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,π]-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) = | λ(n,+) - λ(n,-)|, where λ(..) are the eigenvalues of L=L(v) considered on [0,π] 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)=Aexp(2ix)+Bexp(-2ix) with complex a, b, A, B \neq 0, then the system of root functions of L consists eventually of eigenfunctions. For antiperiodic bc this system is a Riesz basis if |aA|=|bB| (then v \in X), and it is not a basis if |aA| \neq |bB|. For periodic bc the system of root functions is a Riesz basis (and v \in X) always.
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.
A perturbation decaying to 0 at infinity and not too irregular at 0 introduces at most a discrete set of eigenvalues into the spectral gaps of a one-dimensional Dirac operator on the half-line. We show that the number of these eigenvalues in a compact subset of a gap in the essential spectrum is given by a quasi-semiclassical asymptotic formula in the slow-decay limit, which for power-decaying perturbations is equivalent to the large-coupling limit. This asymptotic behaviour elucidates the origin of the dense point spectrum observed in spherically symmetric, radially periodic three-dimensional…
We prove that limit-periodic Dirac operators generically have spectra of zero Lebesgue measure and that a dense set of them have spectra of zero Hausdorff dimension. The proof combines ideas of Avila from a Schrödinger setting with a new commutation argument for generating open spectral gaps. This overcomes an obstacle previously observed in the literature; namely, in Schrödinger-type settings, translation of the spectral measure corresponds to small $L^\infty$-perturbations of the operator data, but this is not true for Dirac or CMV operators. The new argument is much more model-independent. …
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…
A generalized two-dimensional periodic Dirac operator is considered, with $L^{\infty}$-matrix-valued coefficients of the first order derivatives and with complex matrix-valued potential. It is proved that if the matrix-valued potential has zero bound relative to the free Dirac operator, then the spectrum of the operator in question contains no eigenvalues.
The one-dimensional Dirac operator with periodic potential $V=\begin{pmatrix} 0 & \mathcal{P}(x) \\ \mathcal{Q}(x) & 0 \end{pmatrix}$, where $\mathcal{P},\mathcal{Q}\in L^2([0,π])$ subject to periodic, antiperiodic or a general strictly regular boundary condition $(bc)$ has discrete spectrums. It is known that, for large enough $|n|$ in the disc centered at $n$ of radius 1/4, the operator has exactly two (periodic if $n$ is even or antiperiodic if $n$ is odd) eigenvalues $λ_n^+$ and $λ_n^-$ (counted according to multiplicity) and one eigenvalue $μ_n^{bc}$ corresponding to the boundary conditio…
In this paper, for d > 2, we prove the absolute continuity of the spectrum of a d-dimensional periodic Dirac operator with some discontinuous magnetic and electric potentials. In particular, for d = 3, electric potentials from Zygmund classes $L^3\ln ^{1+δ}L(K)$, $δ>0$, and also ones with Coulomb singularities, with constraints on charges depending on the magnetic potential, are admitted (here K is the fundamental domain of the period lattice).
One of the main tools used to understand both qualitative and quantitative spectral behaviour of periodic and almost periodic Schrödinger operators is the method of gauge transform. In this paper, we extend this method to an abstract setting, thus allowing for greater flexibility in its applications that include, among others, matrix-valued operators. In particular, we obtain asymptotic expansions for the density of states of certain almost periodic systems of elliptic operators, including systems of Dirac type. We also prove that a range of periodic systems including the two-dimensional Dirac…
We study the system of root functions (SRF) of Hill operator $Ly = -y^{\prime \prime} +vy $ with a singular potential $v \in H^{-1}_{per}$ and SRF of 1D Dirac operator $ Ly = i {pmatrix} 1 & 0 0 & -1 {pmatrix} \frac{dy}{dx} + vy $ with matrix $L^2$-potential $v={pmatrix} 0 & P Q & 0 {pmatrix},$ 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 o…
We consider the first order periodic systems perturbed by a $2N\ts 2N$ 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-p…
The absolute continuity of the spectrum for the periodic Dirac operator $$ \hat D=\sum_{j=1}^n(-i\frac {\partial}{\partial x_j}-A_j)\hat α_j + \hat V^{(0)}+\hat V^{(1)}, x\in R^n, n\geq 3, $$ is proved given that either $A\in C(R^n;R^n)\cap H^q_{loc}(R^n;R^n)$, 2q > n-2, or the Fourier series of the vector potential $A:R^n\to R^n$ is absolutely convergent. Here, $\hat V^{(s)}=(\hat V^{(s)})^*$ are continuous matrix functions and $\hat V^{(s)}\hat α_j=(-1}^s\hat α_j\hat V^{(s)}$ for all anticommuting Hermitian matrices $\hat α_j$, $\hat α_j^2=\hat I$, s=0,1.
Dirac operators on periodic planar graphs: the right definition advocated by flat bands
The Open University of Japan（日本开放大学）；Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío（比奥比奥大学理学院数学系）；Institute for Liberal Arts and Sciences & Graduate School of Mathematics, Nagoya University（名古屋大学人文学部与研究生院数学研究院）
Spectral Pollution and How to Avoid It (With Applications to Dirac and Periodic Schrödinger Operators)
CNRS and Laboratoire de Mathématiques (CNRS UMR 8088), Université de Cergy-Pontoise（法国国家科学研究中心与数学实验室（CNRS UMR 8088），塞吉蓬图瓦兹大学）；Ceremade (CNRS UMR 7534), Université Paris-Dauphine（巴黎道夫菲纳大学与决策数学实验室（CNRS UMR 7534））
Let $L$ be the Hill operator or the one dimensional Dirac operator on the interval $[0,π].$ If $L$ is considered with Dirichlet, periodic or antiperiodic boundary conditions, then the corresponding spectra are discrete and for large enough $|n|$ close to $n^2 $ in the Hill case, or close to $n, \; n\in \mathbb{Z}$ in the Dirac case, there are one Dirichlet eigenvalue $μ_n$ and two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $λ_n^-, \, λ_n^+ $ (counted with multiplicity). We give estimates for the asymptotics of the spectral gaps $γ_n = λ_n^+ - λ_n^-$ and deviations $ …
We consider 2- and 3-dimensional Schrödinger or generalized Schrödinger-Pauli operators with the non-degenerating magnetic field in the open domain under certain non-degeneracy assumptions we derive pointwise spectral asymptotics. We also consider asymptotics of some related expressions (see below). For all asymptotics loops rather than periodic trajectories play important role.
