### Author

Vaclav Kosar

### Abstract

This work deals with quantum systems determined by time-dependent Hamilton operators. Family of quantum systems, whose Hamilton operators take form H(t) = H 0 + V (t), where V (t) is perturbation and H 0 is self-adjoint with pure-point spectrum and constant gaps between eigenvalues in spectrum σ(H 0 ). Existing theory dealing with stability of quantum systems with Hamilton operators of above form, where H 0 is self-adjoint with pure-point spectrum and growing or shrinking gaps between eigenvalues in spectrum σ(H 0 ) is given in corresponding chapter. Because of non-applicability of existing theory to the studied cases the author attempts to device a new approach based on term ”mean of Hamilton operator over infinite time interval“. Lemma 7 can be interpreted as quantum variant of ergodic theorem in a very nice form that the author have not encountered before. In the last chapter is devoted to the study of simple example and to the application of devised theory.

### Key words

time-dependent Hamiltonians, stability of a quantum system