Isospectral Dirac operators

We give the description of self-adjoint regular Dirac operators, on $[0, \pi]$, with the same spectra.


Remark 1.4.
It is natural to call this a Marchenko theorem, since it is an analogue of the famous theorem of V. A. Marchenko [15], in the case for Sturm-Liouville problem. The proof of this theorem for the case p, q ∈ AC[0, π] there is in the paper [18]. The detailed proof for the case p, q ∈ L 2 R [0, π] there is in [7] (see also [4-6, 8, 10, 19]).
Let us fix some Ω ∈ L 2 R [0, π] and consider the set of all canonical potentialsΩ = pq q −p , with the same spectra as Ω: Our main goal is to give the description of the set M 2 (Ω) as explicit as it possible. From the uniqueness theorem the next corollary easily follows.
SinceΩ ∈ M 2 (Ω), then a n (Ω) have similar to (1.8) asymptotics. Since a n (Ω) and a n (Ω) are positive numbers, there exist real numbers t n = t n (Ω), such that a n (Ω) a n (Ω) = e t n . From the latter equality and from (1.8) follows that It is easy to see, that the sequence {t n ; n ∈ Z} is also from l 2 , i.e. ∑ ∞ n=−∞ t 2 n < ∞. Since all a n (Ω) are fixed, then from the corollary 1.5 and the equality a n (Ω) = a n (Ω)e −t n we will get the following corollary.
Thus, each isospectral potential is uniquely determined by a sequence {t n ; n ∈ Z}. Note, that the problem of description of isospectral Sturm-Liouville operators was solved in [3,11,12,17].
For Dirac operators the description of M 2 (Ω) is given in [8]. This description has a "recurrent" form, i.e. at the first in [8] is given the description of a family of isospectral potentials Ω(x, t), t ∈ R, for which only one norming constant a m (Ω(·, t)) different from a m (Ω) (namely, a m (Ω(·, t)) = a m (Ω)e −t ), while the others are equal, i.e. a m (Ω(·, t)) = a m (Ω), when n = m.
. Then, for arbitrary t ∈ R, λ n (Ω, t) = λ n (Ω) for all n ∈ Z, a n (Ω, t) = a n (Ω) for all n ∈ Z\{m} and a m (Ω, t) = a m (Ω)e −t . The normalized eigenfunctions of the problem L(Ω(·, t), α) are given by the formulae: Theorem 1.7 shows that it is possible to change exactly one norming constant, keeping the others. As examples of isospectral potentials Ω andΩ we can present Ω(x) ≡ 0 = 0 0 0 0 and where t ∈ R is an arbitrary real number and m ∈ Z is an arbitrary integer.
Changing successively each a m (Ω) by a m (Ω)e −t m , we can obtain any isospectral potential, corresponding to the sequence {t m ; m ∈ Z} ∈ l 2 . It follows from the uniqueness Theorem 1.3 that the sequence, in which we change the norming constants, is not important.
In [8] were used the following designations:  Then We see, that each potential matrix Ω(x, T m ) defined by normalized eigenfunctions hm(x, Ω(x, T m−1 )) of the previous operator L(Ω(·, T m−1 ), α). This approach we call "recurrent" description.
In this paper, we want to give a description of the set M 2 (Ω) only in terms of eigenfunctions h n (x, Ω) of the initial operator L(Ω, α) and sequence T ∈ l 2 . With this aim, let us denote by N(T m ) the set of the positions of the numbers in T m , which are not necessary zero, i.e.
where δ ij is a Kronecker symbol. By S (k) p (x, T m ) we denote a matrix which is obtained from the matrix S(x, T m ) by replacing the kth column of S(x, T m ) by H p (x, T m ) = {−(e t k −1)h k p (x)} k∈N(T m ) column, p = 1, 2, Now we can formulate our result as follows. Theorem 1.9. Let T = {t k } k∈Z ∈ l 2 and Ω ∈ L 2 R [0, π]. Then the isospectral potential from M 2 (Ω), corresponding to T, is given by the formula and det S(x, T) = lim m→∞ det S(x, T m ) (the same for det S k p (x, T), p = 1, 2). In addition, for p(x, T) and q(x, T) we get explicit representations: 2 Proof of Theorem 1.9 The spectral function of an operator L(Ω, α) defined as 1 a n (Ω) , i.e. ρ(λ) is left-continuous, step function with jumps in points λ = λ n equals 1 a n and ρ(0) = 0. Let Ω,Ω ∈ L 2 R [0, π] and they are isospectral. It is known (see [1,2,6,13]), that there exists a function G(x, y) such that: It is also known (see, e.g. [1,6,13]), that the function G(x, y) satisfies to the Gelfand-Levitan integral equation: If the potentialΩ from M 2 (Ω) is such that only finite norming constants of the operator L(Ω, α) are different from the norming constants of the operator L(Ω, α), i.e. a n (Ω) = a n (Ω)e −t n , n ∈ N(T m ) and the others are equal, then it means, that (2.4) where δ is Dirac δ-function. In this case the kernel F(x, y) can be written in a form of a finite sum (using notation (1.7)): and consequently, the integral equation (2.2) becomes to an integral equation with degenerated kernel, i.e. it becomes to a system of linear equations and we will look for the solution in the following form: where g k (x) = g k 1 (x) g k 2 (x) is an unknown vector-function. Putting the expressions (2.5) and (2.6) into the integral equation (2.2) we will obtain a system of algebraic equations for determining the functions g k (x): It would be better if we consider the equations (2.7) for the vectors g k = g k 1 g k 2 by coordinates g k 1 and g k 2 to be a system of scalar linear equations: The systems (2.8) might be written in matrix form where the column vectors g p (x, T m ) = {g k p (x, T m )} k∈N(T m ) , p = 1, 2, and the solution can be found in the form (Cramer's rule): Thus we have obtained for g k (x) the following representation: (2.10) and then by putting (2.10) into (2.6) we find the G(x, y, T m ) function. If the potential Ω is from L 1 R , then such is also the kernel G(x, x, T m ) (see [8]), and the relation between them gives as follows: On the other hand we have Ω(x, T k ). (2.12) So, using the Theorem 1.8 and the equality (2.12) we can pass to the limit in (2.11), when m → ∞: Ω(x, T) = Ω(x) + G(x, x, T)B − BG(x, x, T). (2.13) The potentials Ω(x, T) in (1.10) and (2.13) have the same spectral data {λ n (T), a n (T)} n∈Z , and therefore they are the same and Ω(·, T) defined by (2.13) is also from M 2 (Ω). Using (2.6) and (2.10) we calculate the expression G(x, x, T m )B − BG(x, x, T m ) and pass to the limit, obtaining for the p(x, T) and q(x, T) the representations: p (x, T)h k p (x). Theorem 1.9 is proved. For example, when we change just one norming constant (e.g. for T 0 ) we get two independent linear equations: (1 + s 00 (x))g 0 1 (x) = −(e t 0 − 1)h 0 1 (x), (1 + s 00 (x))g 0 2 (x) = −(e t 0 − 1)h 0 2 (x).