A new look at the Helmholtz equation: Lefschetz thimbles and the einbein action

Abstract

Picard-Lefschetz theory is applied to solutions of the Helmholtz equation, formulated in terms of sums of integrals of a proper-time, or `einbein', wave function $\Psi(\Lambda) = \exp(i\mathbb S(\Lambda))$ along complex contours bounded by essential singularities of $\Psi$. There is a one to one map between steepest descent paths connecting essential singularities and real or complex eigenrays. Residues of finite $\Lambda$ poles of $\mathbb S(\Lambda)$ are shown to vanish at spatial points corresponding to sources, provided that the pole bounds only one steepest descent path. If the sum includes two such paths, with one beginning and the other ending at the same pole, points of vanishing residue are not sources, but are argued to be the locus on which caustic curves may have singularities such as cusp points. The map between $\mathbb S$ and the generating function in the Thom--Arnold classification of catastrophes is discussed. Monodromies of the solution set with respect to complexified parameters defining the index of refraction, or spatial endpoints of Green's functions, are trivially determined from the singularities of $\mathbb S(\Lambda)$. We construct a variant of a Laurent series expansion of $\mathbb S$ about a pole at finite $\Lambda$. Expressions for the coefficients of each order in this expansion can often be given exactly. Based on the Laurent series expansion, we propose a variation of a Pad\'e approximant for $\mathbb S$, with the intent of capturing additional poles and the associated cusp caustics which are not visible in the Laurent series expansion.