## Strong nonresonance of Schrödinger operators and an averaging theorem.(English)Zbl 0885.35119

Summary: We prove that linear Schrödinger operators $$-\Delta+q$$ on a torus or on a bounded smooth domain in $$\mathbb{R}^d$$ considered with Dirichlet boundary conditions, have a strongly nonresonant spectrum for any potential q of generic type (generic in the sense of Kolmogorov measure). As a consequence, a Krylov-Bogolubov averaging theorem holds for nonlinear perturbations of the corresponding Schrödinger evolution equations.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35J10 Schrödinger operator, Schrödinger equation 35P20 Asymptotic distributions of eigenvalues in context of PDEs
Full Text:

### References:

 [1] Courant, R.; Hilbert, D., Methods of modern mathematical physics, Vol. 1, (1962), New York [2] Arnold, V.I.; Kozlov, V.V.; Neistadt, A.I., Mathematical aspects of classical and celestial mechanics, () · Zbl 0885.70001 [3] Colin de Verdière, Y., Construction de laplaciens dont une partie finie du spectre est donnée, Ann. sci. ENS., 20, 599-615, (1987) · Zbl 0636.58036 [4] Kappeler, T., Multiplicities of the eigenvalues of the Schrödinger equation in any dimension, J. func. anal., 77, 346-351, (1988) · Zbl 0647.35017 [5] Kappeler, T.; Ruf, B., On the nodal line of the second eigenfunction of elliptic operators, J. reine angew. math., 396, 1-13, (1989) · Zbl 0658.35068 [6] Kuksin, S.B., An averaging theorem for distributed conservative systems and its applications to the von karman equations, Prikl. matem. mekhan., 53, 2, 196-205, (1988), [English translation in P.M.M. USSR 53:2 (1989) 150-157] [7] Kuksin, S.B., Nearly integrable infinite-dimensional Hamiltonian systems, () · Zbl 0667.58060 [8] Ramis, J.P., Sous-ensembles analytiques d’une variété banachique complexe, () · Zbl 0212.42802
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.