AN ASYMPTOTIC FORMULA FOR THE NUMBER OF EIGENVALUES OF A DIFFERENTIAL OPERATOR

In this work, it is proved that the spectrum of an differential operator with unbounded operator coefficients in elliptic type with partial derivatives is pure discrete and an asymptotic formula is found for the number of eigenvalues of this operator. 66 E. Adiguzelov, H. Avci and E. Gul


Introduction
The asymptotic behavior of the spectrum of Sturm-Liouville operator with operator coefficient had been firstly studied by A.G. Kostyuchenko and B.M. Levitan, [1].Some works about this subject had been done later by M.L. Gorbacuk , [2], V.. Gorbacuk and M.L. Gorbacuk , [3], [4], M. Otelbayev , [5], M.Z.Solomyak , [6], F.G. Maksudov, M. Bairamoglu and E.E.Adiguzelov , [7], and etc.In this work, the spectrum of self adjoint operator L forming with differential expression (1.1) defined in H 1 = L 2 (H; IR n ) is studied and an asymptotic formula for the number of eigenvalues of L is found , where H denotes the Hilbert space.Here, we will accept that the coefficients a ij (x) (i, j = 1, . . ., n) and Q(x) in (1.1) satisfy the following conditions: 1) Real valued functions a ij (x) = a ji (x) have bounded derivatives 2) There are positive numbers ν and µ such that 3) Q(x) : D(Q(x)) → H is a self adjoint operator, where D(Q(x)) ⊂ H for each x ∈ IR n .The set D = ∩ x∈IR n D(Q(x)) is dense in H. Q(x) ≥ I and Q −1 (x) ∈ σ ∞ (H).4) There are constants 0 < a < 3  2 and B > 0 such that There is a constant l > 0 satisfying Q −l (x) ∈ σ 1 (H) and Here, trA = traceA is the sum of the eigenvalues of a kernel operator A. 8) Let α 1 (x) ≤ α 2 (x) ≤ . . .≤ α m (x) ≤ . . .be eigenvalues of operator Q(x).Assume that the functions α 1 (x), α 2 (x), . . .are measurable functions.Let Suppose that there is a constant a 0 > 0 such that for positive big values of λ.Let D denotes the set of elements in the form Σ m k=1 ϕ k (x)f k of the space H 1 , where ϕ 1 (x), ϕ 2 (x), . . ., ϕ m (x), are scaler functions having compact support and continuous derivatives of second order defined in IR n and {f k } m k=1 ⊂ D. It is known that D = H 1 .The linear operator L defined in the form is a symmetric and positive operator in H 1 .Suppose that the operator L which is closure of L is pure discrete.Let To solve our main problem, we shall firstly study the Green function of Cauchy problem in the following: We shall look this for the Green function given by the form We will use parametric method for this.Here G 1 (x − y, y, t) is a function of the form which has continuous derivatives of second order, where And G 0 (x − y, η, t) denotes the Green function of the problem The function ϕ(x, y, t) in (1.3) is an unknown operator function that it must be here found.If the equality is true, then we can take G(x, y, t) as a Green function of problem (1.2).And so, the integral equation An asymptotic formula for the number of eigenvalues of a ... 69 is obtained, where (1.9)

Preliminaries
Now, before we give main results, we remind some of its definitions used in this paper as below: About Measurability Let f (x) and {f i (x)} ∞ i=1 be any functions defined on Euclidean space IR n that their values belong to a separable Hilbert space H. Definition 2.1 A function defined on IR n whose range is a countable set, is said to be countable valued if it takes on a measurable set its all values except zero.
If the inner product of two element f and g of the set H 1 is defined by the formula (f, g) then the set H 1 forms a separable Hilbert space, [8].
The spaces σ ∞ (H) and σ 1 (H) The set of all complete continuous operators from Here every eigenvalues is repeated according to multiplicity number.Moreover,s 1 , s 2 , ..., s i , ... are the positive numbers.These are called the s-numbers of operator A. We can write the s-numbers of A in the form s i (A) and show the number of the s-numbers of A with ν(A).Not that ν(A) can be finite or infinite.We shall denote the union of the operator "0" and the set of all operators A ∈ σ ∞ (H) satisfying with σ 1 or σ 1 (H).σ 1 is a Banach space, [9].In this space, every operator A = 0 is defined by and we accept that 0 σ 1 (H) = 0, [9].Along this study, we shall denote positive constants with c which may not be same and the norm of the kernel operator from H to H in space σ 1 (H) with • 1 as in [9].Moreover, here we shall use the inequalities which were proved in [9,10] respectively and we shall suppose t ∈ (0, 1).

The spectrum of operator L asymptotic formula for the number of its eigenvalues
In this section, the spectrum of operator L is studied and an asymptotic formula is found for the number of eigenvalues which is less then λ, where λ is a positive variable.First of all, to solve integral equation (1.8), we need to estimate the operator function K(x, y, t).
In a similar form of the proof of this theorem, the inequality can be shown.From theorem 3.1, (3.7) and the inequalities t which were proved in [10], it can be proved that the series which is a solution of the equation (1.8), is convergent with respect to the norm in σ 1 (H) of such that Q −l (x) ∈ σ 1 (H) for every x ∈ IR n and the inequalities are satisfied.
Since Q(x) satisfies the condition 6) and from the inequality (3.8) and the formula for B (1) 1 we find In the similar form, we can estimate B (2) 1 : From (3.13), (3.14) and the last relation we find This time, let us estimate the expression B 2 in (3.12) using the inequalities .
From (3.12), (3.15) and (3.16) we obtain This completes the proof.Now, we can prove that the spectrum of L is pure discrete and find an asymptotic formula for the number of its eigenvalues.Theorem 3.3.If the coefficients of differential expression (1.1) satisfy the conditions 1) -7) then the spectrum of L is pure discrete.Proof.The operator function G 0 (x − y, y, t) is in the form where I : H → H is an identity operator.By translating s i = t − 1 2 ξ i we have From here, we find (3.17) From the formulas (1.4), (3.17) and theorem 3.2 we obtain (3.18) By hypothesis, for every x ∈ IR n , Q(x) ≥ I and hence On the other hand, since Q(x) satisfies the conditions 5) -7) and from (3.18) we find Since the expression Σ n i,j=1 a i j(x)ξ i ξ j satisfies the condition 2) there exist positive constants M 1 and M 2 such that Thus, from (3.19) we obtain Since u = e −tL ψ(x) and u = IR n G(x, y, t)ψ(y)dy, G(x, y, t) is the kernel of positive bounded operator e −tL (t > 0).In this case, for every t > 0, e −tL is the kernel operator, [8].For the spectrums of operators e −tL and L s{e −tL } = e −ts{L} .
Since for every t > 0, e −tL is a complete continuous operator and from the last formula we obtain that the spectrum of L is pure discrete.