Keywords

Spectral Theory, Harmonic Analysis, Functional Analysis

Abstract

The connection between spectral properties and tiling properties of domains can be explored by mimicking an early approach of Fuglede. This approach focuses on self-adjoint extensions of differential operators and associated strongly continuous, one-parameter unitary groups. We take this approach in this thesis, determining an explicit formula for such a unitary group and use this to create new connections between spectral and tiling properties of a domain. Dutkay and Jorgensen showed it is enough to consider the case when the domain is a disjoint union of open intervals, so we restrict the scope of this thesis to this case as well. In the first part, we follow the exposition of Dutkay and Jorgensen in the creation of these unitary groups. Each self-adjoint extension of the momentum operator on the domain is simultaneously associated to a unique unitary matrix of boundary conditions and a strongly continuous, one parameter, unitary group. Spectrality of the domain yields a unique spectral boundary matrix and associated unitary group. Not only can the spectrum of a spectral domain be determined with these tools, but this unitary group must be a unitary group of local translation, acting as a translation operator on the domain. This part of the thesis culminates in the development of an explicit formula for the unitary group. The second part of the thesis uses the explicit formula for the unitary group and the local translation property to describe certain geometric properties that spectral domains must possess. Namely, the gap between any two intervals of a spectral domain must be a sum of lengths of intervals in the domain. Moreover, we show that under the additional assumption of multiplicativity of the unitary group, a power of the spectral boundary matrix is forced to be a permutation matrix, yielding questions for future research.

Completion Date

2025

Semester

Spring

Committee Chair

Dutkay, Dorin

Degree

Master of Science (M.S.)

College

College of Sciences

Department

Mathematics

Identifier

DP0029295

Document Type

Dissertation/Thesis

Campus Location

Orlando (Main) Campus

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