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,
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