In the present paper, we prove an abstract functional analytic criterion for a class of linear partial differential operators acting on a domain $Ω\subseteq\Bbb R^n$ which are elliptic in the interior to have compact resolvent. This extends known results for magnetic Schrödinger operators to more general differential operators. We point out the relationship between the Dirac operator in real dimension two and the $\bar\partial$-Laplacian on a certain weighted space on $\Bbb C$ and we use this connection to prove a non-compactness result for its resolvent.
Resolvent estimates for one-dimensional Dirac operators with imaginary potentials
Institute of Applied Mathematics, Graz University of Technology（格拉茨工业大学应用数学研究所）；Analytical and Algebraic Methods in Optimization Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University（同德大学数学与统计学院分析与代数优化方法研究组）
The paper concerns with infinite symmetric block Jacobi matrices $\bf J$ with $p\times p$-matrix entries. We present new conditions for general block Jacobi matrices to be selfadjoint and have discrete spectrum. In our previous papers there was established a close relation between a class of such matrices and symmetric $2p\times 2p$ Dirac operators $\mathrm{\bf D}_{X,α}$ with point interactions in $L^2(\Bbb R; \Bbb C^{2p})$. In particular, their deficiency indices are related by $n_\pm(\mathrm{\bf D}_{X,α})= n_\pm({\bf J}_{X,α})$. For block Jacobi matrices of this class we present several cond…
For certain Dirac operators $ð_ϕ$ associated to a fibered boundary metric $g_ϕ$, we provide a pseudodifferential characterization of the limiting behavior of $(ð_ϕ+kγ)^{-1}$ as $k\searrow 0$, where $γ$ is a self-adjoint operator anti-commuting with $ð_ϕ$ and whose square is the identity. This yields in particular a pseudodifferential characterization of the low energy limit of the resolvent of $ð_ϕ^2$, generalizing a result of Guillarmou and Sher about the low energy limit of the resolvent of the Hodge Laplacian of an asymptotically conical metric. As an application, we use our result to give …
We consider two-dimensional Pauli and Dirac operators with a polynomially vanishing magnetic field. The main results of the paper provide resolvent expansions of these operators in the vicinity of their thresholds. It is proved that the nature of these expansions is fully determined by the flux of the magnetic field. The most important novelty of the proof is a comparison between the spatial asymptotics of the zero modes obtained in two different manners. The result of this matching allows to compute explicitly all the singular terms in the associated resolvent expansions. As an application we…
We prove Lieb-Thirring-type bounds for fractional Schrödinger operators and Dirac operators with complex-valued potentials. The main new ingredient is a resolvent bound in Schatten spaces for the unperturbed operator, in the spirit of Frank and Sabin.
The paper is concerned with the stability property under perturbation $Q\to\widetilde Q$ of different spectral characteristics of a BVP associated in $L^2([0,1];\Bbb C^2)$ with the following $2\times2$ Dirac type equation $$L_U(Q)y=-iB^{-1}y'+Q(x)y=λy,\quad B={\rm diag}(b_1,b_2),\quad b_1<0<b_2,\quad y={\rm col}(y_1,y_2),\quad(1)$$ with a potential matrix $Q\in L^p=L^p([0,1];\Bbb C^{2\times2})$ and subject to regular boundary conditions $Uy=\{U_1,U_2\}y=0$. Our approach to spectral stability relies on the existence of triangular transformation operators $K_Q^\pm$ for system (1) with $Q\in L^1$…
This note aims to give prominence to some new results on the absence and localization of eigenvalues for the Dirac and Klein-Gordon operators, starting from known resolvent estimates already established in the literature combined with the renowned Birman-Schwinger principle.
In this work we prove that the eigenvalues of the $n$-dimensional massive Dirac operator $\mathscr{D}_0 + V$, $n\ge2$, perturbed by a possibly non-Hermitian potential $V$, are localized in the union of two disjoint disks of the complex plane, provided that $V$ is sufficiently small with respect to the mixed norms $L^1_{x_j} L^\infty_{\widehat{x}_j}$, for $j\in\{1,\dots,n\}$. In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on $V$, and in particular the spectrum is the same of the unperturbed operator, namely $σ(\mathscr{D}_0+V)=σ(\m…
The spectral theory on the S-spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional heat diffusion and to the spectral theory for the Dirac operator on manifolds. In this seminal paper we introduce the harmonic functional calculus based on the S-spectrum and on an integral representation of axially harmonic functions. This calculus can be seen as a bridge between harmonic analysis and the spectral theory. The resolvent operator of the harmon…
Elementary commutator method for the Dirac equation with long-range perturbations
Graduate School of Mathematical Sciences, The University of Tokyo（东京大学大学院数理科学研究科）；Department of Mathematics, Graduate School of Science, Kobe University（神户大学理学研究科数学系）
MIT bag model and infinite mass limit in non-smooth domains
National Higher School of Mathematics（国立高等数学学校）；Carl von Ossietzky Universität Oldenburg（奥尔登堡卡尔·冯·奥西茨基大学）；POEMS, CNRS, Inria, ENSTA, Institut Polytechnique de Paris（巴黎理工学院 POEMS 联合实验室（CNRS、Inria、ENSTA））
We study massless 1-dimensional Dirac-Coulomb Hamiltonians, that is, operators on the half-line of the form $D_{ω,λ}:=\begin{bmatrix}-\frac{λ+ω}{x}&-\partial_x \\ \partial_x & -\frac{λ-ω}{x}\end{bmatrix}$. We describe their closed realizations in the sense of the Hilbert space $L^2(\mathbb R_+,\mathbb C^2)$, allowing for complex values of the parameters $λ,ω$. In physical situations, $λ$ is proportional to the electric charge and $ω$ is related to the angular momentum. We focus on realizations of $D_{ω,λ}$ homogeneous of degree $-1$. They can be organized in a single holomorphic family of clos…
We derive a limiting absorption principle on any compact interval in $\mathbb{R} \backslash \{0\}$ for the free massless Dirac operator, $H_0 = α\cdot (-i \nabla)$ in $[L^2(\mathbb{R}^n)]^N$, $n \geq 2$, $N=2^{\lfloor(n+1)/2\rfloor}$, and then prove the absence of singular continuous spectrum of interacting massless Dirac operators $H = H_0 +V$, where $V$ decays like $O(|x|^{-1 - \varepsilon})$. Expressing the spectral shift function $ξ(\,\cdot\,; H,H_0)$ as normal boundary values of regularized Fredholm determinants, we prove that for sufficiently decaying $V$, $ξ(\,\cdot\,;H,H_0) \in C((-\in…
Using the resolvent operator, we develop an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators. The algorithm can achieve arbitrarily high-orders of convergence in terms of a smoothing parameter for computing spectral measures of general differential, integral, and lattice operators. Explicit pointwise and $L^p$-error bounds are derived in terms of the local regularity of the measure. We provide numerical examples, including a partial differential operator, a magnetic tight-binding model of graphene, and compute one thousand eigenvalues …
