrained platforms to Dirac debris and nonrelativistic QED.

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**Sample text**

For Γ ⊂ Z we will use the abbreviations Γq := q ∈ Rd : (q, p) ∈ Γ for some p ∈ Rd , Γp := p ∈ Rd : (q, p) ∈ Γ for some q ∈ Rd . Let the phase space support Γ of the initial wave function be such that Γq ⊂ Λ − δ. Then the maximal time interval for which the x-support of the wave function of the nuclei stays in Λ up to errors of order ε can be written as δ (Γ, Λ) := [T−δ (Γ, Λ), T+δ (Γ, Λ)] , Imax where the “ﬁrst hitting times” T± are deﬁned by the classical dynamics through T+δ (Γ, Λ) := sup t ≥ 0 : Φt (Γ ) q ⊆ Λ − δ ∀ t ∈ [0, t] and T−δ (Γ, Λ) analogously for negative times.

Thus the theory to be developed in this chapter can be seen as a last resort when the general scheme can not be applied directly. 3 in order to demonstrate the ﬂexibility of the present approach. On the other hand we will return to the setting of perturbations of ﬁbered Hamiltonians once we remove the gap condition in Chapter 6. There we will establish a general result, the proof of which can easily be translated back to the case with gap. We remark that the idea to be developed in this chapter was applied in a variety of diﬀerent physical contexts: motion of electrons in periodic potentials with a weak external electric ﬁeld [HST], the dynamics of dressed electrons under the inﬂuence of a slowly varying external potential [TeSp] and the Born-Oppenheimer approximation [SpTe].

Also here we are able to compute the eﬀective Hamiltonian including second order corrections. We close Chapter 4 with two heuristic sections relevant to applications of the theory. 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.