#### Abstract

In this dissertation, we study a self-adjoint integral operator $\hat{K}$ which is defined in terms of finite Hilbert transforms on two adjacent intervals. These types of transforms arise when one studies the interior problem of tomography. The operator $\hat{K}$ possesses a so-called "integrable kernel'' and it is known that the spectral properties of $\hat{K}$ are intimately related to a $2\times2$ matrix function $\Gamma(z;\lambda)$ which is the solution to a particular Riemann-Hilbert problem (in the $z$ plane). We express $\Gamma(z;\lambda)$ explicitly in terms of hypergeometric functions and find the small $\lambda$ asymptotics of $\Gamma(z;\lambda)$. This asymptotic analysis is necessary for the spectral analysis of the finite Hilbert transform on multiple adjacent intervals. We show that $\Gamma(z;\lambda)$ also has a jump in the $\lambda$ plane which allows us to compute the jump of the resolvent of $\hat{K}$. This jump is an important step in showing that the finite Hilbert transforms has simple and purely absolutely continuous spectrum. The well known spectral theory now allows us to construct unitary operators which diagonalize the finite Hilbert transforms. Lastly, we mention some future directions which include the many interval scenario and a bispectral property of $\hat{K}$.

#### Notes

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#### Graduation Date

2019

#### Semester

Summer

#### Advisor

Tovbis, Alexander

#### Degree

Doctor of Philosophy (Ph.D.)

#### College

College of Sciences

#### Department

Mathematics

#### Degree Program

Mathematics

#### Format

application/pdf

#### Identifier

CFE0007602

#### URL

http://purl.fcla.edu/fcla/etd/CFE0007602

#### Language

English

#### Release Date

August 2019

#### Length of Campus-only Access

None

#### Access Status

Doctoral Dissertation (Open Access)

#### STARS Citation

Blackstone, Elliot, "Spectral Properties of the Finite Hilbert Transform on Two Adjacent Intervals Via the Method of Riemann-Hilbert Problem" (2019). *Electronic Theses and Dissertations*. 6454.

https://stars.library.ucf.edu/etd/6454