Quasi-classical asymptotics for pseudodifferential operators with discontinuous symbols: Widom’s conjecture

Author： Sobolev A.V.  

Publisher： Springer Publishing Company

ISSN： 0016-2663

Source： Functional Analysis and Its Applications, Vol.44, Iss.4, 2010-12, pp. : 313-317

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Abstract

In 1982 H. Widom conjectured a multi-dimensional generalization of a well-known two-term quasi-classical asymptotic formula for the trace of the function f(A) of a Wiener—Hopf-type operator A in dimension 1 for a pseudodifferential operator A with symbol a(x, ξ) having jump discontinuities in both variables. In 1990 he proved the conjecture for the special case when the jump in any of the two variables occurs in a hyperplane.This note announces a proof of Widomśs conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces.

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