# Study of the asymptotic eigenvalue distribution and trace formula of a second order operator-differential equation

## Abstract

The purpose of writing this article is to show some spectral properties of the Bessel operator equation, with spectral parameter-dependent 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.

## Keywords

Hilbert space discrete spectrum regularized trace## Introduction

**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 Open image in new window .

*L*

_{2}(

*H*, [0, 1]), where

*A*is a self-adjoint positive-definite operator in

*H*which has a compact inverse operator. Further, suppose the operator-valued function

*q*(

*t*) is weakly measurable, and ||

*q*(

*t*)|| is bounded on [0, 1] with the following properties:

- 1.
*q*(*t*) has a second-order weak derivative on [0, 1], and*q*^{(l)}(*t*) (*l*= 0, 1, 2) are self-adjoint 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 Open image in new window ). If {*φ*_{ n }} is a basis formed by the orthonormal eigenvectors of*B*, then Open image in new window . 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 Open image in new window is true for each

*f*∈*H*.

State that if *q*(*t*) ≡ 0, a self-adjoint 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 self-adjoint 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 self-adjoint. 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 self-adjoint.

In [2], Walter considers a scalar Sturm-Liouville problem with an eigenvalue parameter *λ* in the boundary conditions. He shows that one can associate a self-adjoint operator with that by finding a suitable Hilbert space. Further, he obtains the expansion theorem by reference to the self-adjointness of that operator. His approach was used by Fulton in [3] later on.

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 Open image in new window and Open image in new window 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 positive-definite. 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 Open image in new window , where *μ*_{ n } and *λ*_{ n } are the eigenvalues of perturbed and non-perturbed operators. For definition of {*n*_{ m }}, see also [[8], Lemma 1].

is considered on the interval [0, *π*]. Then, the sum Open image in new window 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 *λ*-dependent boundary conditions arise upon separation of variables in the heat and wave equations. We can also refer to [10, 11, 12, 13, 14, 15, 16, 17], where boundary-value problems for ordinary differential operators with eigenvalue-dependent boundary conditions are studied.

and derived the formula Open image in new window , 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, 10, 11, 12, 13, 14, 15, 16, 17, 18]), as well as differential-operator equations (e.g., [6, 7, 8, 19]) and discrete abstract operators (e.g., [20, 21, 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 self-adjoint 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

*H*(see [30], p. 57) that vanish in the vicinity of zero and are infinitely differentiable in the norm of

*H*. Also, on Open image in new window define the operator Open image in new window :

*L*

_{0}and show that it is self-adjoint. To do that, consider the adjoint operator of Open image in new window as Open image in new window . By definition, vector Open image in new window if for each Open image in new window it holds

and *Z** = {*z** (*t*), *z**} ∈ **L**_{2}. However, using integration by parts from (1.1), it is obvious that Open image in new window with Open image in new window and *l*[*z*] ∈ *L*_{2} (*H*, [0, 1])}. In other words, *z*(*t*) has a first-order 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 Open image in new window .

Now, the vector Open image in new window if and only if for any Open image in new window (1.1) holds, *Z** ∈ **L**_{2} and Open image in new window .

*Z*from Open image in new window must also belong to Open image in new window and Open image in new window . On the other hand, it could be verified that relation (1.1) is also true for

Therefore, Open image in new window . In other words, Open image in new window is a self-adjoint operator. However, we know that Open image in new window . Thus, the closure of Open image in new window is a self-adjoint operator Open image in new window , which we will denote by *L*_{0}.

*A > E*, it follows that

*L*

_{0}is a positive-definite operator. To show that, for each

*Y*∈

*D*(

*L*

_{0}), we have

Since the embedding Open image in new window is continuous ([[31], Theorem 1.7.7], [[32], p. 48]), then, Open image in new window , where *c >* 0 is a constant.

which shows that *L*_{0} is a positive-definite 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 self-adjoint 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*.

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 positive-definiteness 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.

*γ*

_{ R }→ ∞ for

*R*→ ∞, for any given

*ε >*0, we could choose

*R*(

*ε*) such that Open image in new window . Therefore, for this choice of

*R*the inequality

This proves Lemma 1.1.

*Y*∈

**Y**. Denote the set of all vector-functions Open image in new window , by

*E*

_{ R }. Then, from Lemma 1.1 it follows that for the set

*Y*,

*E*

_{ R }is an

*ε*-net 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

*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

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}

*A*are

*γ*

_{ n }~

*an*

^{ α }(

*n*→ ∞,

*a >*0,

*α >*0). Then, by virtue of the spectral expansion of the self-adjoint operator

*A*, we get the following boundary value problem for the coefficients

*y*

_{ k }(

*t*) = (

*y*(

*t*),

*φ*

_{ k }):

*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*

*L*

_{0}which are less than

*γ*

_{ k }. These values correspond to the imaginary roots of Equation 2.5. By taking Open image in new window and using [[35], p. 51]:

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.

*z*=

*iy*and using the asymptotic of

*J*

_{ ν }(

*z*) for imaginary

*z*a large |

*z*| [[37], p. 976]

*L*

_{0}which are less than

*γ*

_{ k }

*γ*

_{ k }, i.e., the real roots of Equation 2.5. By virtue of the asymptotic for a large |

*z*| [[35], p. 222]

where *m* is a large integer. Therefore, we can state the following Lemma 2.1:

*where* Open image in new window . 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 self-adjoint 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 Open image in new window , 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 Open image in new window is an entire function of

*z*, then the number of its zeros inside of

*C*equals:

*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]

*η*_{1,ν} (*z*) and *η*_{2,ν} (*z*) are of order Open image in new window for large |*z*|.

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: Open image in new window for large |*z*|, and Open image in new window .

Consequently, the limit of the integral along the entire contour is Open image in new window . However, as the integral must be an integer, it should be equal to *m*. This completes the proof of Lemma 2.3.

*L*

_{0}. To do that, denote the eigenvalue distribution of the operator

*L*

_{0}by

*N*(

*λ*,

*L*

_{0}). Then:

*x*

_{m, k}, it follows that one can find a number

*c*such that for a large

*m*

*N*

_{2}(

*λ*) is less than Open image in new window , where Open image in new window is the number of the positive integer pairs (

*m*,

*k*) satisfying the inequality

*N*

_{2}(

*λ*) is greater than Open image in new window , where Open image in new window is the number of the positive integer pairs for which

where Open image in new window .

*α >*2

and consequently, Open image in new window , Open image in new window .

For *α >* 2, Open image in new window or, Open image in new window .

For *α =* 2, *N* (*λ*) ~ (*c*_{1} + *c*_{2}) *λ* from which *λ*_{ n }(*L*_{0}) ~ *dn*, *d* = (*c*_{1} + *c*_{2})^{-1}.

*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]

*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:

*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 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 Open image in new window . 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*Open image in new window

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

*ω*+

*δ <*1

*such that*Open image in new window

*is a trace class operator. Then, there exists a subsequence of natural numbers*Open image in new window

*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 Open image in new window is bounded. For Open image in new window and *α >* 2, from asymptotic (2.16), we will have that Open image in new window is a trace class operator. If *α <* 2, then Open image in new window will be a trace class operator for Open image in new window .

*α >*2, we have

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

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

*L*

_{0}in

**L**

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

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 Open image in new window .

To estimate the second series in (3.5), we use the relation Open image in new window .

*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

*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 Open image in new window .

*x*

_{0,k,...,}

*x*

_{N - 1,k}and

*j*

_{1},...,

*j*

_{ N }(

*J*

_{ ν }(

*j*

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

*j*

_{ n }equals

*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.

*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

*C*for Open image in new window , Open image in new window , we have |

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

*N*→ ∞ we have

*J*:

*N*[[35], p. 642]

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

which completes the proof.

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

*q*

^{(l)}(

*t*) ∈

*σ*

_{1}, Open image in new window may be weakened. Namely, we may just require to hold

Then formula (3.8) takes the form Open image in new window . 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:

*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

*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,

*U*(

*r*,

*φ*,

*z*) =

*V*(

*r*,

*z*)ϕ(

*φ*),

*q*(

*r*,

*φ*,

*z*) =

*Q*(

*r*,

*z*), and Open image in new window ,

*ν*= const., we get

*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

*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:

*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 Open image in new window .

*μ*

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

*q*(

*r*) ≡ 0. Now calculate Open image in new window

The authors declare that they have no competing interests

## Notes

### Acknowledgements

The author would like to express his thanks to Dr. Yaghoob Ebrahimi, U.S.Fulbright Scholar assigned to Khazar University during 2009-10 academic year, for the latter's help in editing, interpretation, and modification of the initial version of this study.

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