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Asymptotic expansions of slow perturbations of the periodic Schrödinger operator. (Développements asymptotiques des perturbations lentes de l’opérateur de Schrödinger périodique.) (French) Zbl 0784.35071

The author obtains asymptotic expansions in powers of \(h\) \((h\) small) for partial traces of the operator \(P=(D_ y+A(hy))^ 2+V(y)+\varphi(hy)\), where \(V\) is periodic and \(A,\varphi\) are smooth functions. The principal term of the asymptotics is given in terms of \(\varphi\) and the \(\xi\)- dependent periodic eigenvalues of the operator \((D_ y+\xi)^ 2+V(y)\).

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35P25 Scattering theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
35Q40 PDEs in connection with quantum mechanics
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