CHAPTER 1 Introduction For two domains Λ, Ω Rd, d 1 consider in L2(Rd) the operator defined by the formula (1.1) ( ˜ α (a)u)(x) = α d χΛ(x) Ω Rd eiαξ·(x−y)a(x, ξ)χΛ(y)u(y)dydξ, α 0, for any Schwartz class function u, where χΛ( · ) denotes the characteristic function of Λ, and a( · , · ) is a smooth function, with an appropriate decay in both variables. Clearly, ˜ α is a pseudo-differential operator with a symbol having discontinuities in both variables. We are interested in the asymptotics of the trace tr g( ˜ α ) as α with a smooth function g such that g(0) = 0. In 1982 H. Widom in [41] conjectured the asymptotic formula (1.2) tr g ( ˜ α (a) ) = αd W0 ( g(a) Λ, Ω ) + αd−1 log α W1 ( A(g a) ∂Λ,∂Ω ) + o(αd−1 log α), with the following coefficients. For any symbol b = b(x, ξ), any domains Λ, Ω and any C1-surfaces S, P , let (1.3) W0(b) = W0(b Λ, Ω) = 1 (2π)d Λ Ω b(x, ξ)dξdx, (1.4) W1(b) = W1(b S, P ) = 1 (2π)d−1 S P b(x, ξ)|nS(x) · nP (ξ)|dSξdSx, where nS(x) and nP (ξ) denote the exterior unit normals to S and P at the points x and ξ respectively, and (1.5) A(g b) = 1 (2π)2 1 0 g(bt) tg(b) t(1 t) dt, A(g) := A(g 1). The main objective of the paper is to prove the formula (1.2) for a large class of functions g and bounded domains Λ, Ω. The interest in the pseudo-differential operators with discontinuous symbols goes back to the classical Szeg˝ o formula for the determinant of the truncated Toeplitz matrix, see [37] and [17]. There exists a vast body of literature devoted to various non-trivial generalizations of the Szeg˝ o formula in dimension d = 1, and it is not our intention to review them here. Instead, we refer to the monographs by A. ottcher-B. Silbermann [5], and by N.K. Nikolski [27] for the background read- ing, T. Ehrhardt’s paper [10] for a review of the pre-2001 results, and the recent paper by P. Deift, A. Its, I. Krasovsky [9], for the latest results and references. A multidimensional generalization of the continuous variant of the Szeg˝ o formula was obtained by I.J. Linnik [26] and H. Widom [39], [40]. In fact, paper [40] addressed a more general problem: instead of the determinant, suitable analytic functions 1
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