Multidimensional Baker-Akhiezer functions and Huygens' Principle

TL;DR

This paper introduces rational Baker-Akhiezer functions linked to hyperplane configurations, showing their existence only for special locus configurations, which relate to algebraically integrable Schrödinger equations and Huygens' Principle.

Contribution

It establishes the existence criteria for Baker-Akhiezer functions for locus configurations and connects these to hyperbolic equations satisfying Huygens' Principle, providing explicit solutions.

Findings

BA functions exist only for special locus configurations

Locus configurations determine algebraically integrable Schrödinger equations

Explicit fundamental solutions can be constructed from BA functions

Abstract

A notion of rational Baker-Akhiezer (BA) function related to a configuration of hyperplanes in C^n is introduced. It is proved that BA function exists only for very special configurations (locus configurations), which satisfy certain overdetermined algebraic system. The BA functions satisfy some algebraically integrable Schrodinger equations, so any locus configuration determines such an equation. Some results towards the classification of all locus configurations are presented. This theory is applied to the famous Hadamard's problem of description of all hyperbolic equations satisfying Huygens' Principle. We show that in a certain class all such equations are related to locus configurations and the corresponding fundamental solutions can be constructed explicitly from the BA functions.