Date of Award

Spring 5-18-2018

Author's School

Graduate School of Arts and Sciences

Author's Department


Degree Name

Master of Arts (AM/MA)

Degree Type



Every function possesses an inherent topological property: the number of times it links about the x-axis in the three-dimensional space of its complex solution range crossed with its real domain. The up-and-down oscillations of entirely real-valued functions are a degenerate signature of this winding.

Hermitian and unbroken PT-symmetric Schrodinger equations possess eigenfunctions with winding numbers that are well-ordered with respect to their eigenvalue number. As a system passes through PT-symmetry-breaking singular points, this order breaks down in a characteristic manner. Non-Hermitian systems lacking symmetries do not exhibit well-defined winding order.

It is possible to map the relationship between an initial-value or parameter space of a differential-equation system to the winding numbers of the solutions to which each parameter value gives rise. This topological structure aids in the understanding of certain nonlinear and partial differential equations.


English (en)

Chair and Committee

Carl Bender

Committee Members

Michael Ogilvie, Zohar Nussinov


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