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

TL;DR

This paper proves Widom's conjecture on the asymptotic behavior of pseudo-differential operators with discontinuous symbols in multiple dimensions, extending previous results to more general jump discontinuities on smooth surfaces.

Contribution

It provides a proof of Widom's conjecture for symbols with jumps on arbitrary smooth surfaces, broadening the class of operators for which asymptotic formulas are known.

Findings

Established the multi-dimensional asymptotic formula for operators with discontinuous symbols.

Extended Widom's conjecture to arbitrary smooth surfaces, not just hyperplanes.

Confirmed the conjecture under more general conditions on symbol discontinuities.

Abstract

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.