Derkach, Alexey; (2024) Spectral Analysis of Pseudo-Differentiable Operators with Discontinuous Symbols. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

The spectral theory of Pseudo-Differential Operators (ΨDOs) with smooth symbols is quite mature. Many aspects are well studied including asymptotic formulae of eigenvalues (see [4], [9], [14] and the references therein). There are two types of results: those for unbounded ΨDOs (e.g. [14]) and those for compact ΨDOs [4]. We focus on the latter ones. There are significantly more results for associated operators, for the case when the symbol of a ΨDO is smooth, or when ΨDOs are defined on modulation spaces. However, considerably less is known about the cases with discontinuous symbols, even for the simplest type of discontinuity. This work is devoted to spectral properties of compact ΨDOs with a certain type of symbols, namely Weyl symbols with jump discontinuity. These symbols are considered indicator functions of given bounded regions Λ in phase space. The general goal is to understand how the rate of eigenvalues decay depends on the geometry of the boundary ∂Λ.

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