2 A. V. SOBOLEV

of the operator were considered, and instead of the scalar symbol matrix-valued

symbols were allowed: for Ω =

Rd

and a(x, ξ) = a(ξ) it was shown that

tr g(

˜

T α(a)) =

αdV0

+

αd−1V1

+

o(αd−1),

with some explicitly computable coeﬃcients V0,V1, such that V0 = W0(g(a)) for

the scalar case. Under some mild extra smoothness assumptions on the boundary

∂Λ, R. Roccaforte (see [28]) found the term of order αd−2 in the above asymptotics

of tr g(

˜

T

α

(a)).

The situation changes if we assume that Λ =

Rd

and Ω =

Rd,

i.e. that the

symbol has jump discontinuities in both variables, x and ξ. As conjectured by H.

Widom, in this case the second term should be of order

αd−1

log α, see formula (1.2).

For d = 1 this formula was proved by H. Landau-H. Widom [22] and H. Widom

[41]. For higher dimensions, the asymptotics (1.2) was proved in [42] under the

assumptions that one of the domains is a half-space, and that g is analytic in a

disk of a suﬃciently large radius. After this paper there have been just a few

publications with partial results. Using an abstract version of the Szeg˝ o formula

with a remainder estimate, found by A. Laptev and Yu. Safarov (see [23], [24]),

D. Gioev (see [13, 14]) established a sharp bound

(1.6) tr g(

˜

T α(a)) −

αdW0(g(a))

=

O(αd−1

log α).

In [15] D. Gioev and I.Klich observed a connection between the formula (1.2) and

the behaviour of the entanglement entropy for free Fermions in the ground state.

As explained in [15], the studied entropy is obtained as tr h(

˜

T α) with some bounded

domains Λ, Ω, the symbol a(x, ξ) = 1, and the function

(1.7) h(t) = −t log t − (1 − t) log(1 − t),t ∈ (0, 1).

Since h(0) = h(1) = 0, the leading term, i.e. W0(h(1)), vanishes, and the conjec-

ture (1.2) gives the

αd−1

log α-asymptotics of the trace, which coincides with the

expected quasi-classical behaviour of the entropy. However, the formula (1.2) is not

justified for non-smooth functions, and in particular for the function (1.7). Instead,

in the recent paper [18] R. Helling, H. Leschke and W. Spitzer proved (1.2) for a

quadratic g. With g(t) = t − t2 this gives the asymptotics of the particle num-

ber variance, which provides a lower bound of correct order for the entanglement

entropy.

The operators of the form (1.1) also play a role in Signal Processing. Although

the main object there is band-limited functions of one variable, in [35] D. Slepian

considered some multi-dimensional generalizations. In particular, he derived as-

ymptotic formulas for the eigenvalues and eigenfunctions of

˜

T

α

(1) for the special

case when both Λ and Ω are balls in Rd. Some of those results are used in [36].

These results, however, do not allow to study the trace tr g(

˜

T α(1)).

The main results of the present paper are Theorems 2.3 and 2.4. They establish

asymptotic formulas of the type (1.2) for the operator T (a) = Tα(a), defined in

(2.4), which is slightly different from

˜

T α(a), but as we shall see later, the difference

does not affect the first two terms of the asymptotics (1.2). Theorem 2.3 proves

formula (1.2) for functions g analytic in a disk of suﬃciently large radius. Theorem

2.4 proves (1.2) for the real part of T (a) with an arbitrary

C∞-function

g.

The proof comprises the following main ingredients:

1. Trace class estimates for pseudo-differential operators with discontinuous

symbols,