CHAPTER 1
Introduction
For two domains Λ, Ω Rd, d 1 consider in
L2(Rd)
the operator defined by
the formula
(1.1) (
˜
T 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,
˜
T
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(
˜
T
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
(
˜
T 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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