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Vaclav Kosar

Vaclav Kosar is a senior software developer. This blog is focused on interesting technology, tips and news.

Hamiltonians with constant spectral intervals and time-dependent perturbation

11 Jun 2012

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

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