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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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/

[1] A. Böttcher, B. Silbermann, Analysis of Toeplitz operators, Springer Monographs in Math., Springer-Verlag, Berlin, 2006 | MR

[2] P. Deift, A. Its, I. Krasovsky, arXiv: 0905.0443v2

[3] T. Ehrhardt, Recent Advances in Operator Theory (Groningen, 1998), Oper. Theory Adv. Appl., 124, Birkhäuser, Basel, 2001, 217–241 | MR | Zbl

[4] D. Gioev, Generalizations of Szegő Limit Theorem: Higher Order Terms and Discontinuous Symbols, PhD Thesis, Dept. of Math., Royal Inst. of Technology (KTH), Stockholm, 2001

[5] D. Gioev, Int. Math. Res. Not., 2006, Art. ID 95181 | MR | Zbl

[6] D. Gioev, I. Klich, Phys. Rev. Lett., 96:10 (2006), id. 100503 | DOI | MR

[7] U. Grenander, G. Szegő, Toeplitz Forms and Their Applications, Univ. of California Press, Berkeley–Los Angeles, 1958 | MR | Zbl

[8] R. C. Helling, H. Leschke, W. L. Spitzer, “A special case of a conjecture by Widom with implications to fermionic entanglement entropy”, Int. Math. Res. Not., 2010 | DOI | MR

[9] A. Laptev, Yu. Safarov, Contemporary Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, 175, Amer. Math. Soc., Providence, RI, 1996, 69–79 | MR | Zbl

[10] A. Laptev, Yu. Safarov, J. Funct. Anal., 138:2 (1996), 544–559 | DOI | MR | Zbl

[11] I. Yu. Linnik, Izv. AN SSSR, ser. matem., 39:6 (1975), 1393–1403 | MR | Zbl

[12] N. K. Nikolski, Operators, Functions, and Systems: an Easy Reading. Vol. 1. Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, 92, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl

[13] R. Roccaforte, Trans. Amer. Math. Soc., 285:2 (1984), 581–602 | DOI | MR | Zbl

[14] D. Slepian, Bell System Tech. J., 43 (1964), 3009–3057 | DOI | MR | Zbl

[15] A. V. Sobolev, arXiv: 1004.2576v1

[16] G. Szegő, Meddel. Lunds Univ. Mat. Sem., 1952, Suppl. vol. M. Riesz, 228–238 | MR | Zbl

[17] H. Widom, Trans. Amer. Math. Soc., 94:1 (1960), 170–180 | MR | Zbl

[18] H. Widom, J. Funct. Anal., 39 (1980), 182–198 | DOI | MR | Zbl

[19] H. Widom, Toeplitz Centennial (Tel Aviv, 1981), Oper. Theory Adv. Appl., 4, Birkhäuser, Basel–Boston, Mass., 1982, 477–500 | MR

[20] H. Widom, J. Funct. Anal., 88:1 (1990), 166–193 | DOI | MR | Zbl