Nikita Rastegaev

Junior Researcher

SPbU

**Contacts:**

**Education:**

06.2018 Candidate of Sciences (PhD) in Mathematics (SPbU)

06.2013 Specialist degree in Mathematics (SPbU)

The head of the research project: "Sturm-Liouville problem with singular self-similar weight"

Abstract: We study the spectral asymptotics for the Sturm-Liouville problem with singular self-similar weight measure. Of interest is the periodic component of the main asymptotic term, which emerges in the case of so-called "arithmetic" self-similarity. The question is, could this periodic component degenerate into constant.

The results showing that the periodic component of the main spectral asymptotic term does not degenerate into constant have already been obtained for a wide class of arithmetically self-similar measures, but according to the working conjecture (A.I.Nazarov, 2006) this results should allow a generalization to the case of an arbitrary arithmetically self-similar measure.

Moreover, we consider the application of this results to the theory of small ball deviations of Gaussian random processes. We develop a general approach to translate the results about the periodic component of the main asymptotic term to the corresponding small ball asymptotics.

06.2013 Specialist degree in Mathematics (SPbU)

The head of the research project: "Sturm-Liouville problem with singular self-similar weight"

Abstract: We study the spectral asymptotics for the Sturm-Liouville problem with singular self-similar weight measure. Of interest is the periodic component of the main asymptotic term, which emerges in the case of so-called "arithmetic" self-similarity. The question is, could this periodic component degenerate into constant.

The results showing that the periodic component of the main spectral asymptotic term does not degenerate into constant have already been obtained for a wide class of arithmetically self-similar measures, but according to the working conjecture (A.I.Nazarov, 2006) this results should allow a generalization to the case of an arbitrary arithmetically self-similar measure.

Moreover, we consider the application of this results to the theory of small ball deviations of Gaussian random processes. We develop a general approach to translate the results about the periodic component of the main asymptotic term to the corresponding small ball asymptotics.

**Interests:**

Spectral theory, Asymptotic analysis;