zbMATH — the first resource for mathematics

Eigenvalue interlacing for first order differential systems with periodic \(2 \times 2\) matrix potentials and quasi-periodic boundary conditions. (English) Zbl 1409.34075
The paper deals with the self-adjoint Dirac system in the form \[ JY' + Q Y = \lambda Y, \tag{1} \] where \(J = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\), \(Q = \begin{pmatrix} q_1 & q \\ q & q_2 \end{pmatrix}\), the functions \(q\), \(q_1\) and \(q_2\) are real-valued, integrable and \(\pi\)-periodic. It is shown that the eigenvalues of the boundary value problem for equation (1) with the boundary conditions \[ Y(\pi) = \pm R(\theta) Y(\theta), \] where \(R(\theta) = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{pmatrix}\), coincide with the values of \(\lambda\), such that the discriminant \(\Delta_{\theta} = \text{Tr} (\mathbb Y(\pi)^T R(\theta))\) equals \(\pm 2\). Here \(\mathbb Y(x)\) is the solution of (1), satisfying the initial condition \(\mathbb Y(0) = \mathbb I\). The authors obtain explicit formulas for the \(\lambda\)-derivative of the discriminant \(\Delta_{\theta}\) and monotonicity results for the first and the second \(\lambda\)-derivatives. The main results of the paper are several interlacing theorems for the eigenvalues of (1), corresponding to various boundary conditions.
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)
Full Text: DOI
[1] J. BELLISSARD, B. IOCHUM, E. SCOPPOLA, D. TESTARD, Spectral properties of one dimensional quasi-crystals, Commun. Math. Phys. 125 (1989), 527-543. · Zbl 0825.58010
[2] I. BINDER, M. VODA, On Optimal Separation of Eigenvalues for a Quasiperiodic Jacobi Matrix, Commun. Math. Phys. 325 (2014), 1063-1106. · Zbl 1312.47036
[3] P. A. BINDING, H. VOLKMER, Existence and asymptotics of eigenvalues of indefinite systems of Sturm-Liouville and Dirac type, J. Diff. Eq. 172 (2001) 116-133. · Zbl 0993.34024
[4] M. B. BROWN, M. S. P. EASTHAM, K. M. SCHMIDT, Periodic Differential Operators, Birkh¨auser, 2013.
[5] E. A. CODDINGTON, N. LEVINSON, Theory of ordinary differential equations, McGraw-Hill Publishing, 1955. · Zbl 0064.33002
[6] S. CURRIE, B. A. WATSON, T. T. ROTH, First order systems inC2onR with periodic matrix potentials and vanishing instability intervals, Math. Meth. Appl. Sci. 38 (2015), 4435-4447. · Zbl 1344.34034
[7] L. H. ELIASSON, Discrete one-dimensional quasi-periodic Schr¨odinger operators with pure point spectrum, Acta. Math. 179 (1997), 153-196. · Zbl 0908.34072
[8] S. G. KREIN, Functional Analysis, Nauka, Moskow, 1972.
[9] B. M. LEVITAN, I. S. SARGSJAN, Sturm-Liouville and Dirac operators, Kluwer Academic Publishers, 1991.
[10] E. J. MCSHANE, Integration, Princeton University Press, 1944.
[11] T. V. MISYURA, Characterization of the spectra of the periodic and antiperiodic boundary value problems that are generated by the Dirac operator I, Teor. Funktsi˘i Funktsional. Anal. i Prolozhen., 30 (1978), 90-101.
[12] T. V. MISYURA, Characterization of the spectra of the periodic and antiperiodic boundary value problems that are generated by the Dirac operator II, Teor. Funktsi˘i Funktsional. Anal. i Prolozhen., 31 (1979), 102-109.
[13] I. M. NABIEV, Solution of the Inverse Quasiperiodic Problem for the Dirac System, Matematicheskie Zametki 89 (2011), 885-893. · Zbl 1235.34043
[14] A. S ¨UTO¨, The spectrum of a quasiperiodic Schr¨odinger operator, Commun. Math. Phys. 111 (1987), 409-415.
[15] J. WEIDMANN, Spectral theory of ordinary differential operators, Lecture notes in Mathematics 1258, Springer-Verlag, 1987. · Zbl 0647.47052
[16] C.-F. YANG, X.-P. YANG, Some Ambarzumyan-type theorems for Dirac operators, Inverse Problems, 23 (2007), 2565-2574. · Zbl 1153.34006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.