Chapter

Spectral Theory, Function Spaces and Inequalities

Volume 219 of the series Operator Theory: Advances and Applications pp 211-231

Date:

A Szegő Limit Theorem for Operators with Discontinuous Symbols in Higher Dimensions: Widom’s Conjecture

  • Alexander V. SobolevAffiliated withDepartment of Mathematics, University College London Email author 

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Abstract

Relying on the known two-term asymptotic formula for the trace of the function f(A) of a truncated Wiener–Hopf-type operator A in dimension 1, in 1982 H. Widom conjectured a multi-dimensional generalisation of that formula for a pseudo-differential operator A with a 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 on a hyperplane. The present paper outlines a proof of Widom’s Conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces.

Keyword

Pseudo-differential operators with discontinuous symbols quasi-classical asymptotics Szegő formula