By Graffi S. (Ed)

The consultation was once meant to provide a huge survey of the mathematical difficulties bobbing up within the chaotic transition of deterministic dynamical structures, either in classical and quantum mechanics.The lectures of Mather and Forni completely conceal the world- retaining twist maps, and comprise an updated model of the Aubry-Mather theory.The lectures of Bellissard describe the quantum features: classical restrict, localization, spectral homes of the appropriate Schrödinger operators and hence signify an exhaus- tive creation to the maths of "quantum chaos".The lectures of Degli Esposti et al. stories equidistribu- tion of volatile periodic orbits and classical restrict of quantized toral symplectomorphisms.

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**Extra resources for Transition to Chaos in Classical and Quantum Mechanics: Lectures Given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) Held in Montecatini, Italy, July 6-13, 1991**

**Example text**

1 TIME-INDEPENDENT TREATMENT 27 electronic (adiabatic) eigenfunctions of the electronic Hamiltonian introduced above: (He (se |s) − u j (s)) ζ j (se |s) = 0; j = 1, . . 4) Here u j (s), j = 1, . . , N are the electronic eigenvalues. In this treatment we assume that the Hilbert space is of dimension N. Substituting Eq. 3) in Eq. 2), multiplying it from the left by ζk (se |s)|, and integrating over the electronic coordinates while recalling Eqs. 4) yields the following set of coupled equations: N ζk |Tn ψ j (s) ζ j + (u k (s) − E) ψk (s) = 0 ; k = 1, .

22), respectively. 23) ˜ where A(s|s 0 ) is the matrix to be determined. It is important to emphasize that writing Eq. 23) in this way implies that ζ(se |s0 ) contains the diabatic basis set and ζ(se |s), for s = s0 , contains the adiabatic one (although both are legitimate eigenfunctions of the same Hamiltonian but are calculated at two different points in configuration space). To continue along these lines, we refer the reader to Eq. 2. 24) can be converted into a set of first-order differential equations for the ζ j (se |s) eigenfunctions: N ∇ |ζk (se |s) = − τ k j (s)|ζ j (se |s) ; k = 1, .

In this treatment we assume that the Hilbert space is of dimension N. Substituting Eq. 3) in Eq. 2), multiplying it from the left by ζk (se |s)|, and integrating over the electronic coordinates while recalling Eqs. 4) yields the following set of coupled equations: N ζk |Tn ψ j (s) ζ j + (u k (s) − E) ψk (s) = 0 ; k = 1, . . 6) where m is the mass of the system and ∇ is the gradient (vector) operator expressed in terms of mass-scaled coordinates. Substituting Eq. 6) in Eq. 5) and, performing the corresponding differentiations with respect to the nuclear coordinates and the integrations with respect to the electronic coordinates yield the more explicit form of the Born–Oppenheimer system of coupled equations:4 − h¯ 2 2 h¯ 2 ∇ ψk +(u k − E) ψk − 2m 2m N 2τ k j · ∇ + τ (2) k j ψ j = 0; k = 1, .