Oscillation Theorems for the Dirac Operator with Spectral Parameter in the Boundary Condition

We consider the boundary value problem for the one-dimensional Dirac equation with spectral parameter dependent boundary condition. We give location of the eigenvalues on the real axis, study the oscillation properties of eigenvector-functions and obtain the asymptotic behavior of the eigenvalues and eigenvector-functions of this problem.


Introduction
We consider the following boundary value problem for the one-dimensional Dirac canonical system where λ ∈ C is a spectral parameter, the functions p (x) and r(x) are continuous on the interval [0, π], α, β, a 1 and b 1 are real constants such that 0 ≤ α, β < π and σ = a 1 sin β − b 1 cos β > 0. (1.4) λ is called an eigenvalue with corresponding eigenvector-function w if boundary value problem (1.1)-(1.3)under consideration have a non-trivial solution w(x) = u(x) v(x) for λ.
Corresponding author.Email: z_aliyev@mail.ru In the case where p(x) = V(x) + m, r(x) = V(x) − m, V(x) is a potential function and m is the mass of a particle, (1.1) is called an one-dimensional stationary Dirac system in relativistic quantum theory [15,18].
The basic and comprehensive results (except the oscillation properties) about Dirac operator were given in [15].The oscillatory properties of the eigenvector-functions of the Dirac operator have been studied in a recent work [5] (see also [4,6]).Direct and inverse problems for Dirac operators were extensively studied in [1,7,10,20,21] (see also the references in these works).
Boundary value problems with spectral parameter in boundary conditions often appear in mathematics, mechanics, physics, and other branches of natural sciences.A bibliography of papers in which such problems were considered in connection with specific physical processes can be found in [3,9,17,19,22].Eigenvalue-dependent boundary conditions were examined even before the time of Poisson.Eigenvalue problems for ordinary differential operators with spectral parameter contained in the boundary conditions were considered in various settings in numerous papers [2, 3, 7-9, 12-14, 16, 17, 19, 21, 22].
In [13] and [21] oscillatory properties of eigenvector-functions of the Dirac system with spectral parameter contained in the boundary conditions were studied.It should be noted that these studies did not specify the exact number of zeros of the components of the eigenvectorfunction corresponding n-th eigenvalue (although for sufficiently large n).
In the present paper, we study the general characteristics of the location of the eigenvalues on the real axis, oscillation properties of eigenvector-functions and asymptotic behavior of eigenvalues and eigenvector-functions of the spectral problem (1.1)-(1.3). 3) are real and simple, and form a countable set without finite limit points.
Proof.The proof of this lemma is similar to that of [15,Lemma 10.2 and Lemma 11.2].
In order to study the location of the eigenvalues on the real axis and the oscillation properties of eigenvector-functions of the problem (1.1)-(1.3)alongside with this problem we shall consider the following boundary value problem where µ ∈ [0, 1].It is known (see [15, Ch. 1,  § 11]) that the eigenvalues of this problem are real, simple and the values range is from −∞ to +∞, and they can be enumerated in the following increasing order such that (see [5]) Remark 2.2.From continuous dependence of solutions of a system of differential equations on the parameter we obtain that the eigenvalues η k (µ), k ∈ Z, of problem (2.2) depends continuously on the parameter µ ∈ [0, 1].Hence the map η k (µ) continuously transforms η k (0) to η k (1) for any k ∈ Z (see [11, § 6-7]).
We denote by s(g) the number of zeros of the function g ∈ C([0, π] ; R) in the interval (0, π).
and ω α,γ (x) , x ∈ R, is defined as follows: Moreover, the functions u k, µ (x) and v k, µ (x) have only nodal zeros in the interval (0, π) (by a nodal zero we mean a function that changes its sign at the zero).
We set Note that the function is defined for and is meromorphic function of finite order, and τ k and ν k , k ∈ Z, are poles and zeros of this function, respectively.
Corollary 2.5.The function F(λ) is continuous and strictly decreasing on each interval By m (λ) and n (λ), λ ∈ R, we denote the number of zeros in the interval (0, π) of the functions u(x, λ) and v(x, λ), respectively.We define numbers m k and n k , k ∈ Z, as follows: where the function ω α, γ is defined by formula (2.4).
Then, by Theorem 2.3 (see (2.3)), we have The remaining cases are considered similarly.

Oscillatory properties of eigenvector-functions of problem (1.1)-(1.3)
By virtue of the properties of the function F(λ) (see Lemma 2.4 and Corollary 2.5) and the relations v(π, τ k ) = 0, k ∈ Z, we have lim It follows from the preceding considerations that in the interval (τ k−1 , τ k ), there exists a unique point λ = λ * k such that i.e., condition (1.3) is satisfied.Therefore, λ * k is an eigenvalue of the boundary value problem (1.1)-(1.3)and w(x, λ * k ) is the corresponding eigenvector-function. Assume that β = 0 and − In a similar way, one can show that in each of the intervals (− The case in which β = 0 and − b 1 sin β = τ N can be considered in a similar way; here one uses the fact that τ N is also an eigenvalue of the boundary value problem (1.1)- (1.3).In this case, we have Therefore, it follows from these considerations that there exist an unboundedly decreasing sequence of negative eigenvalues and an unboundedly increasing sequence of nonnegative eigenvalues of the boundary value problem (1.1)-(1.3).Hence, these eigenvalues can be enumerated in increasing order.Remark 3.1.When numbering the eigenvalues of the problem (1.1)-( 1.3) we will proceed from the following consideration: the number zero will be assigned to eigenvalue that is contained in the half-open interval (τ −1 , τ 0 ] and is closest to τ 0 .Thus, the following theorem is proved.

Theorem 3.2.
There exists an infinite set of eigenvalues {λ k } k∈ Z of problem (1.1)-(1.3)with values ranging from −∞ to +∞ which can be enumerated in increasing order: where λ 0 is defined in Remark 3.1.
, corresponding to the eigenvalues λ k of the problem (1.1)-(1.3),for |k| > k * have the following oscillation properties: Proof.Let β = 0.In this case it follows from the proof of the Theorem 3.2 and the Remark 3.
Then, again, from the proof of the Theorem 3.2 and the Remark 3.
the case where N > 0. Hence, by virtue of the Theorem 2.6 we have m
[4]the definition of number h −1 , we have λ < ν h −1 .It follows from[15, formulas (11.12)and(11.13)] that the number of zeros of u(x, λ) on (0, π) grows unboundedly as |λ| → +∞.By Corollary 2.1 from[4], the number of zeros of u(x, λ) is a nondecreasing function of λ.By [4, Lemma 2.1], the roots of equation u(x, λ) = 0 continuously depend on λ.On the other hand, by [4, Corollary 2.1], as λ decreases, every zero of u moves to the left but cannot pass through 0, since the number of zeros does not decrease.By [4, Corollary 2.2], zeros enter through the point π.Since m(ν k the case where N > 0. Hence, by virtue of the Theorem 2.6 we have m(λ k the case where N > 0. Hence, by virtue of the Theorem 2.6 we have m(λ k