Determination of Dirac operator with eigenparameter-dependent boundary conditions from interior spectral data

Abstract

In this article, inverse spectra problems for Dirac operator with eigenparameter-dependent boundary conditions are studied. By using the approach similar to those in Hochstadt and Lieberman [H. Hochstadt and B. Lieberman, An inverse Sturm–Liouville problem with mixed given data, SIAM J. Appl. Math. 34 (1978), pp. 676–680] and Ramm [A.G. Ramm, Property C for ODE and applications to inverse problems, Operator theory and applications, Vol. 25, AMS, Providence, RI, 2000, pp. 15–75], we prove that (1) a set of values of eigenfunctions at the mid-point of the interval [0, 1] and one spectrum suffice to determine the potential Q(x) on the interval [0, 1] and all parameters in the boundary conditions; (2) some information on eigenfunctions at an internal point and parts of two spectra suffice to determine the potential Q(x) on the interval [0, 1] and all parameters in the boundary conditions.

Keywords:

• Dirac operator
• eigenparameter-dependent boundary conditions
• inverse problem
• eigenvalue
• interior spectral data

AMS Subject Classifications::

• Primary 34A55
• Secondary 34L05
• 34L40
• 45C05

1. Introduction

Inverse spectral analysis has been an important research topic in mathematical physics. Inverse problems of spectral analysis involve reconstruction of a linear operator from its spectral characteristics 1–6. For inverse Sturm–Liouville problems, such characteristics are two spectra for different boundary conditions, one spectrum and normalizing constants, spectral functions, nodal points (zeros of eigenfunctions) as given spectral data, scattering data, the Weyl function 1–3,7–22. Such problems play an important role in mathematics and have many applications in natural sciences. Inverse problems are studied for certain special classes of ordinary differential operators. Typically, in inverse eigenvalue problems, one measures the frequencies of a vibrating system and tries to infer some physical properties of the system. An early important result in this direction, which gave vital impetus for the further development of inverse problem theory, was obtained in 23.

Inverse problem for interior spectral data of the differential operator lies in reconstructing this operator by some eigenvalues and information on eigenfunctions at an internal point in the interval considered. The similar problems for the Sturm–Liouville operator were studied in 24,25 and the problem for Dirac operator with boundary conditions independent of eigenparameter was considered in 26.

Dirac systems with spectral parameter in the boundary conditions described the behaviour of a relativistic particle in an electromagnetic field 27. We consider Dirac operator L(Q; α, a₀, b₀; β, a₁, b₁) defined by a differential expression (1) with Here Q(x) is a real-valued function from the class C²[0, 1], and the domain of L(Q; α, a₀, b₀; β, a₁, b₁) (2) where AC([0, 1]) denotes a set of absolutely continuous functions on [0, 1], a_k, b_k(k = 0, 1), α and β are real constants; moreover, and . Throughout this article, we assume that

Generally, the potential Q(x) for Dirac operator L(Q; α, a₀, b₀; β, a₁, b₁) is uniquely determined by known collection of eigenvalues and normalizing constants or two spectra 28. In 29 we show that if the potential Q(x) is prescribed on the half interval [1/2, 1], then a single spectrum of the Dirac operator with eigenparameter dependent boundary conditions on the whole interval [0, 1] suffices to determine Q(x) on another half-interval [0, 1] and the boundary condition at x = 0. As far as I know, the inverse problem of interior spectral data for Dirac systems with spectral parameter contained in the boundary conditions has not been considered before. The aim of this article is to give two uniqueness theorems from some eigenvalues and information on eigenfunctions at an internal point in the interval [0, 1]. Notice that the results obtained are new and natural generalizations of the well known one for the classical Dirac operator which was studied in 26. The novelty of this article lies in a technical expression for products of the initial solutions for Dirac systems, which is more complicated than that of the classical Dirac system, and the representation of certain entire functions that we shall use in the proof of our central results.

2. Main results

The spectrum of the operator L(Q; α, a₀, b₀; β, a₁, b₁) consists of eigenvalues λ_n, n ∈ Z, and from the characteristic Equation (3.3) eigenvalues λ_n satisfy the asymptotic form (3) The fact that eigenvalues of the operator L(Q; α, a₀, b₀; β, a₁, b₁) are real and algebraically simple can be obtained by using the same method as Lemmas 1.1 and 1.2 in 30. We denote by y(x, λ_n) = (y₁(x, λ_n), y₂(x, λ_n))^T an eigenfunction corresponding to the eigenvalue λ_n of L(Q; α, a₀, b₀; β, a₁, b₁). Agree that if a certain symbol δ denotes an object related to Dirac operator L(Q; α, a₀, b₀; β, a₁, b₁), then will denote an analogous object related to another Dirac operator defined by a differential expression (4) with Here is a real-valued function from the class C²[0, 1], and the domain of (5) where , and are real constants; moreover, , and

Let l(n), r(n) be sequences of natural numbers with properties (6) and let μ_n be the eigenvalues of the operator L(Q; α, a₀, b₀; β₁, a₁, b₁), β ≠ β₁ ∈ R.

Now we state the main results of this work.

Theorem 2.1

Let {λ_n: n ∈ Z} and be the eigenvalue set of L(Q; α, a₀, b₀; β, a₁, b₁) and , respectively. If for any n ∈ Z and then and

Note that the solution of inverse problem in Theorem 2.1 is not unique without condition , since a single spectrum cannot determine the potential for Dirac operator. In particular, when , equation is replaced by

Theorem 2.2

Let l(n), r(n) and be such that σ₁ > 2b − 1, σ₂ > 2 − 2b. If for any n ∈ Z, then and

3. Proofs

Before proving the theorems we shall first mention some results which will be needed later.

Denote by the solutions of (1.1) satisfying conditions and , respectively. For each fixed x these solutions are entire functions of order no greater than 1 on the parameter λ. Moreover, the following representations hold 5,30: (7) where φ_1,0(x, λ) = (cos λx, sin λx)^T and (8) where φ_2,0(x, λ) = (−sin λx, cos λx)^T. Moreover, kernels K_i(x, t), i = 1, 2, are symmetric matrix-valued functions whose entries are continuously differentiable in both of its variables.

Denote by the solution of (1.1) satisfying the condition then from (3.1) and (3.2) we have

The characteristic function w(λ) of the problem (1.1) and (1.2) is defined by the relation and zeros of the entire function w(λ) are all simple and coincide with eigenvalues of the problem (1.1) and (1.2). Using the above asymptotic formulae (3.1) and (3.2), one can easily obtain that for sufficiently large |λ|: (9)

Now we can give the proofs of theorems in this work.

Proof of Theorem 2.1

Denote by y(x, λ) = (y₁(x, λ), y₂(x, λ))^T the solution to Equation (1.1) with initial conditions y₁(0, λ) = λ sin α + b₀ and y₂(0, λ) = λ cos α + a₀, and by the solution to Equation (2.2) with initial conditions and .

Suppose 0 ≤ a < b ≤ 1, for all (a₁, a₂)^T and (b₁, b₂)^T in (L²[a, b])², define an inner product where a_ib_i, i = 1, 2, are the usual inner product in L²[a, b].

Denote (10) Multiplying (1.1) with λ_n by and (2.2) with λ_n by y(x, λ_n) (in the sense of scalar product in R²), respectively, and subtracting, we get (11) Integrating the last equality from 0 to with respect to the variable x, we obtain (12) Together with initial conditions at the point 0 and given assumption on eigenfunction, then it yields (13) and (14) We can rewrite the second term of the left-hand side in (3.6) as (15) where

Denote (16)

Using (3.1) and (3.2), we can show that (17) where the functions , are piecewise-continuously differentiable on 0 ≤ t ≤ x ≤ 1, and

Define an entire function of λ (18) Combining (3.6)–(3.9) with (3.12), we get which implies that the set of zeros of the entire function ω(λ) is contained in the set of zeros of H(λ).

From (3.11) and (3.12), we find that for all complex number λ (19) for some positive constant C₁ and τ = |Im λ|. Define Since the set of zeros of the entire function ω(λ) is contained in the set of zeros of H(λ), we see that the function Φ(λ) is an entire function on the parameter λ. It follows from (3.3) and (3.13) that for |λ| large enough. Thus, by Liouville's theorem, we obtain that for all complex number λ where C is a constant.

Let us show that the constant C = 0. We can rewrite the equation H(λ) = Cω(λ) in the form i.e.

Assertion follows immediately from (3.6), (3.7), (3.8) and (3.9) and I_k(λ) = o(1) for λ → ∞ which is a consequence of the Riemann–Lebesgue lemma 31.

Together with (3.12), using the Riemann–Lebesgue lemma, we see that the limit of the left-hand side of the above equality is equal to zero as real λ → +∞. From this we obtain that C = 0. Thus, the entire function (20)

Moreover, it follows that (21)

Denote (22)

Integrating by parts, we have (23)

Integrating by parts up to two times, we have (24) (25) (26) and (27) where

For convenience, denote (28) where and functions here * denotes some continuous function of t and # denotes some continuous function of the variables x and t.

Substituting expressions (3.17) into (3.15) and dividing two sides of (3.15) by λ, one takes the limit of the left-hand side of (3.15) as real λ → +∞, then it yields that the limit of H₁(λ) as λ → ∞ exists and thus we obtain (29) From (3.11), (3.22) and (3.23) it is easy to obtain and

Substituting expressions (3.18–3.21) into (3.15) and taking H₁(λ) = 0 into account, we have (30) Using the Riemann–Lebesgue lemma we see that the limit of H₂(λ) as λ → ∞ exists and which implies that (31) and With account of H₂(λ) = 0 instead of (3.24), we get (32) on the whole λ-plane. From the completeness of vector-valued functions (exp(2iλt), exp(−2iλt))^T in , it follows that which yields (33) With account of (3.23) and (3.25), there hold (34) from which Equation (3.27) can be written as (35)

Introduce and Equations (3.28) and (3.29) can readily be reduced to a vector form But this equation is a homogeneous Volterra integral equation and has only a zero solution. Thus we have obtained which yields that From (3.10), there holds which imply that i.e. (36)

Using assertions f₁(x) = f₂(x) = 0 for 0 < x < 1, from (3.16) β_k = 0 for , from (3.11) , and B = 0 in Equation (3.22), we obtain Similarly, from A = 0 in Equation (3.22), we have Since a₀ sin α − b₀ cos α < 0, it follows from the last two relations that

To prove that and we should repeat arguments for the supplementary problem defined by a differential expression with and the domain of L(Q₁(x); β, a₁, b₁; α, a₀, b₀) Then we obtain and The proof of theorem is finished.

To prove Theorem 2.2 in this article, we first give a lemma.

Let m(n) be a sequence of natural numbers such that (37)

Lemma 3.1

1.	Let m(n) and be such that σ > 2b. If for any n ∈ Z then and
2.	Let m(n) and be such that σ > 2 − 2b. If for any n ∈ Z then and

Proof

(1) Integrating Equation (3.5) from 0 to b with respect to the variable x, by the assumption it follows that where H(λ) is defined by (3.12) with replaced by ∫^b in expressions I_k(λ).

Next, we shall show that H(λ) = 0 on the whole complex plane.

From (3.11) and (3.12), we see that the entire function H(λ) is a function of exponential type ≤2b. One has (38) for some positive constant C₁, λ = re^iθ.

Let us define an indicator of function H(λ) by the formula (39) Since, |ℑλ| = r|sin θ| and θ = arg λ, by virtue of (3.32) and (3.33) one gets (40) It is known 32 that for any entire function H(λ) of exponential type, not identically zero, one has (41) where n(r) is the number of zeros of H(λ) in the disk |λ| ≤ r. From (3.34) one gets (42) By the assumption (3.31) and the asymptotic expression (2.1) of eigenvalues λ_n, n ∈ Z, for the number of zeros of H(λ) in the disk |λ| ≤ r one gets that for sufficiently large r the estimate (43) It follows that in the case σ > 2b (44) Inequalities (3.35) and (3.38) imply that H(λ) ≡ 0 on the whole λ-plane. As we already mentioned, if H(λ) ≡ 0, then the conclusion of lemma is true.

(2) Note that the interval [b, 1] can be converted to an interval [0, 1 − b] by a transformation of variable x ↦ 1 − x.

To prove (2), we should repeat arguments in part (1) for the supplementary problem L(Q₁(x); β, a₁, b₁; α, a₀, b₀). A direct calculation implies that is a solution to the supplementary problem L(Q₁(x); β, a₁, b₁; α, a₀, b₀) and . Note that . Thus, for the supplementary problem, the assumption in the case (1) is satisfied still. If we repeat the above arguments we can obtain the proof of Lemma.

Proof of Theorem 2.2

Since where r(n) satisfies (2.4) and σ₂ > 2 − 2b, by Lemma 3.1, we obtain that on [b, 1] and Thus, we only need to prove that on [0, b] and

Eigenfunctions y(x, λ_n) and satisfy the same boundary condition at 1 and on [b, 1]. This means that (45) on [b, 1], where α_n are constants independent of variable x.

Integrating Equation (3.5) from 0 to b with respect to the variable x and using (3.39), it follows that where H(λ) is defined by (3.12) with replaced by ∫^b in expressions I_k(λ).

In the same way, we obtain that Notice that eigenvalues λ_n and μ_n possess the asymptotic expression (2.1). We can count the number of λ_n and μ_n located inside the disc of radius r; we get of λ_n's and of μ_n's. Thus, the total number of λ_n's and μ_n's are and Using the same method as in Lemma 3.1, by the assumption σ₁ > 2b − 1 we can show that inequality (3.35) does not hold. Repeating the last part in the proof of Lemma 3.1, we can prove that H(λ) = 0 on the whole λ-plane. This implies that on [0, b] and

Since μ_n are eigenvalues of the operator L(Q(x); α, a₀, b₀; β₁, a₁, b₁) and are eigenvalues of the operator , from the asymptotic expression (2.1), it yields and Using the relation , we obtain . The proof is complete.

Acknowledgements

The author would like to thank the referees for valuable comments. This work was supported by the National Natural Science Foundation of China (11171152/A010602), Natural Science Foundation of Jiangsu Province of China (BK 2010489) and the Outstanding Plan-Zijin Star Foundation of Nanjing University of Science and Technology (AB 41366), and NUST Research Funding (No. AE88787).