Nonlinear eigenvalue problems and PT-symmetric quantum mechanics

Abstract

Semiclassical (WKB) techniques are commonly used to find the large-energy behavior of the eigenvalues of linear time-independent Schrödinger equations. In this talk we generalize the concept of an eigenvalue problem to nonlinear differential equations. The role of an eigenfunction is now played by a separatrix curve, and the special initial condition that gives rise to the separatrix curve is the eigenvalue. The Painlevé transcendents are examples of nonlinear eigenvalue problems, and semiclassical techniques are devised to calculate the behavior of the large eigenvalues. This behavior is found by reducing the Painlevé equation to the linear Schrödinger equation associated with a non-Hermitian PT-symmetric Hamiltonian. The concept of a nonlinear eigenvalue problem extends far beyond the Painlevé equations to huge classes of nonlinear differential equations.