Gulden, Tobias2016-10-252016-10-252016-08https://hdl.handle.net/11299/182831University of Minnesota Ph.D. dissertation. August 2016. Major: Physics. Advisor: Alex Kamenev. 1 computer file (PDF); vi, 83 pages.Increased interest in non-Hermitian quantum systems calls for the development of efficient methods to treat these. This interest was sparked by the introduction of PT-symmetry and the study of mathematical mappings which map conventional statistical or quantum mechanics onto non-Hermitian quantum operators. One of the most common methods in quantum mechanics is the semiclassial approximation which requires integration along trajectories that solve classical equations of motion. However in non-Hermitian systems these solutions are rarely attainable. We borrow concepts from algebraic topology to develop methods to avoid solving the equations of motion and avoid straightforward integration altogether. We apply these methods to solve the semiclassical problem for three largely dierent systems and demonstrate their usefulness for Hermitian and non-Hermitian systems alike.enAlgebraic topologyComplex manifoldsNon-Hermitian quantum mechanicsSemiclassical physicsStatistical physicsThesis or Dissertation