Now make use of the theorem proved in [20] for abstract operators. At first, introduce the following notations.

Let *A*_{0} be a self-adjoint positive discrete operator, {*λ*_{
n
}} be its eigenvalues arranged in ascending order, {*φ*_{
j
}} be a basis formed by the eigenvectors of *A*_{0}, *B* be a perturbation operator, and {*μ*_{
n
}} be the eigenvalues of *A*_{0} + *B*. Also, assume that
. For operators *A*_{0} and *B* in [[20], Theorem 1], the following theorem is proved.

**Theorem 3.1**.

*Let the operator B be such that D*(

*A*_{0}) ⊂

*D*(

*B*),

*and let there exist a number δ* ∈ [0, 1)

*such that*
*has a bounded extension, and number ω*∈ [0, 1),

*ω* +

*δ <* 1

*such that*
*is a trace class operator. Then, there exists a subsequence of natural numbers*
*and a subsequence of contours* Γ

_{
m
} ∈

*C, that for ω* ≥

*δ the formula**is true*.

Note that the conditions of this theorem are satisfied for *L*_{0} and *L*. That is, if we take *A*_{0} = *L*_{0}, *B* = *Q*, then
is bounded. For
and *α >* 2, from asymptotic (2.16), we will have that
is a trace class operator. If *α <* 2, then
will be a trace class operator for
.

Thus, by the statement of Theorem 3.1, for

*α >* 2, we have

where *ψ*_{1}(*x*), *ψ*_{2}(*x*),... are the orthonormal eigenvectors of *L*_{0}.

Introduce the following notation:

and investigate the sum of series
, which as will be seen later, is independent of the choice of
. We will call the sum of this series a regularized trace of the operator *L*_{0}.

Now, we calculate the norm for the eigen-vectors of the operator

*L*_{0} in

**L**_{2}. To do this, we will use the following identity obtained from the Bessel equation"

We also consider the following identities:

By the above identities and also by the equation

satisfied by

*x*_{m, k}, we obtain

So, the orthonormal eigen-vectors of

*L*_{0} are

Now, we prove the following lemma.

**Lemma 3.1**.

*If the operator function q*(

*t*)

*has properties 1, 2, and also α >* 0,

*then***Proof**. Assume that *f*_{
k
}(*t*) = (*q*(*t*) *φ*_{
k
}, *φ*_{
k
}). By Lemma 2.1 we have
.

So, in virtue of the inequality

[[

35], p. 666] and properties 1 and 2 we have

To estimate the second series in (3.5), we use the relation
.

By hypothesis of Lemma 3.1

*α >* 0. Therefore, denoting this sum by

*s*, we have

This proves Lemma 3.1.

for small *δ >* 0.

Then, we can state the following theorem.

**Theorem 3.2**.

*Let the conditions of Theorem 2.1, (3.6) and (3.7) hold. If the operator-value function q*(

*t*)

*has properties 1-3, then the following formula is true***Proof**. By virtue of lemma 3.1 we have

At first evaluate the inner sum in the second term on the right hand side of (3.9). To do this, as

*N* → ∞ investigate the asymptotic behavior of the function

To derive a formula for *R*_{
N
}(*t*), show for each fixed value of *k*, the *m* th term of the sum *R*_{
N
}(*t*) as a residue at the point *x*_{m, k}of some complex variable function with poles at
.

For this purpose, consider the following function:

The poles of this function are

*x*_{0,k,...,}*x*_{N - 1,k}and

*j*_{1},...,

*j*_{
N
} (

*J*_{
ν
}(

*j*_{
n
}) = 0). The residue at

*j*_{
n
} equals

Now, compute the residue at

*x*_{m, k}:

Denote the right hand side of (3.10) by

*G*(

*z*). Since

*x*_{m, k}satisfies equation (

2.4), by setting

*z* =

*x*_{m, k}and using the identity

Consider the contour *C* mentioned in Lemma 2.3 as the contour of integration. According to Lemmas 2.1 and 2.3, for a sufficiently large *N*, we have *x*_{N - 1,k}< *A*_{
N
}< *x*_{N, k}and *j*_{
N
}< *A*_{
N
}< *j*_{N+1}.

It could easily be verified that in the vicinity of zero, the function *g*(*z*) is of order *O* (*z*^{
ν
}). By virtue of this asymptotic and because *g*(*z*) is an odd function, the integral along the left-hand side of the contour *C* vanishes when *r* (radius of a semicircle) goes to zero.

Furthermore, if

*z* =

*u* +

*iv*, then for large |

*v*| and

*u* ≥ 0, the integrand will be of order

*O* (

*e*^{|v|(2t- 2)}). That is, for a given value of

*A*_{
N
}, the integrals along the upper and lower sides of

*C* go to zero as

*B* → ∞ (0

*< t <* 1). Thus, we obtain

Also, along the contour

*C* for

,

, we have |

*tz*| → ∞. Therefore, in integral (3.11), we could replace the Bessel functions by their asymptotic at large arguments. Hence, from

Denote the right side of (3.12) by

*J*:

Then the limit of (3.11) becomes:

Using (3.6) and (3.13), we obtain

Moreover, if (3.7) holds, then by virtue of the known relation for a large

*N* [[

35], p. 642]

Using property 2 and the asymptotic of

*x*_{m, k}Earlier it was obtained that under the assumptions 1-3 (see [[

7], Theorem 2.2])

Thus, from (3.14) to (3.18), we have

In a similar way to the one considered above, we get (this time Equation

2.5 has no imaginary roots, so the contour

*C* will only bypass the origin on the right half-plane):

Finally, combining (3.19) and (3.20), we get

which completes the proof.

**Remark**. It should be noted that in condition 1, property

*q*^{(l)}(

*t*) ∈

*σ*_{1},

may be weakened. Namely, we may just require to hold

Then formula (3.8) takes the form
. There exist the bounded functions that are not from the trace class, even compact, but satisfy the above stated condition. Now, introduce an example.

**Example**. We consider the following boundary value problem:

in the cylinder ∂Ω × [0, 1], where Ω is a circle in

*R*^{2} ((

*x*,

*y*) ∈

*R*^{2}) of radius 1. Also, ∂Ω is a circumference of this circle,

*n* is an exterior normal to the surface ∂Ω × [0, 1] and

*h* = const.. Looking for the solution of this problem, which can be represented as

*u*(

*x*,

*y*,

*z*,

*t*) =

*U*(

*x*,

*y*,

*z*)

*T*(

*t*), we have

Thus, the left-hand side of this equality depends only on

*t*, while the right-hand side on

*x*,

*y*,

*z*. This means they are equal to some constant which we will denote by -

*λ*. Therefore,

Using the cylindric coordinates

*x* =

*r* cos

*φ*,

*y* =

*r* sin

*φ*,

*z* =

*z*, we have

The boundary condition in (3.24) becomes

We will solve this problem by separation of variables. Taking

*U*(

*r*,

*φ*,

*z*) =

*V* (

*r*,

*z*)ϕ(

*φ*),

*q*(

*r*,

*φ*,

*z*) =

*Q*(

*r*,

*z*), and

,

*ν* = const., we get

By making

substitution, we get

and (3.25), (3.23) take the form:

where

*Q*(

*r*,

*z*) is a real-valued function which is continuous on [0, 1] × [0, 1], and has second partial derivative with respect to

*r* on [0, 1] for each fixed z. Fourier series of this function and its partial derivatives converge, respectively, to their values. Also assume that

Now, rewrite the problem in the differential operator form:

where

*v*(

*r*) =

*V*_{1}(

*r*, ·) is a vector function with the values from

*L*_{2}(0, 1). Operators

*A* and

*q*(

*r*) are defined in the following way:

Obviously, the operator

*A* is self-adjoint, positive-definite, and

*A*^{-1} is a compact operator in

*L*_{2}(0, 1). Also, the eigenvalues of

*A* are of the form:

Then, by virtue of Theorem 2.1, the eigenvalues of this problem behave like
.

Using the statement of Theorem 3.2, we have

where

*μ*_{
i
} are the eigenvalues of problem (3.28) with

*q*(

*r*) ≡ 0. Now calculate

In a similar way, we can find

The authors declare that they have no competing interests