CHAPTER 2 Assumptions and Main Results The purpose of this paper is to investigate the asymptotic behavior as h 0+ of the solutions of the time-dependent Schr¨ odinger equation, (2.1) ih ∂ϕ ∂t = P (h)ϕ with (2.2) P (h) = ω + Q(x) + W (x), where Q(x) (x IRn) is a family of selfadjoint operators on some fix Hilbert space H with same dense domain DQ, ω = |α|≤m cα(x h)(hDx)α is a symmetric semiclassical differential operator of order 0 and degree m, with scalar coefficients depending smoothly on x, and W (x) is a non negative function defined almost everywhere on IRn. Typically, in the case of a molecular system, x stands for the position of the nu- clei, Q(x) represents the electronic Hamiltonian that includes the electron-electron and nuclei-electron interactions (all of them of Coulomb-type), ω is the quantized cinetic energy of the nuclei, and W (x) represents the nuclei-nuclei interactions. Moreover, the parameter h is supposed to be small and, in the case of a molecular system, h−2 actually represents the quotient of electronic and nuclear masses. In more general systems, one can also include a magnetic potential and an exterior electric potential both in ω and Q(x). We refer to Chapter 12 for more details about this case. We make the following assumptions: (H1) For all α, β ZZ+ n with |α|≤ m, ∂βcα(x, h) = O(1) uniformly for x IRn and h 0 small enough. Moreover, setting ω(x, ξ h) := |α|≤m cα(x h)ξα, we assume that there exists a constant C0 1 such that, for all (x, ξ) IR2n and h 0 small enough, Re ω(x, ξ h) 1 C0 ξ m C0. In particular, Assumption (H1) implies that m is even and ω is well defined as a selfadjoint operator on L2(IRn) (and, by extension, on L2(IRn H)) with domain Hm(IRn). Moreover, by the Sharp arding Inequality (see, e.g., [Ma2]), it is uni- formly semi-bounded from below. (H2) W 0 is Dx m -compact on L2(IRn), and there exists γ IR such that, for all x IRn, Q(x) γ on H. 7
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