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Quasi-classical asymptotics for pseudo-differential operators with discontinuous symbols: Widom's Conjecture

Relying on the known two-term quasiclassical asymptotic formula for the trace of the function $f(A)$ of a Wiener-Hopf type operator $A$ in dimension one, in 1982 H. Widom conjectured a multi-dimensional generalization of that formula for a pseudo-differential operator $A$ with a symbol $a(\bx, \bxi)$ 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 on a hyperplane. The present paper gives a proof of Widom's Conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces.