We consider the Dirac system on the interval $[0,1]$ with a spectral parameter $μ\in\mathbb{C}$ and a complex-valued potential with entries from $L_p[0,1]$, where $1\leq p <2$. We study the asymptotic behavior of its solutions in a stripe $|{\rm Im}\,μ|\le d$ for $μ\to \infty$. These results allows us to obtain sharp asymptotic formulas for eigenvalues and eigenfunctions of Sturm--Liouville operators associated with the aforementioned Dirac system.
We characterize the set of rectangular Weyl matrix functions corresponding to Dirac systems with locally square-integrable potentials on a semi-axis and demonstrate a new way to recover the locally square-integrable potential from the Weyl function. Important interconnections between our approach and accelerants of convolution operators are discussed as well.
We prove the Bari-Markus property for spectral projectors of non-self-adjoint Dirac operators on a finite interval with square-integrable matrix-valued potentials and some separated boundary conditions.
We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general Dirac-type operators on half-lines and on $\bbR$. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operators and use it to prove a Borg-type theorem.
We study the non-selfadjoint Dirac system on a finite interval having non-integrable regular singularities in interior points with additional matching conditions at these points. Properties of spectral characteristics are established, and the inverse spectral problem is investigated. We provide a constructive procedure for the solution of the inverse problem, and prove its uniqueness. Moreover, necessary and sufficient conditions for the global solvability of this nonlinear inverse problem are obtained.
We study the non-selfadjoint Dirac system on the line having an non-integrable regular singularity in an interior point with additional matching conditions at the singular point. Special fundamental systems of solutions are constructed with prescribed analytic and asymptotic properties. Behavior of the corresponding Stockes multipliers is established. These fundamental systems of solutions will be used for studying direct and inverse problems of spectral analysis.
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…
In this paper, we study a singular perturbation of a problem used in dimension two to model graphene or in dimension three to describe the quark confinement phenomenon in hadrons. The operators we consider are of the form $H + MβV (x)$, where $H$ is the free Dirac operator, $β$ is a constant matrix, $V (x)$ is a real valued piecewise constant potential having a jump discontinuity across a smooth interface and $M$ is the mass that we can see as a coupling constant. In particular, we perform a complete asymptotic expansion of spectral quantities as the mass $M$ tends to $+\infty$.
In this article, we provide the spectral analysis of a Dirac-type operator on $\mathbb{Z}^2$ by describing the behavior of the spectral shift function associated with a sign-definite trace-class perturbation by a multiplication operator. We prove that it remains bounded outside a single threshold and obtain its main asymptotic term in the unbounded case. Interestingly, we show that the constant in the main asymptotic term encodes the interaction between a flat band and whole non-constant bands. The strategy used is the reduction of the spectral shift function to the eigenvalue counting functio…
In this paper we study direct and inverse problems for discrete and continuous time skew-selfadjoint Dirac systems with rectangular (possibly non-square) pseudo-exponential potentials. For such a system the Weyl function is a strictly proper rational rectangular matrix function and any strictly proper rational matrix function appears in this way. In fact, extending earlier results, given a strictly proper rational matrix function we present an explicit procedure to recover the corresponding potential using techniques from mathematical system and control theory. We also introduce and study a no…
The main focus of this paper is the following matrix Cauchy problem for the Dirac system on the interval $[0,1]:$ \[ D'(x)+\left[\begin{array}{cc} 0 & σ_1(x)\\ σ_2(x) & 0 \end{array} \right] D(x)=iμ\left[\begin{array}{cc} 1 & 0\\ 0 &-1 \end{array} \right]D(x),\quad D(0)=\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right], \] where $μ\in\mathbb{C}$ is a spectral parameter, and $σ_j\in L_2[0,1]$, $j=1,2$. We propose a new approach for the study of asymptotic behaviour of its solutions as $μ\to \infty$ and $|{\rm Im}\,μ|\le d$. As an application, we obtain new, sharp asymptotic formulas for …
I derive sharp semiclassical asymptotics of $\int |e_h(x,y,0)|^2ω(x,y) dx dy$ where $e_h(x,y,τ)$ is the Schwartz kernel of the spectral projector and $ω(x,y)$ is singular as $x=y$. I also consider asymptotics of more general expressions.
I derive sharp semiclassical asymptotics of \int |e_h(x,y,0)|^2ω(x,y)dxdy where e_h(x,y,τ) is the Schwartz kernel of the spectral projector of Magnetic Schroedinger operator and ω(x,y) is singular as x=y. I also consider asymptotics of more general expressions.
Asymptotic behaviors of zero modes of the massless Dirac operator $H=α\cdot D + Q(x)$ are discussed, where $α= (α_1, α_2, α_3)$ is the triple of $4 \times 4$ Dirac matrices, $ D=\frac{1}{i} \nabla_x$, and $Q(x)=\big(q_{jk} (x) \big)$ is a $4\times 4$ Hermitian matrix-valued function with $| q_{jk}(x) | \le C < x >^{-ρ} $, $ρ>1$. We shall show that for every zero mode $f$, the asymptotic limit of $|x|^2f(x)$ as $|x| \to +\infty$ exists. The limit is expressed in terms of an integral of $Q(x)f(x)$.
We prove that the inverse spectral mapping reconstructing the square integrable potentials on [0,1] of Dirac operators in the AKNS form from their spectral data (two spectra or one spectrum and the corresponding norming constants) is analytic and uniformly stable in a certain sense.
This article introduces the notion of good labellings for asymptotic lattices in order to study joint spectra of quantum integrable systems from the point of view of inverse spectral theory. As an application, we consider a new spectral quantity for a quantum integrable system, the quantum rotation number. In the case of two degrees of freedom, we obtain a constructive algorithm for the detection of appropriate labellings for joint eigenvalues, which we use to prove that, in the semiclassical limit, the quantum rotation number can be calculated on a joint spectrum in a robust way, and converge…
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.
We give a proof of the exact asymptotic behaviour in Anderson's Orthogonality Catastrophe for Dirac-$δ$ perturbations. We prove the exact asymptotics of the scalar product of the ground states of two non-interacting Fermi gases confined to a $3$-dimensional ball $B_L$ of radius $L$ in the thermodynamic limit, where the underlying one-particle operators differ by a Dirac-$δ$ perturbation. More precisely, we show the algebraic decay of the correlation determinant $\big|\det\big(\langleφ_j^L, ψ_k^L\rangle\big)_{j,k=1,...,N}\big|^2= L^{-ζ(E)+ \text{o}(1)}$, as $N,L\to\infty$ and $N/|B_L|\to~ ρ>0$,…
We introduce an integrable Hamiltonian system which Lax deforms the Dirac operator D=d+d* on a finite simple graph or compact Riemannian manifold. We show that the nonlinear isospectral deformation always leads to an expansion of the original space, featuring a fast inflationary start. The nonlinear evolution leaves the Laplacian L=D^2 invariant so that linear Schroedinger or wave dynamics is not affected. The expansion has the following effects: a complex structure can develop and the nonlinear quantum mechanics asymptotically becomes the linear relativistic Dirac wave equation u''=Lu. While …
We study the asymptotic behavior, as Planck's constant $\hbar\to 0$, of the number of discrete eigenvalues and the Riesz means of Pauli and Dirac operators with a magnetic field $μ\mathbf{B}(x)$ and an electric field. The magnetic field strength $μ$ is allowed to tend to infinity as $\hbar\to 0$. Two main types of results are established: in the first $μ\hbar\le constant$ as $\hbar\to 0$, with magnetic fields of arbitrary direction; the second results are uniform with respect to $μ\ge 0$ but the magnetic fields have constant direction. The results on the Pauli operator complement recent work o…
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 $ …
