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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