Geodesic
Quasi-Classical Asymptotics for Pseudodifferential Operators with Discontinuous Symbols: Widom's Conjecture
Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 4, pp. 86-90

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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 $a(\mathbf x,\boldsymbol\xi)$ 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 Widom's conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces.
Keywords: pseudodifferential operators with discontinuous symbols, quasi-classical asymptotics, Szegö formula. 
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     author = {A. V. Sobolev},
     title = {Quasi-Classical {Asymptotics} for {Pseudodifferential} {Operators} with {Discontinuous} {Symbols:} {Widom's} {Conjecture}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {86--90},
     publisher = {mathdoc},
     volume = {44},
     number = {4},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2010_44_4_a7/}
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A. V. Sobolev. Quasi-Classical Asymptotics for Pseudodifferential Operators with Discontinuous Symbols: Widom's Conjecture. Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 4, pp. 86-90. http://geodesic.mathdoc.fr/item/FAA_2010_44_4_a7/