2. MAIN RESULT 9

Remark 2.5. Let us make some comments on the integral (1.5), which enters

the coeﬃcient W1 in the above theorems. Note that the integral (1.5) is finite

for any smooth function g such that g(0) = 0 (see Lemmas 2.10, 2.11 for the

estimates). Clearly, A is linear in g, i.e. A(g + ˜; g b) = A(g; b) + A(˜; g b). For the

function gp(t) =

tp,

p = 1, 2,... , we have A(g1; b) = 0 and A(gp, ; b) =

bpA(gp).

Remark 2.6. It would be natural to expect that the variables x and ξ in

the operator Tα(a) have “equal rights”. Indeed, it was shown in [42], p. 173,

by an elementary calculation, that the roles of x, ξ are interchangeable. On the

other hand, the conditions on Λ and Ω in the main theorems above, are clearly

asymmetric. At present it is not clear how to rectify this drawback.

Remark 2.7. As mentioned in the Introduction, the study of the traces of

the form tr g(Tα) goes back to the original Szeg˝ o formula for the determinant of

the truncated Toeplitz matrix. The continuous analogue of the Szeg˝ o Theorem

is known as the Ahiezer-Kac formula, see e.g. [6] and [11] for references, and it

describes the asymptotic behaviour of the determinant det(I + Tα). Due to the

formula log det(I + Tα) = tr log(1 + Tα), under the condition that the symbol a is

suﬃciently small, Theorem 2.3 with g(t) = log(1 + t) leads to a multi-dimensional

generalization of the Ahiezer-Kac formula. Since the symbol in Tα(a) has jump

discontinuities, these asymptotics can be interpreted as a multi-dimensional variant

of the Hartwig-Fisher formula, see e.g., [9], [10]. As explained in [10], [11] and [9]

for the classical truncated Toeplitz matrices, apart from the jump discontinuities

there are other types of Fisher-Hartwig singularities: isolated roots of the symbol,

and points where the symbol tends to infinity in a power-like fashion. In the

presence of such singularities the asymptotic formula for the Toeplitz determinant

is known to acquire an extra factor. It would be natural to expect a similar effect

in the Ahiezer-Kac formula, both in the one- and multi-dimensional situations.

However, Theorem 2.3 with g(t) = log(1 + t) does not cover this case, since a is a

bounded function, and the smallness of a guarantees that 1 + a(x, ξ) has no roots.

Remark 2.8. Denote by n(λ1,λ2; α) with λ1λ2 0,λ1 λ2 the number of

eigenvalues of the operator Sα(a) which are greater than λ1 and less than λ2. In

other words,

n(λ1,λ2; α) = tr χI (Sα(a)), I = (λ1,λ2).

Since the interval I does not contain the point 0, this quantity is finite. Theorem

2.4 can be used to find the leading term of the asymptotics of the counting function

n(λ1,λ2; α), by approximating the characteristic function χI with smooth functions

g. Suppose for instance that a− a(x, ξ) a+, x ∈ Λ, ξ ∈ Ω, with some positive

constants a−,a+, and that [a−,a+] ⊂ I. Then it follows from Theorem 2.4 that

n(λ1,λ2; α) =

α

2π

d

|Λ| |Ω| +

αd−1

log α W1

(

A(χI ; a)

)

+

o(αd−1

log α).

A straightforward calculation shows that

A

(

χI ; a(x, ξ)

)

=

1

(2π)2

log

a(x, ξ)

λ1

− 1 .

Another interesting case is when [λ1,λ2] ⊂ (0,a−). This guarantees that W0(χI (a))

= 0, and the asymptotics of n(λ1,λ2; α) are described by the second term in (2.22):

n(λ1,λ2; α) =

αd−1

log α W1

(

A(χI ; a)

)

+

o(αd−1

log α), α → ∞.