Abstract. For one-dimensional Dirac operators of the form (original) (raw)
Related papers
1D Dirac operators with special periodic potentials
Bulletin of the Polish Academy of Sciences Mathematics
For 1D Dirac operators Ly= i J y' + v y, where J is a diagonal 2x2 matrix with entrees 1,-1 and v(x) is an off-diagonal matrix with L^2 [0,\pi]-entrees P(x), Q(x) we characterize the class X of pi-periodic potentials v such that: (i) the smoothness of potentials v is determined only by the rate of decay of related spectral gaps gamma (n) = | \lambda (n,+) - \lambda (n,-)|, where \lambda (..) are the eigenvalues of L=L(v) considered on [0,\pi] with periodic (for even n) or antiperiodic (for odd n) boundary conditions (bc); (ii) there is a Riesz basis which consists of periodic (or antiperiodic) eigenfunctions and (at most finitely many) associated functions. In particular, X contains symmetric potentials X_{sym} (\overline{Q} =P), skew-symmetric potentials X_{skew-sym} (\overline{Q} =-P), or more generally the families X_t defined for real nonzero t by \overline{Q} =t P. Finite-zone potentials belonging to X_t are dense in X_t. Another example: if P(x)=a exp(2ix)+b exp(-2ix), Q(x...
Riesz bases consisting of root functions of 1D Dirac operators
Proceedings of the American Mathematical Society, 2012
For one-dimensional Dirac operators \[ L y = i ( 1 a m p ; 0 0 a m p ; − 1 ) d y d x + v y , v = ( 0 a m p ; P Q a m p ; 0 ) , y = ( y 1 y 2 ) , Ly= i \begin {pmatrix} 1 & 0 \\ 0 & -1 \end {pmatrix} \frac {dy}{dx} + v y, \quad v= \begin {pmatrix} 0 & P \\ Q & 0 \end {pmatrix}, \;\; y=\begin {pmatrix} y_1 \\ y_2 \end {pmatrix}, \] subject to periodic or antiperiodic boundary conditions, we give necessary and sufficient conditions which guarantee that the system of root functions contains Riesz bases in L 2 ( [ 0 , π ] , C 2 ) . L^2 ([0,\pi ], \mathbb {C}^2). In particular, if the potential matrix v v is skew-symmetric (i.e., Q ¯ = − P \overline {Q} =-P ), or more generally if Q ¯ = t P \overline {Q} =t P for some real t ≠ 0 , t \neq 0, then there exists a Riesz basis that consists of root functions of the operator L . L.
Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators
Journal of Functional Analysis, 2012
We study the system of root functions (SRF) of Hill operator Ly = −y ′′ + vy with a singular potential v ∈ H −1 per and SRF of 1D Dirac operator Ly = i 1 0 0 −1 dy dx + vy with matrix L 2-potential v = 0 P Q 0 , subject to periodic or anti-periodic boundary conditions. Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property of SRF in L p-spaces and other rearrangement invariant function spaces.
On the basis property of root function systems of Dirac operators with regular boundary conditions
2019
where B is a nonsingular diagonal n× n matrix, B = diag(b−1 1 In1, . . . , b −1 r Inr) ∈ Cn×n, n = n1 + . . . nr, with complex entries bj 6= bk, and Q(x) is a potential matrix takes its origin in the paper by Birkho and Langer [4]. Afterwards their investigations were developed in many directions. Malamud and Oridoroga in [20] established rst general results on completeness of root function systems of boundary value problems for di erential systems (1). A little bit later Lunyov and Malamud in [17] obtained rst general results on Riesz basis property (Riesz basis property with parentheses) for mentioned boundary value problems with a potential matrix Q(x) ∈ L∞. There is an enormous literature related to the spectral theory outlined above, and we refer to [6, 7, 16, 22, 25] and their extensive reference lists for this activity. In the present paper, we study the Dirac system
Characterization of the periodic and antiperiodic spectra of non-self-adjoint Dirac operators
2021
The necessary and sufficient conditions are given for a sequence of complex numbers to be the periodic (or antiperiodic) spectrum of non-self-adjoint Dirac operator. 1. Intoduction One of the important classes of inverse spectral problems is the problem of recovering a system of differential equations from spectral data. The solution of such problems are considered in many papers [12,18, 29-35, and the references therein]. The most studied are such problems for Dirac and Dirac type differential operator. In particular, such problems for canonical Dirac system on a finite interval By′ + V y = λy, (1.1) where y = col(y1(x), y2(x)), B = ( 0 1 −1 0 ) , V (x) = ( p(x) q(x) q(x) −p(x) ) , in selfadjoint case have been studied in detail. In the cases of the Dirichlet and the Newmann boundary conditions reconstruction of a continuous potential from two spectra was carried out in [6], from one spectrum and the norming constants in [5], and from the spectral function in [15]. The analogous re...
Periodic Dirac operator on the half-line
2019
We consider the Dirac operator with a periodic potential on the half-line with the Dirichlet boundary condition at zero. Its spectrum consists of an absolutely continuous part plus at most one eigenvalue in each open gap. The Dirac resolvent admits a meromorphic continuation onto a two-sheeted Riemann surface with a unique simple pole on each open gap: on the first sheet (an eigenvalue) or on the second sheet (a resonance). These poles are called states and there are no other poles. If the potential is shifted by real parameter t, then the continuous spectrum does not change but the states can change theirs positions. We prove that each state is smooth and in general, non-monotonic function of t. We prove that a state is a strictly monotone function of t for a specific potential. Using these results we obtain formulas to recover potentials of special forms.
Spectral estimates for matrix-valued periodic Dirac operators
Asymptotic Analysis, 2008
We consider the first order periodic systems perturbed by a 2N × 2N matrix-valued periodic potential on the real line. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. The Lyapunov function has branch points, which we call resonances. We prove the existence of real or complex resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of the resonances at high energy (in terms of the Fourier coefficients of the potential). We show that there exist two types of gaps: i) stable gaps, i.e., the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, i.e., the endpoints are resonances (real branch points). Moreover, we determine various new trace formulae for potentials and the Lyapunov exponent.
On the Schrödinger operator with a periodic PT-symmetric matrix potential
Journal of Mathematical Physics
In this article, we obtain asymptotic formulas for the Bloch eigenvalues of the operator L generated by a system of Schrödinger equations with periodic PT-symmetric complex-valued coefficients. Then, using these formulas, we classify the spectrum σ(L) of L and find a condition on the coefficients for which σ(L) contains all half line [H, ∞) for some H.
Mathematical Notes, 2009
The paper deals with the Sturm-Liouville operator Ly = −y ′′ + q(x)y, x ∈ [0, 1], generated in the space L 2 = L 2 [0, 1] by periodic or antiperiodic boundary conditions. Several theorems on Riesz basis property of the root functions of the operator L are proved. One of the main results is the following. Let q belong to Sobolev space W p 1 [0, 1] with some integer p ≥ 0 and satisfy the conditions q (k) (0) = q (k) (1) = 0 for 0 ≤ k ≤ s − 1, where s≤ p. Let the functions Q and S be defined by the equalities Q(x) = x 0 q(t) dt, S(x) = Q 2 (x) and let q n , Q n , S n be the Fourier coefficients of q, Q, S with respect to the trigonometric system {e 2πinx } ∞ −∞. Assume that the sequence q 2n − S 2n + 2Q 0 Q 2n decreases not faster than the powers n −s−2. Then the system of eigen and associated functions of the operator L generated by periodic boundary conditions forms a Riesz basis in the space L 2 [0, 1] (provided that the eigenfunctions are normalized) if and only if the condition q 2n − S 2n + Q 0 Q 2n ≍ q −2n − S −2n + 2Q 0 Q −2n , n > 1, holds.