Date of Award

Summer 8-15-2018

Author's School

Graduate School of Arts and Sciences

Author's Department


Degree Name

Doctor of Philosophy (PhD)

Degree Type



Space-time reflection symmetry, or PT symmetry, first proposed in quantum mechanics by Bender and Boettcher in 1998 [2], has become an active research area in fundamental physics. This dissertation contains several research problems which are more or less related to this field of study. After an introduction on complementary topics for the main projects in Chap.1, we discuss about an idea which is originated from the remarkable paper by Chandrasekar et al in Chap.2. They showed that the (second-order constant-coefficient) classical equation of motion for a damped harmonic oscillator can be derived from a Hamiltonian having one degree of freedom. We gives a simple derivation of their result and generalizes it to the case of an nth-order constant-coefficient differential equation.

In Chap.3 we studied the analytical continuation of the coupling constant g of a coupled quantum theory. We get to this conclusion that one can, at least in principle, arrive at a state whose energy is lower than the ground state of the theory. The idea is to begin with the uncoupled g = 0 theory in its ground state, to analytically continue around an exceptional point (square-root singularity) in the complex-coupling-constant plane, and finally to return to the point g = 0. In the course of this analytic continuation, the uncoupled theory ends up in an unconventional state whose energy is lower than the original ground-state energy. However, it is unclear whether one can use this analytic continuation to extract energy from the conventional vacuum state; this process appears to be exothermic but one must do work to vary the coupling constant g.

PT-symmetric quantum mechanics began with a study of the Hamiltonian H=p^2+x^2(ix)^\epsilon. When \epsilon\geq 0, this portion of parameter space is known as the region of unbroken PT symmetry. The region of unbroken PT symmetry has been studied but the region of broken PT symmetry which is related to the negative epsilon has thus far been unexplored. In Chap.4 we present a detailed numerical and analytical examination of the behavior of the eigenvalues for _4<\epsilon<0. In particular, it reports the discovery of an infinite-order exceptional point at \epsilon =_1, a transition from a discrete spectrum to a partially continuous spectrum at \epsilon=_2, a transition at the Coulomb value \epsilon =_3, and the behavior of the eigenvalues as \epsilon approaches the conformal limit \epsilon =_4.

Finally in Chap.5 we devised a simple and accurate numerical technique for finding eigenvalues, node structure, and expectation values of PT-symmetric potentials. The approach involves expanding the solution to the Schrdinger equation in series involving powers of both the coordinate and the energy. The technique is designed to allow one to impose boundary conditions in PT-symmetric pairs of Stokes sectors. The method is illustrated by using many examples of PT-symmetric potentials in both the unbroken- and broken-PT-symmetric regions.


English (en)

Chair and Committee

Carl M. Bender

Committee Members

Mark Alford, Machael Ogilvie, Jung-Tsung Shen, Erik Henriksen,


Permanent URL: 2018-08-15