# A TALE OF TWO DIAGONALIZATIONS: METHODS TO DIAGONALIZE A 1-D PIECEWISE CONSTANT INDEFINITE SCHRÖDINGER OPERATOR

Summer 8-1-2023

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## School

Mathematics and Natural Sciences

## Committee Chair

Dr. James Lambers

## Committee Chair School

Mathematics and Natural Sciences

Dr. Haiyan Tian

## Committee Member 2 School

Mathematics and Natural Sciences

Dr. Zhifu Xie

## Committee Member 3 School

Mathematics and Natural Sciences

Dr. Huiqing Zhu

## Committee Member 4 School

Mathematics and Natural Sciences

## Abstract

We present two numerical methods for computing the solution of a partial differential equation (PDE) for modeling acoustic pressure, known as an extra-wide angle parabolic equation, that features the square root of a differential operator. The differential operator is the negative of an indefinite Schrödinger operator with a piecewise constant potential. This work primarily deals with the 3-piece case; however, a generalization is made to the case of an arbitrary number of pieces. In the first method, the Rayleigh-Secant Method, through restriction to a judiciously chosen lower-dimensional subspace, approximate eigenfunctions are used to obtain estimates for the eigenvalues of the operator. Then, the estimated eigenvalues are used as initial guesses for the Secant Method to find the exact eigenvalues, up to roundoff error. An eigenfunction expansion of the solution is then constructed. The computational expense of obtaining each eigenpair is independent of the grid size. The accuracy, efficiency, and scalability of this method is shown through numerical experiments and comparisons with other methods. The second method described in this dissertation is an eigenfunction expansion method based on the Fokas transform. This method takes advantage of the Fokas transform’s inner product form that observes the boundary and continuity conditions of the piecewise operator. The method can be simplified and applied more efficiently upon analyzing the singularity of the characteristic matrix of the operator. This method is called the Singular Fokas method, which will be described and applied to construct the eigenfunction expansion of the solution to the extra-wide angle parabolic equation.

## ORCID ID

0000-0002-4350-4749