## Masters Theses and Doctoral Dissertations

#### Committee Chair

Nichols, Roger A.

#### Committee Member

Barioli, Francesco; Belinskiy, Boris P.; van der Merwe, Lucas C.

#### Department

Dept. of Mathematics

#### College

College of Arts and Sciences

#### Publisher

University of Tennessee at Chattanooga

#### Place of Publication

Chattanooga (Tenn.)

#### Abstract

We consider self-adjoint extensions of the minimal operator generated by the differential expression $\cL = - d^2/dx^2+V$ on the half-line $[0,\infty)$, where $V$ is a real-valued function integrable with respect to the weight $1+x$. The self-adjoint extensions are of Schr\"odinger-type and form a one-parameter family formally given by $H_{\alpha}=-d^2/dx^2+V$, $\alpha\in[0,\pi)$, with the boundary condition $\sin(\alpha)f'(0) = \cos(\alpha)f(0)$ at $x=0$. We derive a formula that relates the resolvent operator of $H_{\alpha}$ to the resolvent operator of $H_0$ in terms of the Jost solution corresponding to the underlying differential equation $\mathcal{L}u=zu$. Combining this resolvent formula and properties of the Jost solution, we compute the trace of the difference of the resolvents of $H_{\alpha}$ and the free operator $H_{\alpha}^{(0)}$ with $V\equiv 0$ in terms of the parameter $\alpha$ and the Jost function for $\cL u=zu$.

#### Acknowledgments

I am grateful to the Department of Mathematics of the University of Tennessee at Chattanooga for the support and numerous experiences it has given me. I would especially like to thank my advisor, Roger Nichols, for his guidance during my time in graduate school and, in particular, throughout the research process.

#### 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.

5-2017

#### Subject

Differential operators; Schrödinger operator; Operator theory; Kreĭn spaces; Differential equations

#### Keyword

Krein identity; Resolvent; Schrodinger operator; Jost solution; Jost function

Masters theses

vi, 54 leaves

English