Multidimensional Baker–Akhiezer Functions and Huygens' Principle

Abstract:

A notion of the rational Baker–Akhiezer (BA) function related to a configuration of hyperplanes in C ⁿ is introduced. It is proved that the BA function exists only for very special configurations (locus configurations), which satisfy a certain overdetermined algebraic system. The BA functions satisfy some algebraically integrable Schrödinger 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 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.