Vincent Moncrief: Euclidean-signature semi-classical asymptotic methods for Schrödinger operators

Microlocal analysts have long realized that the classical (physics textbook) W.K.B. ansatz for Schrödinger eigenfunctions can be significantly enhanced by posing, instead, the (non-standard) Euclidean-signature ansatz \(\psi=\varphi e^{-S/\hbar}\). To leading order in a (formal) expansion in powers of \(\hbar\), \(S=S_{(0)}+\hbar S_{(1)}+\frac{\hbar^{2}}{2}S_{(2)}+\dots,\) \(S_{(0)}\) is found to satisfy the inverted-potential-vanishing-energy Hamilton Jacobi equation,

\(\frac{1}{2m}\nabla S_{(0)}\cdot\nabla S_{(0)}-V=0 \),

in place of the conventional Hamilton-Jacobi equation

\(\frac{1}{2m}\nabla S_{(0)}\cdot\nabla S_{(0)}+V=E\)

of the W.K.B. approach. Given an appropriate solution \(S_{(0)}\) one can proceed to compute higher order quantum corrections to the eigenfunctions and eigenvalues through the integration of a sequence of linear 'transport' equations along the 'flow' generated by \(S_{(0)}\). Together with A. Marini (Yeshiva) and R. Maitra (Wentworth I. T.) I have been developing different techniques for solving this Euclidean-signature Hamilton-Jacobi equation that, in contrast to the original microlocal ones, can even be applied to a class of (bosonic) field theories and thus used to further extend the scope of these elegant microlocal techniques. I'll present the basic ideas first in a quantum mechanical context and then finally discuss ongoing research on the application of these methods to quantum field theory.