### 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 the stability of quantum systems with Hamilton operators of the 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 the corresponding chapter. Because of non-applicability of existing theory to the studied cases, the author attempts to device a new approach based on the 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 has not encountered before. In the last chapter is devoted to the study of simple example and to the application of devised theory.

### Preface

The subject of this thesis is the study of time-dependent quantum systems, i.e., systems on which the Hamiltonian H(t) depends. The discussion is primarily on cases where the Hamiltonian of the quantum system takes the form H(t) = H0 + εV (t), with H0 being a semi-bounded operator with a pure point spectra with constant gaps between distinct eigenvalues and εV (t) as the periodically time-dependent perturbation.

This topic is complex and even the very existence of time evolution is nontrivial as shall be demonstrated in Chapter 2. As such, most attempts for an analytical study only tackle the fundamental problems, such as stability of evolution, i.e., the boundedness of energy in time for particular sets of initial states. Existing results regarding stability problems are largely based on the assumptions on gaps between distinct eigenvalues in the pure point spectra. However, the results are only applicable to cases where the gaps between distinct eigenvalues are either shrinking or growing fast enough.

The main results of previous research are included in Chapter 6 of this work. Since the core of the existing theory is the assumption that the gaps between distinct eigenvalues are either shrinking or growing fast enough, expectations to obtain any results for the case where the gaps between distinct eigenvalues are constant through simple modification of previously obtained results are low. Therefore, the author tried a new approach to the problem.

Chapter 7 is devoted to studying the simplest case of the problem which is well-defined for any dimension of the separable Hilbert space, including the infinite one. Here, the author presents his results: the first order perturbation of the time evolution for any dimension, the applicable discrete symmetries, and a numerical analysis. A natural introduction in the study of the finite dimensional problem was the operator “mean of Hamiltonian over an infinite time interval”. The introduction of a similar result for an unbounded operator was complex and non-intuitive.

Hence, the definition of positive operator-valued measure in chapter 4 and researching existing results in this area. The correct definition of all terms regarding the integral with respect to positive operator valued measure has been a challenge. This is primarily due to the author being unaware of any complete and suitable source on positive operator measures and source on this subject. It is worth noting that some terms and relations regarding positive operator measures and integration with respect to positive operator measure may be a reinvention of existing terms.

The author’s main theoretical results are included in Chapter 5. Lemma 7 can be interpreted as a quantum equivalent of the ergodic theorem in a form the author has not encountered before.

The theorem 14 is of special importance. But because of time constraints, it remains uncertain whether the theorem 14 would yield any results about the stability of some families of quantum systems. There are suggestions for potential theorems about the stability of the infinite dimensional instance of the simple problem studied in the last section 7.7 of Chapter 7.

## Conclusion

The author studied existing theory of time-dependent Hamiltonians, particularly theoretical problems appearing in the cases, where gaps between points in spectra of the Hamiltonian are constant. The author studied analytically and numerically a simple case of the system and used gathered knowledge attempted to enrich existing theory proposing a new approach based on time-mean of Hamilton operator using theory of positive operator measures and integration with respect to positive operator measure. The author then proposed a possible way for continuation of the research regarding the studied simple case

### Key words

time-dependent Hamiltonians, stability of a quantum system