We consider the Schrόdinger operator P(h) for a polyatomic molecule in the semiclassical limit where the mass ratio h 2 of electronic to nuclear mass tends to zero. We obtain WKB-type expansions of eigenvalues and eigenfunctions of P(h) to all orders in h. This allows to treat the splitting of the ground state energy of a non-planar molecule. Our class of potentials covers the physical case of thp Coulomb interaction. We use methods of /ι-pseudodifferential operators with operator valued symbols, which by use of appropriate coordinate changes in local coordinate patches covering the classically accessible region become applicable even to our class of singular potentials.
We describe a rigorous mathematical reduction of the spectral study for a class of periodic problems with perturbations which gives a justification of the method of effective Hamiltonians in solid state physics. We study the partial differential operators of the form P = P(hy,y,D y + A(hy}} on R" (when h>0 is small enough), where P(x, y, η) is elliptic, periodic in y with respect to some lattice Γ, and admits smooth bounded coefficients in (x,y). A(x) is a magnetic potential with bounded derivatives. We show that the spectral study of P near any fixed energy level can be reduced to the study of a finite system of /i-pseudodifferential operators ^(x, hD x , h\ acting on some Hubert space depending on 7". We then apply it to the study of the Schrδdinger operator when the electric potential is periodic, and to some quasiperiodic potentials with vanishing magnetic field.
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