Abstract
In the article, spectrum of operator generated by differential operator expression given on semi axis is investigated and proved formula for regularized trace of this operator.
Introduction
Let H be a separable Hilbert space with a scalar product (.,.) and norm ∥.∥. Consider in L_{2}((0, ∞), H) the problem
where A is a selfadjoint positivedefinite operator in H which has a compact inverse operator and A > E (E is an identity operator in H). Denote the eigenvalues and eigenvectors of the operator A by γ_{1 }≤ γ_{2 }≤ ..., and φ_{1}, φ_{2}, ..., respectively.
Suppose that operatorvalued function q(x) is weakly measurable, ∥q(x)∥ is bounded on [0, ∞), q*(x) = q(x)∀x ∈ [o, ∞). The following properties hold:
(2) is summable on (0, ∞), for
In the case q(x) ≡ 0 in L_{2}(H, (0, ∞)) associate with problems (1), (2) a selfadjoint operator L_{0 }whose domain is
In the case q(x) ≠ 0 denote the corresponding operator by L, so L = L_{0 }+ q.
In this article the asymptotics of eigenvalues and the trace formula of operator L will be studied.
In [1] the regularized traces of all orders of the operator generated by the expression
and the boundary conditions
are obtained.
In [2] the sum of eigenvalue differences of two singular SturmLiouville operators is studied.
The asymptotics of eigenvalues and trace formulas for operators generated by differential expressions with operator coefficients are studied, for example, in [37]. We could also refer to papers [810] where trace formulas for abstract operators are obtained. Trace formulas are used for evaluation of first eigenvalues, they have application to inverse problems, index theory of operators and so forth. For further detailed discussions of the subject refer to [11].
1 The asymptotic formula for eigenvalues of L_{0 }and L
One could easily show that under conditions A > E, A^{1 }∈ σ_{∞}, the spectrum of L_{0 }is discrete.
Suppose that γ_{k }~ ak^{α }(k → ∞, a > 0, α > 0). Denote y_{k}(x) = (y(x), φ_{k}). Then by virtue of the spectral expansion of the selfadjoint operator A we get the following boundaryvalue problem for the coefficients y_{k}(x):
In the case x + γ_{k }> λ solution of problem (1.1) from L_{2}(0, ∞) is
and in the case x + γ_{k}< λ we can write it as a function of real argument as
For this solution to satisfy (1.2) 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
Prove the following two lemmas which we will need further.
Lemma 1.1. Equation (1.6) has only real roots.
Proof. Suppose that z = iα, α ∈ R, α ≠ 0. Then the operator associated with problem
is positive and its eigenvalues are squares of the roots of Equation (1.6). So,
But
which is contradiction. Then z can be only real, otherwise, the selfadjoint operator corresponding to (1.7), (1.8) will have nonreal eigenvalues, which is impossible. The lemma is proved.
Now, find the asymptotics of the solutions of Equation (1.6). By virtue of the asymptotics for large z [[12], p. 975]
we get
Hence
where m is a large integer. Therefore, the statement of the following lemma is true.
Lemma 1.2. For the eigenvalues of L_{0 }the following asymptotic is true
For large z consider the rectangular contour l with vertices at the points
which bypasses the origin along the small semicircle on the right side of the imaginary axis.
The following lemma is true.
Lemma 1.3. For a sufficiently large integer N the number of the roots of the equation inside l is N + O(1).
Proof. For large z we have
Denote the function in braces on the right hand side of (1.12) by F(z). Then for large z by Rouches' theorem the number of the zeros of F(z) inside the contour equals the number of the zeros . Therefore, the number of the zeros of function
inside l is N + O(1).
Now, by using the above results, derive the asymptotic formula for the eigenvalue distribution of L_{0}.
Denote the distribution function of L_{0 }by N(λ). Then
So, N(λ) is a number of positive integer pairs (m,k) for which
By Lemma 1.2 for the great values of m
From the asymptotics of γ_{k }we have
Hence, by virtue of Lemmas 1.1 and 1.3
where N"(λ) is the number of the positive integer pairs for which
N'(λ) is the number of the positive integer pairs (m, k) satisfying the inequality
Thus by using (1.14), (1.15) in (1.13) as in [[13], Lemma 2] we come to the following statement.
Lemma 1.4. If γ_{k }~ ak^{α}, (0 < a, α > 0) then
where
2 Trace formula
The following lemma is true.
Lemma 2.1. Let the conditions of Lemma 1.4 hold. Then for there exists such a subsequence {n_{m}} of natural numbers that the relation
holds.
Proof. In virtue of Lemma 1.4 for , from which it follows that
That is why one could choose a subsequence n_{1 }< n_{2 }< ....n_{m }< ..., that for each k ≥ n_{m }holds , or . The lemma is proved.
We will call a regularized trace of the operator L. It will be shown later it is independent of the choice of {n_{m}} satisfying the hypothesis of Lemma 2.1.
From (1.16) it is obvious that for α > 2 resolvents R(L_{0}) and R(L) are trace class operators. By using Lemma 2.1 for α > 2 one can prove the following lemma.
Lemma 2.2. Let ∥q(x)∥ < const on the interval [0, ∞) and also the conditions of Lemma 1.6 hold. Then for α > 2
where {ψ_{n}} are orthonormal eigenvectors of the operator L_{0}.
The proof of this lemma is analogous to the proof of Lemma 2 and Theorem 2 from [8]. For this reason we will not derive it here.
The orthogonal eigenvectors of the operator L_{0 }in L_{2}((0, ∞), H) are
Calculate their norm. We have
Take in Equation (1.7) z^{2 }= α^{2 }and z^{2 }= β^{2}. The solutions corresponding to these values denote by ψ(x, α^{2}) and ψ(x, β^{2}). Multiplying the first of the obtained equations by ψ (x, β^{2}), the second by ψ (x, α^{2}), subtracting the second one from the first one and integrating from zero to infinity we get
Going to limit as α → β, we get
By making use of identities (12, p.981)
we have
Finally by equation
we get
So, the orthonormal eigenvectors of L_{0 }are
Lemma 2.3. If the operatorvalued function q(x) has property 1 and , then
Proof. Take (q(x)φ_{k}, φ_{k}) = q_{k}(x). Let ε > 0 be sufficiently small number. If then . For we have and, finally, for it will be .
then
For ε → 0 we have
From asymptotic by using (2.10), (2.11) and property 1 we get
The lemma is proved.
By using Lemma 2.3 prove the following theorem.
Theorem 2.1. Let the conditions of Lemma 1.6 hold. If the operatorvalued function q (x) has properties 13, then it holds the formula
Proof. In virtue of Lemma 2.1
Denote
Show that for each fixed value of k the mth term of the sum T_{N }(x) is a residue at the point α_{m }of some function of complex variable which has poles at points .
For this purpose consider the following function
By taking in place of zero x in (2.6) one can show that
Note that all zeros of the function are simple, otherwise
and by virtue of (2.7) the norm of the eigenvectors equals zero, which is contradiction.
Denote z^{2 } x = f (x, z) and the right hand side of (2.14) by G(f(x, z). Then
Then from (2.14), (2.15)
The function g(z) has poles of second order at the points α_{m}. By using identities (2.15), (2.16) show that residues at this points equal the terms of sum T_{N}(x). Denoting , write Taylor expansion of this function in the vicinity α_{m}:
Show that the coefficient of the expansion of function zu^{2 }(z) at (z  α_{m})^{3 }equals zero. So,
Therefore,
On the other hand, and satisfy the Bessel equation for . So their difference also satisfies this equation
If , then the right hand side (2.21) vanishes. Hence,
which shows that the coefficient at (z  α_{m})^{3 }in (2.17) vanishes.
Consequently, by (2.16), (2.17), (2.22) and the relation
we have
Take as a contour of integration a rectangular contour C with vertices at the points ±A_{N}, ±A_{N }+ +iB, which bypasses points α_{m }above real axis, α_{m }below it.
Consider the right hand side of the contour with vertices at A_{N }and A_{N }+ iB. By using the asymptotics
For x > 0, N → ∞ taking B = A_{N}, z = u + iv we have
From condition 2
By conditions 23 as N → ∞
On the side of the contour with the vertices at ±A_{N }+ iB
In the same way as it is done in (2.25), (2.26) we get that
Similarly, one may show that the integral along the left hand side of the contour converges to zero:
So, by the Cauchy theorem we finally get
which completes the proof of the theorem.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This study was supported by the Science Development Foundation under the President of the Republic of AzerbaijanGrant No EIF20111(3)82/181.
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