#### Date of Award

7-1-2002

#### Degree Type

Thesis

#### University or Center

Clark Atlanta University(CAU)

#### School

School of Arts and Sciences

#### Degree Name

M.S.

#### Department

Physics

#### First Advisor

Professor Carlos R. Handy

#### Abstract

The research objective of this thesis is to extend the Eigenvalue Moment Method (EMM) to bound state problems for non-Hermitian potentials, extended to the complex plane in configuration space. The confirmation of this objective enables the generation of converging lower and upper bounds to the discrete state energies, thereby, defining a very accurate computational tool for analyzing such difficult problems. The basic approach adopted here combines several important developments. The first is that one can transform the one dimensional Schrodinger equation, on any complex contour, into a fourth order, linear differential equation for the probability density. This then defines a Nonnegativity Quantization Representation (NQR) in which the physical solution is uniquely associated with a nonnegative, bounded, configuration/solution. For the problems of interest, one can then transform the NQR differential representation into a Moment Equation representation (involving the moments of the physical solution) which assumes the form of a linear recursion relation in which the energy appears as an unknown parameter. Because of the underlying nonnegativity (positivity) of the physical solution, it is then possible to impose the positivity constraints emanating from the famous Moment Problem theorems in mathematics. Its linear programming implementation defines the algorithmic structure of the EMM approach. The generated constraints then serve to define a converging sequence of approximants which define lower and upper bounds for the physical values of the discrete state energies.

#### Recommended Citation

Yan, Zhenyu, "Extension of a spectral bounding method to the PT-invariant states of the - (ix)N non-hermitian potential" (2002). *ETD Collection for AUC Robert W. Woodruff Library*. 677.

http://digitalcommons.auctr.edu/dissertations/677

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