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

|Λ| |Ω| +
log α W1
A(χI ; a)
log α).
A straightforward calculation shows that
χI ; a(x, ξ)
a(x, ξ)
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) =
log α W1
A(χI ; a)
log α), α ∞.
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