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Study of the asymptotic eigenvalue distribution and trace formula of a second order operatordifferential equation
Boundary Value Problems volume 2011, Article number: 7 (2011)
Abstract
The purpose of writing this article is to show some spectral properties of the Bessel operator equation, with spectral parameterdependent boundary condition. This problem arises upon separation of variables in heat or wave equations, when one of the boundary conditions contains partial derivative with respect to time. To illustrate the problem and the proof in detail, as a first step, the corresponding operator's discreteness of the spectrum is proved. Then, the nature of the eigenvalue distribution is established. Finally, based on these results, a regularized trace formula for the eigenvalues is obtained.
MSC: 34B05; 34G20; 34L20; 34L05; 47A05; 47A10.
Introduction
Let L_{2} = L_{2} (H, [0, 1]) ⊕ H, where H is a separable Hilbert space with a scalar product (·, ·) and a norm · inside of it. By definition, a scalar product in L_{2} is
where Y = {y (t), y_{1}}, Z = {z (t), z_{1}} and y(t), z(t) ∈ L_{2} (H, [0, 1]) for which L_{2} (H, [0, 1]) is a space of vector functions y(t) such that .
Now, consider the equation:
in L_{2} (H, [0, 1]), where A is a selfadjoint positivedefinite operator in H which has a compact inverse operator. Further, suppose the operatorvalued function q(t) is weakly measurable, and q(t) is bounded on [0, 1] with the following properties:

1.
q(t) has a secondorder weak derivative on [0, 1], and q^{(l)}(t) (l = 0, 1, 2) are selfadjoint operators in H for each t ∈ [0, 1], [q(^{l)}(t)]* = q^{(l)}(t), q^{(l)}(t) ∈ σ_{1}(H). Here σ_{1}(H) is a trace class, i.e., a class of compact operators in separable Hilbert space H, whose singular values form a convergent series (denoting the compact operator by B, then its singular values are the eigenvalues of ). If {φ_{ n }} is a basis formed by the orthonormal eigenvectors of B, then . For simplicity, denote the norm in σ_{1}(H) by ·_{1}.

2.
The functions q^{(l)}(t)_{1} (l = 0, 1, 2) are bounded on [0, 1].

3.
The relation is true for each f ∈ H.
State that if q(t) ≡ 0, a selfadjoint operator denoted by L_{0} can be associated with problem (2), (3) whose definition will be given later.
If q(t) ≢ 0, the operators L and Q are defined by L = L_{0} + Q, and Q : Q {y (t), y_{1}} = {q(t) y(t), 0} which is a bounded selfadjoint operator in L_{2}.
After the above definitions and the assumptions, the asymptotic of the eigenvalue distribution and regularized trace of the considered problem will be studied. It is clear that because of the appearance of an eigenvalue parameter in the boundary condition at the end point, the operator associated with problem (2), (3) in L_{2} (H, [0, 1]) is not selfadjoint. Introduce a new Hilbert space L_{2} (H, [0, 1]) ⊕ H with the scalar product defined by formula (1) similar to one used in [1]. Then, in this space, the operator becomes selfadjoint.
In [2], Walter considers a scalar SturmLiouville problem with an eigenvalue parameter λ in the boundary conditions. He shows that one can associate a selfadjoint operator with that by finding a suitable Hilbert space. Further, he obtains the expansion theorem by reference to the selfadjointness of that operator. His approach was used by Fulton in [3] later on.
As for the differential operator equations, to the best of this author's knowledge in the articles [1, 4–6], an eigenvalue parameter appears in the boundary conditions. In [4], the following problem is considered:
where A = A* > E, and u(x) ∈ L_{2} (H, (0, b)). It is proved that the operator associated with this problem has a discrete spectrum, iff : A has a discrete spectrum. The eigenvalues of this problem form two sequences like and where n, k ∈ N, and μ_{ k } is an eigenvalue of A. This is obtained from appearance of λ in the boundary condition.
In [5], both boundary conditions depend on λ. It is shown that the operator defined in the space L_{2} (H, (0, 1)) ⊕ H ⊕ H is symmetric positivedefinite. Further, the asymptotic formulas for eigenvalues are obtained.
In this author's previous study [6], for the operator considered in [4], the trace formula has been established.
If h = 0 in (3), then the boundary condition takes the form y(1) = 0. This problem is considered in [[7], Theorem 2.2], where the trace formula is established. It is proved that there exists a subsequence of natural numbers {n_{ m }} such that , where μ_{ n } and λ_{ n } are the eigenvalues of perturbed and nonperturbed operators. For definition of {n_{ m }}, see also [[8], Lemma 1].
For a scalar case, please refer to [9], where the following problem
is considered on the interval [0, π]. Then, the sum is calculated.
In comparison with the above mentioned articles, here we consider a differential operator equation which has a singularity at 0, and the boundary condition at 1 involves both the eigenvalue parameter λ and physical parameter h < 0.
Problems with λdependent boundary conditions arise upon separation of variables in the heat and wave equations. We can also refer to [10–17], where boundaryvalue problems for ordinary differential operators with eigenvaluedependent boundary conditions are studied.
In 1953, Gelfand and Levitan [18] considered the SturmLiouville operator
and derived the formula , where μ_{ n } are the eigenvalues of the above operator. For q (x) ≡ 0 the eigenvalues of the operator are given by λ_{ n } = n^{2}.
It is worthwhile to note that, several studies are devoted to searching a regularized trace for the concrete operators (e.g., [9–18]), as well as differentialoperator equations (e.g., [6–8, 19]) and discrete abstract operators (e.g., [20–22]). For further detailed discussion of the subject, please refer to [23].
Trace formulas are used for the approximation of the first eigenvalues of the operators [24, 25] to solve inverse problems [26, 27]. They are also applied to index theory of linear operators [28, 29].
To summarize this study, in Section 1, it is proved that the operator associated with (2), (3) is selfadjoint and has a discrete spectrum. In Section 2, we establish an asymptotic formula for the eigenvalues. To do this, the zeros of the characteristic equation (Lemmas 2.1, 2.2, 2.3) are searched in detail. In Section 3, by using the asymptotic for the eigenvalues, we prove that the series called "a regularized trace" converges absolutely (Lemma 3.1). This enables us to arrange the terms of the series in a suitable way for calculation as in (3.9). To calculate the sum of this series, we introduce a function whose poles are zeros of the characteristic equation, the residues at poles of which are the terms of our series. Finally, we establish a trace formula by integrating this function along the expanded contours.
In conclusion, we apply the results of our study to a boundary value problem generated by a partial differential equation.
1 Definition of L_{0}and proof of discreteness of the spectrum
Let , where is a set of vector functions with values in H (see [30], p. 57) that vanish in the vicinity of zero and are infinitely differentiable in the norm of H. Also, on define the operator :
Using integration by parts it is easy to see that is symmetric. Denote its closure by L_{0} and show that it is selfadjoint. To do that, consider the adjoint operator of as . By definition, vector if for each it holds
and Z* = {z* (t), z*} ∈ L_{2}. However, using integration by parts from (1.1), it is obvious that with and l[z] ∈ L_{2} (H, [0, 1])}. In other words, z(t) has a firstorder derivative on [0, 1] which is absolutely continuous in the norm of H and z (0) = z'(0) = 0, Az(t) ∈ L_{2} (H, [0, 1]) and .
Now, the vector if and only if for any (1.1) holds, Z* ∈ L_{2} and .
By virtue of , , we can state that any vector Z from must also belong to and . On the other hand, it could be verified that relation (1.1) is also true for
Therefore, . In other words, is a selfadjoint operator. However, we know that . Thus, the closure of is a selfadjoint operator , which we will denote by L_{0}.
By virtue of all as stated above, L_{0} is defined as
By the properties of ν ≥ 1, A > E, it follows that L_{0} is a positivedefinite operator. To show that, for each Y ∈ D (L_{0}), we have
Since the embedding is continuous ([[31], Theorem 1.7.7], [[32], p. 48]), then, , where c > 0 is a constant.
Thus,
which shows that L_{0} is a positivedefinite operator.
To prove the discreteness of the spectrum, we will use the following Rellich's theorem (see [[33], p. 386]).
Theorem 1.1. Let B be a selfadjoint operator in H satisfying (Bφ, φ) ≥ (φ, φ), φ ∈ D_{ B }, where D_{ B } is a domain of B.
Then, the spectrum of B is discrete if and only if the set of all vectors φ ∈ D_{ B }, satisfying (Bφ, φ) ≤ 1 is precompact.
Let γ_{1} ≤ γ_{2} ≤ · · · ≤ γ_{ n } ≤ · · · be the eigenvalues of A counted with multiplicity and φ_{1}, φ_{2},..., φ_{ n },... be the corresponding orthonormal eigenvectors in H.
Take y_{ k }(t) = (y (t), φ_{ k }). Then
Hence, using the Rellich's theorem, we come to the following theorem:
Theorem 1.2. If the operator A^{1} is compact in H, then the operator L_{0} has a discrete spectrum.
Proof. By virtue of positivedefiniteness of L_{0}, by Rellich's theorem, it is sufficient to show that the set of vectors
is precompact in L_{2}.
To prove this theorem, consider the following lemma.
Lemma 1.1. For any given ε > 0, there is a number R = R(ε), such that
Proof. From (1.1) for Y ∈ Y :
Since γ_{ R } → ∞ for R → ∞, for any given ε > 0, we could choose R(ε) such that . Therefore, for this choice of R the inequality
holds. On the other hand, by virtue of (1.3):
From (1.4) and the above, it follows that
This proves Lemma 1.1.
Now, turn to the proof of Theorem 1.2. Assume, Y ∈ Y. Denote the set of all vectorfunctions , by E_{ R }. Then, from Lemma 1.1 it follows that for the set Y, E_{ R } is an εnet in L_{2}. Therefore, to prove the precompactness of the set Y, we must prove the precompactness of E_{ R } in L_{2}. Since y_{ k } (1) ≤ 1 (k = 1,..., R), it is sufficient to show that y_{ k }(t) (k = 1,..., R) satisfies the criteria of precompactness in L_{2} (0, 1) [[34], p. 291]. In other words, y_{ k } (t), (k = 1,..., R) must be equicontinuous and bounded with respect to the norm in L_{2} (0, 1). To show that, using (1.3) results in
which proves the boundedness of the functions y_{ k } (t) (k = 1,..., R). Assume that y_{ k } (t) is a zero outside the interval (0, 1). Then, by using the following relation
we have
From the above, for η < ε we have
This shows the equicontinuity of E_{ R }, and it completes the proof of the discreteness of the spectrum of L_{0}.
2 The derivation of the asymptotic formula for eigenvalue distribution of L_{0}
Suppose that the eigenvalues of A are γ_{ n } ~ an^{α} (n → ∞, a > 0, α > 0). Then, by virtue of the spectral expansion of the selfadjoint operator A, we get the following boundary value problem for the coefficients y_{ k }(t) = (y(t), φ_{ k }):
The solution to problem (2.1) from L_{2} (0, 1) is
For this solution to satisfy (2.2), it is necessary and sufficient to hold
at least for one γ_{ k }(λ ≠ γ_{ k }). Therefore, the spectrum of the operator L_{0} consists of those real values of λ ≠ γ_{ k }, such that at least for one k
where . Then, by using (2.4) and identity [[35], p. 56], we get
Find the eigenvalues of the operator L_{0} which are less than γ_{ k }. These values correspond to the imaginary roots of Equation 2.5. By taking and using [[35], p. 51]:
we get
or equivalently
Now, consider the quadratic equation whose roots are given as
Therefore, the coefficients for y^{n} in (2.6) become positive for
Further, let N be the number of positive roots of the function in (2.6), and W be the number of sign changes in its coefficients. Because the radius of convergence of this series is ∞, then by Descartes' rule of signs [[36], p. 52] W  N is a nonnegative even number. From (2.7), W = 1, therefore N = 1. Hence, beginning with some k, Equation 2.6 has exactly one positive root corresponding to the imaginary root of Equation 2.5.
Now, find the asymptotic of the imaginary roots of Equation 2.5. For z = iy and using the asymptotic of J_{ ν } (z) for imaginary z a large z [[37], p. 976]
This means (2.4) is equivalent to
from which
Using (2.8) in , we come up with the asymptotic formula for the eigenvalues of L_{0} which are less than γ_{ k }
Now, find the asymptotic of those solutions of Equation 2.3 which are greater than γ_{ k }, i.e., the real roots of Equation 2.5. By virtue of the asymptotic for a large z [[35], p. 222]
Equation 2.5 becomes
Hence,
where m is a large integer. Therefore, we can state the following Lemma 2.1:
Lemma 2.1. The eigenvalues of the operator L_{0} form two sequences
where . Denote the imaginary and real roots of Equation 2.2 by x_{0,k}and x_{m, k}, respectively.
State the following two lemmas.
Lemma 2.2. Equation 2.5 has no complex roots except the pure imaginary or real roots.
Proof. λ is real since it is eigenvalue of selfadjoint operator associated with problem (2.1), (2.2). γ_{ k } is real by our assumption (A* = A). Hence, the roots of (2.5) are square roots of real numbers. Lemma 2.2 is proved.
Let C be a rectangular contour with vertices at ±iB, ±iB + A_{ m }, where , and B is a large positive number. Further, assume that this contour bypasses the origin and the imaginary root at ix_{0,k}along the small semicircle on the right side of the imaginary axis and ix_{0,k}on the left.
Then, we claim that the following lemma is true.
Lemma 2.3. For a sufficiently large integer m, the number of zeros of the function
inside of C is exactly m.
Proof. Since is an entire function of z, then the number of its zeros inside of C equals:
In the above, we have used the following identities:
As the integrand is an odd function. the order of its numerator in the vicinity of zero is O(z^{ν+1}), and the order of its denominator is O(z^{ν}), the integral along the left part of contour vanishes. Now, consider the integrals along the remaining three sides of the contour. On these sides [[35], p. 221, p. 88]
where
η_{1,ν} (z) and η_{2,ν} (z) are of order for large z.
For simplicity, denote the integrand by f(z), then
One can analogously show that the integral along the lower side tends to the same number.
To calculate the integral along the fourth side, use the relations: for large z, and .
Since is bounded on the righthand side of the contour, we get
Consequently, the limit of the integral along the entire contour is . However, as the integral must be an integer, it should be equal to m. This completes the proof of Lemma 2.3.
By using the above results, derive the asymptotic formula for the eigenvalue distribution of L_{0}. To do that, denote the eigenvalue distribution of the operator L_{0} by N (λ, L_{0}). Then:
where
Since γ_{ k } ~ a · k^{α}, then . That is
From Lemmas 2.2 and 2.3 and the asymptotic of x_{m, k}, it follows that one can find a number c such that for a large m
From this inequality, it follows that N_{2}(λ) is less than , where is the number of the positive integer pairs (m, k) satisfying the inequality
Also, N_{2}(λ) is greater than , where is the number of the positive integer pairs for which
To summarize, we have
Thus, by (2.12) and (2.13) as in [[38], Section 3, Lemma 2] we have:
where .
From the above, we have
Therefore, by virtue of (2.11) and (2.15), we have
For α > 2
and consequently, , .
For α > 2, or, .
For α = 2, N (λ) ~ (c_{1} + c_{2}) λ from which λ_{ n }(L_{0}) ~ dn, d = (c_{1} + c_{2})^{1}.
Then, as Q is a bounded operator in L_{2}, it follows from the relation for the resolvents of the operators L_{0} and L [[30], p. 219]
that the spectrum of L is also discrete. By virtue of the last equality and the properties that hold for s numbers of compact operators [[30], pp. 44, 49] as in [[38], Section 3, Lemma 2], for the eigenvalues of L denoted by μ_{ n }(L), we have
Therefore, we can state the following theorem:
Theorem 2.1. If γ_{ n } ~ an^{α} (0 < a, α > 0), then
where
For simplicity, we will denote the eigenvalues of L_{0} and L by λ_{ n } and μ_{ n }, respectively.
3 Regularized trace of the operator L
Now make use of the theorem proved in [20] for abstract operators. At first, introduce the following notations.
Let A_{0} be a selfadjoint 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 eigenvectors of the operator L_{0} in L_{2}. To do this, we will use the following identity obtained from the Bessel equation"
As α → β, we get
We also consider the following identities:
By the above identities and also by the equation
satisfied by x_{m, k}, we obtain
Therefore,
So, the orthonormal eigenvectors 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.
Now, assume that
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 operatorvalue function q(t) has properties 13, 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
we have
Therefore,
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 lefthand 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
where
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
as N → ∞ we have
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]
Hence, we will have
Using property 2 and the asymptotic of x_{m, k}
Earlier it was obtained that under the assumptions 13 (see [[7], Theorem 2.2])
Thus, from (3.14) to (3.18), we have
Consequently,
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 halfplane):
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 lefthand side of this equality depends only on t, while the righthand side on x, y, z. This means they are equal to some constant which we will denote by λ. Therefore,
and (3.22) becomes like
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 realvalued 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 selfadjoint, positivedefinite, 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
Therefore,
The authors declare that they have no competing interests
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Acknowledgements
The author would like to express his thanks to Dr. Yaghoob Ebrahimi, U.S.Fulbright Scholar assigned to Khazar University during 200910 academic year, for the latter's help in editing, interpretation, and modification of the initial version of this study.
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Aslanova, N.M. Study of the asymptotic eigenvalue distribution and trace formula of a second order operatordifferential equation. Bound Value Probl 2011, 7 (2011). https://doi.org/10.1186/1687277020117
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Keywords
 Hilbert space
 discrete spectrum
 regularized trace