## 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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