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Baker functions in many dimensions

Alexander Veselov

Department of Mathematical Sciences

Loughborough University

Loughborough Leics LE11 3TU UK and

Landau Institute for Theoretical Physics

Moscow, Russia

Abstract

A notion of a rational Baker-Akhiezer (BA) function related to a configuration of hyperplanes in C

n

was introduced and discussed. It was shown that such a function exists only for very special configurations (so-called locus configurations), which satisfy certain overdetermined algebraic system. Some results towards the classification of the locus configurations have been presented. The important property of the BA functions is that they satisfy some algebraically integrable Schr¨odinger equation. It was shown that the hyperbolic equations with the corresponding potentials satisfy the Huygens’Principle in the narrow Hadamard sense. This relation explains the importance of the BA functions and may lead to the solution of the famous

Hadamard problem in certain classes of the hyperbolic equations. The talk was based on the results of the joint investigations of O.Chalykh, M.Feigin

and the speaker.

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