## Masters Theses and Doctoral Dissertations

Nichols, Roger

#### Committee Member

Barioli, Francesco; van der Merwe, Lucas; Wang, Jin

#### Department

Dept. of Mathematics

#### College

College of Arts and Sciences

#### Publisher

University of Tennessee at Chattanooga

#### Place of Publication

Chattanooga (Tenn.)

#### Abstract

We prove weak and vague convergence results for spectral shift functions associated with self-adjoint one-dimensional Schr\"odinger operators on intervals of the form $(-\ell,\ell)$ with periodic boundary conditions to the full-line spectral shift function in the infinite volume limit $\ell\to \infty$. The approach employed relies on the use of a Krein-type resolvent identity to relate the resolvent of the operator with periodic boundary conditions to the corresponding operator with Dirichlet boundary conditions in combination with various operator theoretic facts.

#### Acknowledgments

I am indebted to my advisor, Roger Nichols, for the help he provided me in completing this thesis. I am extremely lucky to have a professor as patient and knowledgable as he is. I thank my mom, Denise Murphy, for being a great mother. I also thank Doug and Nancy Murphy for believing in me. I also thank my siblings, Ashley and Thomas, for always being there for me. Also, I thank all my friends, family, professors, teachers, and coaches that have made me who I am today. I would like to thank the members of my committee, Drs. Barioli, van der Merwe, and Wang for their advice and encouragement.

#### Degree

M. S.; A thesis submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Master of Science.

12-2016

#### Subject

Schrödinger operator; Operator theory

#### Keyword

Spectral shift function; Schrodinger operator; Vague convergence; Periodic boundary conditions

Masters theses

Text

vii, 67 leaves

English

#### Rights

https://rightsstatements.org/page/InC/1.0/?language=en