On the Spectrum of Two-Point Boundary Value Problems for the Dirac Operator

We consider the spectral problem for a Dirac operator with arbitrary two-point boundary conditions and an arbitrary complex-valued integrable potential. The existence of nontrivial boundary value problems of this type with an unbounded growth of the multiplicity of eigenvalues is established.


INTRODUCTION
In the present paper, we study the Dirac system where y = col (y 1 (x), y 2 (x)), λ ∈ C is the spectral parameter, , and the functions p, q ∈ L 1 (0, π) are complex-valued, with the two-point boundary conditions U (y) ≡ Cy(0) + Dy(π) = 0, where C = a 11 a 12 a 21 a 22 , D = a 13 a 14 a 23 a 24 , the coefficients a ij can be any complex numbers, and the rows of the matrix A = (CD) = a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 are linearly independent.
It is well known that the entries of the matrix E(x, λ) are related by for any x and λ. Let J ij be the determinant formed by the ith and jth columns of A. Set J 0 = J 12 + J 34 , J 1 = J 14 − J 23 , and J 2 = J 13 + J 24 . It was shown in [1] by the transformation operator method that the characteristic determinant ∆(λ) of problem (1), (2), which is equal to can be reduced to the form where the function , is the characteristic determinant of the unperturbed problem By = λy, U (y) = 0 (7) and the functions r j belong to the space L 1 (0, π), j = 1, 2. If p, q ∈ L 2 (0, π) (for short, we write V ∈ L 2 (0, π)), then r j ∈ L 2 (0, π). It follows that the function ∆(λ) is an entire function of exponential type; therefore, we only have the following possibilities for the operator L of problem (1), (2): 1. The spectrum is empty. 2. The spectrum is a finite nonempty set. 3. The spectrum is a countable set without finite limit points. 4. The spectrum fills the entire complex plane. Relations (5) and (6) imply that case 1 is realized for problem (7), for example, with the boundary conditions defined by the matrix and case 4, with the boundary conditions defined by the matrix Let us prove that case 2 is impossible. Let the equation have finitely many roots λ k , k = 1, . . . , n. If C 1 C 2 = 0, then conditions (2) are regular and problem (1), (2) has a countable set of eigenvalues; therefore, C 1 C 2 = 0. Set P (λ) = n k=1 (λ − λ k ). By [2], where a and b are some constants. Assume, for example, that C 2 = 0. Setting λ = −iy in relation (5), where y > 0, we obtain which implies that J 0 e −πy + C 1 + e −πy R(−iy) = P (−iy)e b−i Re ay e (Im a−π)y .
According to [3, p. 36], the expression on the left-hand side in relation (8) tends to C 1 as y → ∞.
If Im a−π ≥ 0, then the expression on the right-hand side in relation (8) tends to infinity in absolute value, and if Im a − π < 0, then it tends to zero. It follows that Obviously, the left-hand side of relation (9) is bounded on the real axis, while the right-hand side is not; that is, we arrive at a contradiction.
Definition. We say that problem (1), (2) has the classical spectral asymptotics if its spectrum is a countable set and the multiplicities of the eigenvalues are uniformly bounded.
The present paper is aimed at constructing problems (1), (2) for which case 3 is realized and the multiplicities of the eigenvalues grow unboundedly, i.e., problems with nonclassical spectral asymptotics.

MAIN RESULTS
Set c j (λ) = c j (π, λ) and s j (λ) = s j (π, λ), j = 1, 2. In addition, let P W σ be the class of entire functions f (z) of the exponential type ≤ σ such that f L2(R) < ∞. It is well known [4] that the functions c j (λ) and s j (λ) admit the representation where g j , h j ∈ P W π , j = 1, 2.
Lemma 1 [5]. The functions u(λ) and v(λ) admit the representations where h, g ∈ P W π , if and only if where λ n = n + ε n and {ε n } ∈ l 2 , and where λ n = n − 1/2 + κ n and {κ n } ∈ l 2 . Consider the Dirac system with the boundary conditions defined by the matrix We will assume that V ∈ L 2 (0, π). It follows from the representation (4) that the characteristic determinant ∆(λ) of problem (1), (2) with matrix A defined in (10) can be reduced to the form where r ∈ L 2 (0, π), and f ∈ P W π . The converse statement holds true as well.
Theorem. For each function f ∈ P W π , there exists a potential V ∈ L 2 (0, π) such that the characteristic determinant ∆(λ) of problem (1), (2) with the matrix A defined by relation (10) and the potential V (x) is identically equal to f (λ).
Since the functionġ belongs to the class P W π , we have, according to [6], Based on this, by the definition of the numbers λ n , we obtaiṅ where ∞ n=−∞ |ρ n | 2 < ∞. Consequently, for all even n sufficiently large in modulus one has the inequalityċ(λ n ) > 0. One can readily see that the inequalityċ(λ n )ċ(λ n+1 ) < 0 holds for all n ∈ Z. It follows that for all n ∈ Z. Note that (15) implies the relation where ∞ n=−∞ |σ n | 2 < ∞.
Consider the quadratic equation It has the roots By Γ(z, r) we denote the disk of radius r centered at point z. One can readily see that all numbers s + n lie inside the disk Γ(1, 1/10) and all numbers s − n lie inside the disk Γ(−1, 1/10). Let s n = s + n if n is odd and s n = s − n if n is even. Since [6] {f (λ n )} ∈ l 2 , it follows from the definition of the numbers s n that where {ϑ n } ∈ l 2 . It also follows from the definition of the numbers s n and inequality (16) that all numbers z n = s n /ċ(λ n ) lie strictly to the left of the imaginary axis, while (17) and (19) imply the relation where {ρ n } ∈ l 2 . Let β n = s n − sin(πλ n ); then {β n } ∈ l 2 in view of (19). Set According to [7, p. 120], the function h belongs to the class P W π , and h(λ n ) = β n . Set s(λ) = sin(πλ) + h(λ); then s(λ n ) = s n = 0, and consequently, the functions s(λ) and c(λ) do not have common roots. Set .
The proof of the theorem is complete.
Examples of functions in the class P W π with roots of arbitrarily high multiplicity are known in the literature (see, e.g., [10,11]). Note that the existence of one-dimensional boundary value problems with an unboundedly increasing multiplicity of eigenvalues was previously established for the Sturm-Liouville operator and an ordinary differential operator of any even order [10][11][12].

OPEN ACCESS
This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.