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.
AMS Subject Classifications::
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 Citation1–6. 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 Citation1–3,Citation7–22. 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 Citation23.
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 Citation24,Citation25 and the problem for Dirac operator with boundary conditions independent of eigenparameter was considered in Citation26.
Dirac systems with spectral parameter in the boundary conditions described the behaviour of a relativistic particle in an electromagnetic field Citation27. 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], ak, bk(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 Citation28. In Citation29 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 Citation26. 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 λn, n ∈ Z, and from the characteristic Equation (3.3) eigenvalues λn 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 Citation30. We denote by y(x, λn) = (y1(x, λn), y2(x, λn))T an eigenfunction corresponding to the eigenvalue λn 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 μn 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 {λn: 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 Citation5,Citation30:
(7)
where φ1,0(x, λ) = (cos λx, sin λx)T and
(8)
where φ2,0(x, λ) = (−sin λx, cos λx)T. Moreover, kernels Ki(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 aibi, i = 1, 2, are the usual inner product in L2[a, b].
Denote
(10)
Multiplying (1.1) with λn by
and (2.2) with λn by y(x, λn) (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 Ik(λ) = o(1) for λ → ∞ which is a consequence of the Riemann–Lebesgue lemma Citation31.
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) βk = 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 | ||||
2. | Let m(n) 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 Ik(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 = reiθ.
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 Citation32 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 λn, 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, λn) and satisfy the same boundary condition at 1 and
on [b, 1]. This means that
(45)
on [b, 1], where αn 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 Ik(λ).
In the same way, we obtain that
Notice that eigenvalues λn and μn possess the asymptotic expression (2.1). We can count the number of λn and μn located inside the disc of radius r; we get
of λn's and
of μn's. Thus, the total number of λn's and μn'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 μn 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).
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