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.
Highlights
By extending conventional physical theories into the complex domain we can understand and tame instabilities: WHY ARE INSTABILITIES TAMED? If you extend the real numbers to the complex numbers, you lose the ordering property of the real numbers
Second arbitrary constant is invisible because it is contained in the subdominant solution: Physical solution is Unstable under small changes in E
For large n the nth eigenvalue grows like the square root of n times a constant A, and we used Richardson extrapolation to show that
Summary
Rev. D 74, 025016 (2006) [arXiv: hep-th/0605066]. (on the real axis) How can it possibly have bound states? This upside-down potential is unstable! (on the real axis) How can it possibly have bound states?
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