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

2π

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. B¨ 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