Oscillation theorems for the Dirac operator with spectral parameter in the boundary condition

Transcription

1 Electronic Journal of Qualitative Theory of Differential Equations 216, No. 115, 1 1; doi: /ejqtde Oscillation theorems for the Dirac operator with spectral parameter in the boundary condition Ziyatkhan S. Aliyev 1, 2 and Parvana R. Manafova 1 1 Department of Mathematical Analysis, Baku State University Z. Khalilov Str. 23, Baku, AZ-1148, Azerbaijan 2 Institute of Mathematics and Mechanics NAS of Azerbaijan F. Agaev Str. 9, Baku, AZ-1141, Azerbaijan Received 2 September 216, appeared 9 December 216 Communicated by Jeff R. L. Webb Abstract. 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. Keywords: one dimensional Dirac equation, eigenvalue, eigenvector-function, oscillatory properties, asymptotic behavior of the eigenvalues and eigenfunctions. 21 Mathematics Subject Classification: 34B5, 34B9, 34B24. 1 Introduction We consider the following boundary value problem for the one-dimensional Dirac canonical system v {λ + px} u =, u + {λ + rx} v =, < x < π, 1.1 v cos α + u sin α =, 1.2 λ cos β + a 1 vπ + λ sin β + b 1 u π =, 1.3 where λ C is a spectral parameter, the functions p x and rx are continuous on the interval [, π], α, β, a 1 and b 1 are real constants such that α, β < π and σ = a 1 sin β b 1 cos β >. 1.4 λ is called an eigenvalue with corresponding eigenvector-function w if boundary value problem under consideration have a non-trivial solution wx = for λ. Corresponding author. z_aliyev@mail.ru ux vx

2 2 Z. S. Aliyev and P. R. Manafova In the case where px = Vx + m, rx = Vx m, Vx 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 [1,7,1,2,21] 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 Several auxiliary facts and some properties of the solution of problem 1.1, 2.1 Lemma 2.1. The eigenvalues of the boundary value problem 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 1.2 and Lemma 11.2]. Consider the boundary condition vπ cos γ + uπ sin γ =, 2.1 where γ, π. In order to study the location of the eigenvalues on the real axis and the oscillation properties of eigenvector-functions of the problem alongside with this problem we shall consider the following boundary value problem v {λ + µp x} u =, u + {λ + µrx} v =, < x < π, v cos α + u sin α =, vπ cos γ + u π sin γ =. 2.2 where µ [, 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 < η k µ < < η 1 µ < η µ < η 1 µ < < η k µ <,

3 Oscillation theorems for the Dirac operator 3 such that see [5] η k = k + α β / π, k Z. 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 [, 1]. Hence the map η k µ continuously transforms η k to η k 1 for any k Z see [11, 6 7]. We denote by sg the number of zeros of the function g C[, π] ; R in the interval, π. For problem 2.2 with a suitable interpretation see Remark 2.2 the following oscillation theorem holds. Theorem 2.3. [5, Theorem 3.1] The eigenvector-functions w k, µ x = uk, µ x, corresponding to v k, µ x the eigenvalues η k µ, k Z, have the following oscillation properties except for k = the cases α = β = and α = β = π/2 : suk, µ k 1 + H α π/2 ωα,γ k + H π / 2 γ ω = α,γ k sv k, µ k 1 + sgn α H ω α,β k + sgn γ H ω α,γ k where Hx is the Heaviside function, i.e. and ω α,γ x, x R, is defined as follows: Hx = { if x, 1 if x >, 2.3 ω α,γ x = { 1, if x < or x =, α < γ, 1, if x > or x =, α γ. 2.4 Moreover, the functions u k, µ x and v k, µ x have only nodal zeros in the interval, π by a nodal zero we mean a function that changes its sign at the zero. We denote by τ k µ and ν k µ, k Z, the eigenvalues of problem 2.2 for γ = and γ = π/2, respectively. By virtue of [5, formula 3.1] the eigenvalues of problem 2.2 have the following location on the real axis: if γ, π / 2, then... < τ 2 µ < ν 1 µ < η 1 µ < τ 1 µ < ν µ < η µ < τ µ < ν 1 µ < η 1 µ < τ 1 µ <, 2.5 and if γ π / 2, π, then... < τ 2 µ < η 1 µ < ν 1 µ < τ 1 µ < η µ < ν µ < τ µ < η 1 µ < ν 1 µ < τ 1 µ <. One can readily show that there exists a unique solution wx, λ = satisfying the initial condition 2.6 ux,λ of system 1.1 vx,λ u, λ = cos α, v, λ = sin α; 2.7

4 4 Z. S. Aliyev and P. R. Manafova moreover, for each fixed x [, π], the functions ux, λ and vx, λ are entire functions of the argument λ. The proof of this assertion repeats that of Theorem 1.1 in [15, Ch. 1, 1] with obvious modifications. By 2.7 the functions ux, λ and vx, λ satisfy the boundary condition 1.2, so that to find the eigenvalues of the boundary value problem we have to insert the functions ux, λ and vx, λ in the boundary condition 1.3 and find the roots of this equation. Thus, the eigenvalues of problem are the roots of the following equation λ cos β + a 1 vπ, λ + λ sin β + b 1 u π, λ =. 2.8 Obviously, the eigenvalues η k 1, k Z, of problem 1.1, 1.2, 2.1 or 2.2 for µ = 1 are zeros of the entire function vπ, λ cos γ + uπ, λ sin γ =. We set τ k = τ k 1 and ν k = ν k 1, k Z. Note that the function is defined for Fλ = u π, λ vπ, λ λ D C\R + k= τ k 1, τ k and is meromorphic function of finite order, and τ k and ν k, k Z, are poles and zeros of this function, respectively. Lemma 2.4. The following formula holds: λ u π, λ = vπ, λ Proof. By 1.1 for any µ, λ Z, we obtain π {u2 x, λ + v 2 x, λ} dx v 2, λ D. 2.9 π, λ d {vx, µ u x, λ ux, µ vx, λ} = µ λ {u x, µ u x, λ + vx, µ vx, λ}. dx Integrating this relation from to π and taking into account condition 1.2 we find π vπ, µ u π, λ u π, µ vπ, λ = µ λ {u x, µ u x, λ + vx, µ vx, λ} dx 2.1 By 2.1 for any µ, λ D we have u π, µ vπ, µ u π, λ = µ λ vπ, λ π {u x, µ u x, λ + vx, µ vx, λ} dx. vπ, µ vπ, λ Dividing this equality by µ λ and passing to the limit as µ λ we obtain 2.9. Corollary 2.5. The function Fλ is continuous and strictly decreasing on each interval τ k 1, τ k, k Z. By m λ and n λ, λ R, we denote the number of zeros in the interval, π of the functions ux, λ and vx, λ, respectively. We define numbers m k and n k, k Z, as follows: m k = k 1 + H α π ω 2 α, π 2 k, n k = k 1 + sgn α Hω α, k,

5 Oscillation theorems for the Dirac operator 5 where the function ω α, γ is defined by formula 2.4. Then, by Theorem 2.3 see 2.3, we have Let mν k = m k and nτ k = n k. h 1 = max {k Z : η k,1 + p x <, η k,1 + rx <, x [, π]}, h 1 = min {k Z : η k,1 + p x >, η k,1 + rx >, x [, π]}. Theorem 2.6. If λ [ν k 1, ν k for k < h 1, then mλ = m k 1, and if λ ν k 1, ν k ] for k > h 1, then mλ = m k. If λ [τ k 1, τ k for k < h 1, then nλ = n k 1, and if λ τ k 1, τ k ] for k > h 1, then nλ = n k. Proof. Let λ [ν k 1, ν k for k < h 1. Then, by the definition of number h 1, we have λ < ν h 1. It follows from [15, formulas and 11.13] that the number of zeros of ux, λ on, π grows unboundedly as λ +. By Corollary 2.1 from [4], the number of zeros of ux, λ is a nondecreasing function of λ. By [4, Lemma 2.1], the roots of equation ux, λ = 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, since the number of zeros does not decrease. By [4, Corollary 2.2], zeros enter through the point π. Since mν k = m k, k Z and uπ, λ = for λ ν k 1, ν k it follows from these considerations that mλ = m k 1 for λ [ν k 1, ν k. The remaining cases are considered similarly. 3 Oscillatory properties of eigenvector-functions of problem For β = let N = if b 1 sin β = τ, N < be an integer such that τ N b 1 sin β < τ N+1 if b 1 sin β < τ, N > be an integer such that τ N 1 < b 1 sin β τ N if b 1 sin β > τ, and for β = π 2 let M = if a 1 cos β = ν, M < be an integer such that ν M a 1 cos β < ν M+1 if a 1 cos β < ν, M > be an integer such that ν M 1 < a 1 cos β ν M if a 1 cos β > ν. By virtue of the properties of the function Fλ see Lemma 2.4 and Corollary 2.5 and the relations vπ, τ k =, k Z, we have lim Fλ = +, lim λ τ k 1 + Fλ = ; λ τ k moreover, the function Fλ takes each value in, + at a unique point in the interval τ k 1, τ k. For the function Gλ = λ cos β+a 1 λ sin β+b 1 we have G λ = σ / λ sin β + b 1 2. Since σ > see 1.4, it follows that for β = the function Gλ is strictly increasing in the interval, + ; for β, π the function Gλ is increasing in both intervals, b 1 / sin β and b 1 / sin β, + ; moreover, lim Gλ = +, λ b 1/sin β lim Gλ =. λ b 1/sin β+ Assume that either β =, or β = and b 1 sin β = τ, or β = and b 1 sin β / [τ k, τ k+1 if k <, b 1 sin β / τ k 1, τ k ] if k >. 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 Fλ = Gλ, 3.1

6 6 Z. S. Aliyev and P. R. Manafova i.e., condition 1.3 is satisfied. Therefore, λ k is an eigenvalue of the boundary value problem and wx, λ k is the corresponding eigenvector-function. Assume that β = and b 1 sin β τ k, τ k+1 if k <, b 1 sin β τ k 1, τ k if k >. In a similar way, one can show that in each of the intervals b 1 sin β, τ k+1 and τ k, b 1 sin β if k <, τ k 1, b 1 sin β and b 1 sin β, τ k if k >, there exists a unique value λ k, 1 and λ k, 2, respectively such that relation 3.1 is valid. The case in which β = 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 In this case, we have λ k, 1 τ N, τ N+1 if N <, λ k, 1 τ N 1, τ N if N >, and λ k, 2 = τ N. 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 Hence, these eigenvalues can be enumerated in increasing order. Remark 3.1. When numbering the eigenvalues of the problem 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 ] and is closest to τ. Thus, the following theorem is proved. Theorem 3.2. There exists an infinite set of eigenvalues {λ k } k Z of problem with values ranging from to + which can be enumerated in increasing order: where λ is defined in Remark 3.1. < λ k < < λ 1 < λ < λ 1 < < λ k <, Let k = max { h 1, h 1, N + 1, M + 1}. Theorem 3.3. The eigenvector-functions w k x = wx, λ k = ux,λk = uk x, corresponding vx,λ k v k x to the eigenvalues λ k of the problem , for k > k have the following oscillation properties: a if β =, then mλ k = m k, nλ k = n k+hk 1 ; b if β, π 2, then mλk = m k HN+1, nλ k = n k HN ; c if β = π 2, then mλ k = m k+hk HN, nλ k = n k HN ; d if β π 2, π, then mλ k = m k HN, nλ k = n k HN. Proof. Let β =. In this case it follows from the proof of the Theorem 3.2 and the Remark 3.1 that λ k τ k 1, τ k for any k Z ; moreover, λ k ν k, ν k+1 for k < k, λ k ν k 1, ν k for k > k. Hence, by Theorem 2.6 we obtain that mλ k = m k for k > k, nλ k = n k 1 for k < k and nλ k = n k for k > k. Let β, π 2. Then, again, from the proof of the Theorem 3.2 and the Remark 3.1 it follows that λ k τ k, τ k+1 for k < k and λ k τ k 1, τ k for k > k in the case N ; λ k τ k 1, τ k for k < k and λ k τ k 2, τ k 1 for k > k in the case where N > ; moreover, λ k ν k+1, ν k+2 for k < k and λ k ν k, ν k+1 for k > k in the case N ; λ k ν k, ν k+1 for k < k and λ k ν k 1, ν k for k > k in the case where N >. Hence, by virtue of the Theorem 2.6 we have mλ k = m k+1, nλ k = n k for k > k in the case N ; mλ k = m k, nλ k = n k 1 for k > k in the case N >.

7 Oscillation theorems for the Dirac operator 7 Let β = π 2. Then, λ k τ k, τ k+1 for k < k and λ k τ k 1, τ k for k > k in the case N ; λ k τ k 1, τ k for k < k and λ k τ k 2, τ k 1 for k > k in the case where N > ; moreover, λ k ν k, ν k+1 for k < k and λ k ν k, ν k+1 for k > k in the case N ; λ k ν k 1, ν k for k < k and λ k ν k 1, ν k for k > k in the case where N >. Hence, by virtue of the Theorem 2.6 we have mλ k = m k for k < k, mλ k = m k+1 for k > k, nλ k = n k for k > k in the case N ; mλ k = m k 1 for k < k, mλ k = m k for k > k, nλ k = n k 1 for k > k in the case N. Let β π 2, π. Then, λ k τ k, τ k+1 for k < k and λ k τ k 1, τ k for k > k in the case N ; λ k τ k 1, τ k for k < k and λ k τ k 2, τ k 1 for k > k in the case where N > ; moreover, λ k ν k, ν k+1 for k < k and λ k ν k 1, ν k for k > k in the case N ; λ k ν k 1, ν k for k < k and λ k ν k 2, ν k 1 for k > k in the case where N >. Hence, by virtue of the Theorem 2.6 we have mλ k = m k, nλ k = n k for k > k in the case N ; mλ k = m k 1, nλ k = n k 1 for k > k in the case N >. 4 Asymptotic formulas for the eigenvalues and eigenvector-functions of problem By [15, Ch. 1, Lemma 11.1] for λ + the following estimates hold uniformly with respect to x, in x, x [, π]: u x, λ = cos ξx, λ α + O1 / λ, 4.1 where vx, λ = sin ξx, λ α + O1 / λ, 4.2 x ξx, λ = λx + 1/2 {p t + rt} dt. 4.3 Remark 4.1. Note that the formula from [15, Ch. 1] has an error. This is due to the fact that in [15, Ch. 1, formula 11.9] the expression for the function βx to be of minus sign, whereby the formula from [15, Ch. 1] must be of the form 4.3. Moreover, the asymptotic formula from [15, Ch. 1] for the eigenvalues of the boundary problem 2.2 as µ = 1 is incorrect, and this formula by [5, formula 3.26] should be in the following form η k 1 = k + α γ 1/2 π {p t + rt} dt π + O k By 2.5 and 2.6 the following location on the real axis of eigenvalues of problem 1.1, 1.2, 2.1 i.e. of problem 2.2 for µ = 1 is valid: if γ, π / 2, then < τ 2 < ν 1 < η 1 1 < τ 1 < ν < η 1 < τ < ν 1 < η 1 1 < τ 1 <, 4.5 if γ π / 2, π, then < τ 2 < η 1 1 < ν 1 < τ 1 < η 1 < ν < τ < η 1 1 < ν 1 < τ 1 <. 4.6 Theorem 4.2. The following asymptotic formulas hold for sufficiently large k k > k λ k = k + 1 HN sgn β Hk + α β 1/2 π {p t + rt} dt π + O k

8 8 Z. S. Aliyev and P. R. Manafova Proof. Recall that the eigenvalues of problem are the roots of the equation 2.8. Substituting uπ, λ and vπ, λ from the estimates 4.1 and 4.2, we obtain which is implied by 4.3 that sin sin ξπ, λ α + β + O1 / λ =, π λπ α + β + 1/2 { p t + rt} dt + O 1 =. 4.8 λ It is obvious that for a large λ, Eq. 4.8 has solutions of the form see [15, p. 57] λ k π α + β + 1/2 π { p t + rt} dt = k + τπ + δ k, k Z, where τ is some integer which dependence of β and sgn k. Inserting these values in 4.8, we see that sin δ k = O 1 k, so that δk = O 1 k. Therefore for the eigenvalues of the problem we obtain the following asymptotic formula λ k = k + τ + α β 1/2 π { p t + rt}dt + O π k By location of eigenvalues of the problem for large k > k see proof of the Theorem 3.3, relations 4.5 and 4.6 and formula 4.4 it follows that Now inserting 4.1 in 4.9, we obtain 4.7. τ = 1 HN sgn β Hk. 4.1 By for sufficiently large k > k we obtain the following asymptotic formulas for the components of the eigenvector-functions = of problem : References ux,λk vx,λ k uk x v k x x u k x = cos λ k x + 1/2 {p t + rt} dt α x v k x = sin λ k x + 1/2 {p t + rt} dt α + O1 / k, + O1 / k. [1] S. Albeverio, R. Hryniv, Ya. Mykytyuk, Inverse spectral problems for Sturm Liouville operators in impedance form, J. Funct. Anal , MR ; url [2] Z. S. Aliev, Basis properties in L p of systems of root functions of a spectral problem with spectral parameter in a boundary condition, Differ. Equ , No. 6, MR ; url [3] Z. S. Aliev, A. A. Dunyamalieva, Defect basis property of a system of root functions of a Sturm Liouville problem with spectral parameter in the boundary conditions, Differ. Equ , No. 1, url [4] Z. S. Aliev, H. Sh. Rzayeva, Oscillation properties of the eigenvector-functions of the one-dimensional Dirac s canonical system, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 4214, Special issue: In memory of M. G. Gasymov on his 75-th birthday,

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