Jussi Behrndt, Markus Holzmann, Andrea Mantile, Andrea Posilicano
We provide a limiting absorption principle for self-adjoint realizations of Dirac operators with electrostatic and Lorentz scalar $δ$-shell interactions supported on regular compact surfaces. Then we show completeness of the wave operators and give a representation formula for the scattering matrix.
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In this survey we gather recent results on Dirac operators coupled with $δ$-shell interactions. We start by discussing recent advances regarding the question of self-adjointness for these operators. Afterward we switch to an approximation question: can these operators be recovered as limits of Dirac operators coupled with squeezing potentials ? We also discuss spectral features of these models. Namely, we recall the main spectral consequences of a resolvent formula and conclude the survey by commenting a result of asymptotic nature for the eigenvalues in the gap of a Dirac operator coupled wit…
We develop an approach to prove self-adjointness of Dirac operators with boundary or transmission conditions at a $\mathcal{C}^2$-compact surface without boundary. To do so we are lead to study the layer potential induced by the Dirac system as well as to define traces in a weak sense for functions in the appropriate Sobolev space. Finally, we introduce Calderón projectors associated with the problem and illustrate the method in two special cases: the well-known MIT bag model and an electrostatic $δ$-shell interaction.
In this note the two dimensional Dirac operator $A_η$ with an electrostatic $δ$-shell interaction of strength $η\in\mathbb R$ supported on a straight line is studied. We observe a spectral transition in the sense that for the critical interaction strengths $η=\pm 2$ the continuous spectrum of $A_η$ inside the spectral gap of the free Dirac operator $A_0$ collapses abruptly to a single point.
We consider limiting Carleman weights for Dirac operators and prove corresponding Carleman estimates. In particular, we show that limiting Carleman weights for the Laplacian also serve as limiting weights for Dirac operators. As an application we consider the inverse problem of recovering a Lipschitz continuous magnetic field and electric potential from boundary measurements for the Pauli Dirac operator.
We study a static, spherically symmetric system of (2j+1) massive Dirac particles, each having angular momentum j, j=1,2,..., in a classical gravitational and SU(2) Yang-Mills field. We show that for any black hole solution of the associated Einstein-Dirac-Yang/Mills equations, the spinors must vanish identically outside of the event horizon.
General $δ$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation
该研究针对二维带闭曲线支撑一般局部奇异相互作用的Dirac算子，将相互作用分解为四类基本相互作用的线性组合开展系统分析，探讨了算子的自伴性、谱性质及正则势Dirac算子的逼近方法。
We establish a limiting absorption principle for some long range perturbations of the Dirac systems at threshold energies. We cover multi-center interactions with small coupling constants. The analysis is reduced to study a family of non-self-adjoint operators. The technique is based on a positive commutator theory for non self-adjoint operators, which we develop in appendix. We also discuss some applications to the dispersive Helmholzt model in the quantum regime.
On the approximation of the $δ$-shell interaction for the 3-D Dirac operator
本文研究三维Dirac算子与δ-壳层相互作用的逼近问题，证明在一般局部势V下算子强预解收敛到δ-壳层Dirac哈密顿量，且耦合常数非线性依赖于V。
Given a bounded smooth domain $Ω\subset\mathbb{R}^3$, we explore the relation between couplings of the free Dirac operator $-iα\cdot\nabla+mβ$ with pure electrostatic shell potentials $λδ_{\partialΩ}$ ($λ\in\mathbb{R}$) and some perturbations of those potentials given by the normal vector field $N$ on the shell $\partialΩ$, namely $\{λ_e+λ_n(α\cdot N)\}δ_{\partialΩ}$ ($λ_e$, $λ_n\in\mathbb{R}$). Under the appropiate change of parameters, the couplings with perturbed and unperturbed electrostatic shell potentials yield unitary equivalent self-adjoint operators. The proof relies on the construct…
Spectral asymptotics of the Dirac operator in a thin shell
本文研究薄管状邻域内带无限质量边界条件的 Dirac 算子谱，证明邻域收缩时特征值渐近由含几何性电势与 Yang-Mills 势的 Schrödinger 算子决定。
On two-dimensional Dirac operators with $δ$-shell interactions supported on unbounded curves with straight ends
本文研究支撑在具有直端点无界曲线上的二维Dirac算子，刻画了带静电、Lorentz标量和反常磁δ-壳相互作用的自伴实现与本质谱，并证明了几何诱导离散特征值的存在性。
In this paper the two-dimensional Dirac operator with a general hermitian $δ$-shell interaction supported on a straight line is introduced as a self-adjoint operator and its spectral properties are investigated in detail. In particular, it is demonstrated that the singularly continuous spectrum is always empty and that by switching a certain $δ$-shell interaction on, it is possible to generate an eigenvalue in the gap of the spectrum of the free operator or to partially or even fully close the gap. This suggests that the studied operators may serve as interesting continuum toy-models for Dirac…
This paper deals with the massive three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction supported on a compact smooth surface. The rigorous definition of the operator involves suitable transmission conditions along the surface. After showing the self-adjointness of the resulting operator we switch to the investigation of its spectral properties, in particular, to the existence and non-existence of eigenvalues. In the case of an attractive coupling, we study the eigenvalue asymptotics as the mass becomes large and show that the behavior of the individual eigenvalues a…
In this paper we consider Dirac operators in $\mathbb R^n$, $n\geq2$, with a potential $V$. Under mild decay and continuity assumptions on $V$ and some spectral assumptions on the operator, we prove a limiting absorption principle for the resolvent, which implies a family of Strichartz estimates for the linear Dirac equation. For large potentials the dynamical estimates are not an immediate corollary of the free case since the resolvent of the free Dirac operator does not decay in operator norm on weighted $L^2$ spaces as the frequency goes to infinity.
Spinor quantum states of the Dirac's core/shell at fm-space
本文提出球形核心/壳层中狄拉克粒子在张量效应下的模型，研究约1.0 fm尺度内能级随阱宽变化，并发现同质量双能级及重质量下出现反粒子型反常解。
We prove a global limiting absorption principle on the entire real line for free, massless Dirac operators $H_0 = α\cdot (-i \nabla)$ for all space dimensions $n \in \mathbb{N}$, $n \geq 2$. This is a new result for all dimensions other than three, in particular, it applies to the two-dimensional case which is known to be of some relevance in applications to graphene. We also prove an essential self-adjointness result for first-order matrix-valued differential operators with Lipschitz coefficients.
Let $Ω\subset\mathbb{R}^3$ be an open set, we study the spectral properties of the free Dirac operator $\mathcal{H}$ coupled with the singular potential $V_κ=(εI_4 +μβ+η(α\cdot N))δ_{\partialΩ}$. The open set $Ω$ can be either a $\mathcal{C}^2$-bounded domain or a locally deformed half-space. In both cases, self-adjointness is proved and several spectral properties are given. In particular, we give a complete description of the essential spectrum of $\mathcal{H}+V_κ$ for the so-called critical combinations of coupling constants, when $Ω$ is a locally deformed half-space. Finally, we introduce …
In this article we investigate spectral properties of the coupling $H+V_λ$, where $H=-iα\cdot\nabla +mβ$ is the free Dirac operator in $\mathbb R^3$, $m>0$ and $V_λ$ is an electrostatic shell potential (which depends on a parameter $λ\in\mathbb R$) located on the boundary of a smooth domain in $\mathbb R^3$. Our main result is an isoperimetric-type inequality for the admissible range of $λ$'s for which the coupling $H+V_λ$ generates pure point spectrum in $(-m,m)$. That the ball is the unique optimizer of this inequality is also shown. Regarding some ingredients of the proof, we make use of th…
On the discrete spectrum of Dirac operators with Lorentz-scalar $δ$-shell interactions supported on unbounded curves
研究支撑于折线局部形变曲线上的洛伦兹标量δ壳相互作用的狄拉克算子，证明其能隙内离散本征值有限，且当τ足够小或绝对值足够大时离散谱非空，该谱由几何结构诱导。
In this article Dirac operators $A_{η, τ}$ coupled with combinations of electrostatic and Lorentz scalar $δ$-shell interactions of constant strength $η$ and $τ$, respectively, supported on compact surfaces $Σ\subset \mathbb{R}^3$ are studied. In the rigorous definition of these operators the $δ$-potentials are modelled by coupling conditions at $Σ$. In the proof of the self-adjointness of $A_{η, τ}$ a Krein-type resolvent formula and a Birman-Schwinger principle are obtained. With their help a detailed study of the qualitative spectral properties of $A_{η, τ}$ is possible. In particular, the e…
The Landau-Dirac operator with shell interactions: self-adjointness and clustering
本文研究带壳相互作用的二维朗道-狄拉克算子，分析其自伴性，证明非临界情形下本征值随曲线对数容量在朗道-狄拉克能级处积累，还得到外边界值问题的本征值聚类结果。
We discuss the limiting behavior (using the notion of Γ-limit) of the 3d nonlinear elasticity for thin shells around an arbitrary smooth 2d surface. In particular, under the assumption that the elastic energy of deformations scales like h^4 (where h is the thickness of a shell), we derive a limiting theory which is a generalization of the von Kármán theory for plates.
On two-dimensional Dirac operators with critical delta-shell interactions
研究二维Dirac算子在临界δ壳相互作用下的本质谱特性，揭示直线和圆几何中孤立点的不同本征值行为。
Non-local relativistic $δ$-shell interactions
该研究构造了二维、三维Dirac算子的新自伴实现，将其对应含超曲面单层分布的形式表达式并解释为标度非局域势Dirac算子的极限，同时分析了其谱性质。
We review some results on the spectral theory of Schr{ö}dinger and Dirac operators. We focus on two aspects: the existence of embbedded eigen-values in the essential spectrum and the limiting absorption principle. They both are important for Physics, in general, and for Scattering theory, in particular. We chose to include a special form of oscillations in the potential of the considered operators to illustrate the diversity of behaviours that can exist in the two selected topics. Concerning the limiting absorption principle, we discuss several methods to prove it. To known, old or recent resu…
Eigenvalue estimate for the Dirac-Witten operator on locally reducible Riemannian manifolds
本文研究局部可约类空子流形上Dirac-Witten算子的特征值问题，利用内蕴与外蕴表达式给出最优下界，并分析了极限情形。
Convergence of generalized MIT bag models to Dirac operators with zigzag boundary conditions
本文研究广义MIT袋算子向锯齿型边界条件Dirac算子的预解式收敛，证明强收敛成立而范数收敛不成立，并指出唯一障碍是极限算子存在无限重特征值，投影到其正交补后可得范数收敛。
Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. Limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary.
Congruence of Dirac operators with applications to generalized MIT bag models
研究提出带壳相互作用Dirac算子的同余变换，为广义MIT袋模型的自伴性与无穷质量诠释提供新观测结果
We get optimal lower bounds for the eigenvalues of the submanifold Dirac operator on locally reducible Riemannian manifolds in terms of intrinsic and extrinsic expressions. The limiting-cases are also studied. As a corollary, one gets several known results in this direction.
We investigate the self-adjointness of the two-dimensional Dirac operator $D$, with quantum-dot and Lorentz-scalar $δ$-shell boundary conditions, on piecewise $C^2$ domains with finitely many corners. For both models, we prove the existence of a unique self-adjoint realization whose domain is included in the Sobolev space $H^{1/2}$, the formal form domain of the free Dirac operator. The main part of our paper consists of a description of the domain of $D^*$ in terms of the domain of $D$ and the set of harmonic functions that verify some mixed boundary conditions. Then, we give a detailed study…
In this paper, we present the Dirac equation for the Mobius-square-Yukawa potentials including the tensor interaction term within the framework of pseudospin and spin symmetry limit with arbitrary spin-orbit quantum number . We obtained the energy eigenvalues and the corresponding wave functions using the supersymmetry method. The limiting cases of this potential model reduce to the Deng-Fan, Yukawa and Coulomb potentials, respectively
We carry out the spectral analysis of matrix valued perturbations of 3-dimensional Dirac operators with variable magnetic field of constant direction. Under suitable assumptions on the magnetic field and on the pertubations, we obtain a limiting absorption principle, we prove the absence of singular continuous spectrum in certain intervals and state properties of the point spectrum. Various situations, for example when the magnetic field is constant, periodic or diverging at infinity, are covered. The importance of an internal-type operator (a 2-dimensional Dirac operator) is also revealed in …
We carry out the spectral analysis of singular matrix valued perturbations of 3-dimensional Dirac operators with variable magnetic field of constant direction. Under suitable assumptions on the magnetic field and on the perturbations, we obtain a limiting absorption principle, we prove the absence of singular continuous spectrum in certain intervals and state properties of the point spectrum. Constant, periodic as well as diverging magnetic fields are covered, and Coulomb potentials up to the physical nuclear charge Z<137 are allowed. The importance of an internal-type operator (a 2-dimensiona…
Physical self-adjoint extensions and their spectra of the one-dimensional Dirac type Hamiltonian operator in which both the mass and velocity are constant except for a finite jump at one point of the real axis are correctly found. Different boundary conditions on envelope wave functions are studied, and the limiting case of equal masses (with no mass jump) is reviewed. Transport across one-dimensional heterostructures described by the Dirac equation is considered.
With this paper we provide a mathematical review on the initial-value problem of the one-particle Dirac equation on space-like Cauchy hypersurfaces for compactly supported external potentials. We, first, discuss the physically relevant spaces of solutions and initial values in position and mass shell representation; second, review the action of the Poincaré group as well as gauge transformations on those spaces; third, introduce generalized Fourier transforms between those spaces and prove convenient Paley-Wiener- and Sobolev-type estimates. These generalized Fourier transforms immediately all…
A dynamic scheme basing on equation for T-matrix momentum transfer spectral density and integral representation for Jost function is proposed for local Dirac Hamiltonians in arbitrary N- dimension spaces and for Schrodinger one with singular or nonlocal generalized Yukawa-type potentials. A generalization of the off-shell-Jost function method for that Hamiltonians and universal renormalization procedure of Jost function calculation for singular and nonlocal potentials is proposed.
The Lagrangian of the causal action principle is computed in Minkowski space for Dirac wave functions interacting with classical electromagnetism and linearized gravity in the limiting case when the ultraviolet cutoff is removed. Various surface layer integrals are computed in this limiting case.
In this paper we find spectral properties in the large $N$ limit of Dirac operators that come from random finite noncommutative geometries. In particular for a Gaussian potential the limiting eigenvalue spectrum is shown to be universal regardless of the geometry and is given by the convolution of the semicircle law with itself. For simple non-Gaussian models this convolution property is also evident. In order to prove these results we show that a wide class of multi-trace multimatrix models have a genus expansion.
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