A semiclassical theory for non-Hermitian operators

Increased interest in non-Hermitian quantum systems calls for the development of efficient mathematical methods to treat these. This interest was sparked by the study of mathematical mappings which map conventional statistical or quantum mechanics onto non-Hermitian quantum operators, and the introduction of PT-symmetry. In this talk I will focus on one of the most common methods in quantum mechanics, the semiclassial approximation. It requires integration along trajectories that solve classical equations of motion. However in non-Hermitian systems these solutions are rarely attainable. By studying Picard-Fuchs equations and monodromy, two concepts from algebraic topology, we circumvent solving the equations of motion and straightforward integration. I present the main ideas by means of two largely different systems, and demonstrate their usefulness for Hermitian and non-Hermitian Hamiltonians alike.