| Abstract: | As is known, in mathematical physics there are differential operators with constant coefficients whose fundamental solutions
can be constructed explicitly; such operators are said to be exactly solvable. In this paper, the problem of adding lower-order
terms with variable coefficients to exactly solvable operators in such a way that the new operators (deformations) admit constructing
fundamental solutions in explicit form is posed. This problem is directly related to Hadamard’s problem of describing differential
operators satisfying the Huygens’ principle. On the basis of the Fourier method of separation of variables and the method
of gauge-equivalent operators, an effective method for finding exactly solvable deformations depending on one variable is
constructed. An application of such deformations to constructing Huygens’ differential operators associated with the cone
of real symmetric positive-definite matrices is suggested.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal
Conference-2004, Part 3, 2006. |
