Let $D_t$, $t \in [0,1]$ be an arbitrary 1-parameter family of Dirac type operators on a two-dimensional disk with $m-1$ holes. Suppose that all operators $D_t$ have the same symbol, and that $D_1$ is conjugate to $D_0$ by a scalar gauge transformation. Suppose that all operators $D_t$ are considered with the same locally elliptic boundary condition, given by a vector bundle over the boundary. Our main result is a computation of the spectral flow for such a family of operators. The answer is obtained up to multiplication by an integer constant depending only on the number of the holes in the disk. This constant is calculated explicitly for the case of the annulus ($m=2$).
It is shown that a spin-1/2 fermion coupled to the axially symmetric electromagnetic vector potential has the same matrix structure as that one for the planar Dirac oscillator. In particular, the planar Dirac oscillator can be interpreted as a charged particle minimally coupled to a transverse homogeneous magnetic field.
Consider a Dirac operator defined on the whole plane with a mass term of size m supported outside a domain Omega. We give a simple proof for the norm resolvent convergence, as m goes to infinity, of this operator to a Dirac operator defined on Omega with infinite mass boundary conditions. The result is valid for bounded and unbounded domains and gives estimates on the speed of convergence. Moreover, the method easily extends when adding external matrix-valued potentials.
We study the effect of curvature and torsion on the solitons of the nonlinear Dirac equation considered on planar and space curves. Since the spin connection is zero for the curves considered here, the arc variable provides a natural setting to understand the role of curvature and then we can obtain the transformation for the 1+1 dimensional Dirac equation directly from the metric. Depending on the curvature, the soliton profile either narrows or expands. Our results may be applicable to yet-to-be-synthesized curved quasi-one dimensional Bose condensates.
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）
The planar Dirac and the topologically massive vector gauge fields are unified into a supermultiplet involving no auxiliary fields. The superPoincaré symmetry emerges from the $osp(1|2)$ supersymmetry realized in terms of the deformed Heisenberg algebra underlying the construction. The non-relativistic limit yields spin 1/2 as well as new, spin 1 "Lévy-Leblond-type" equations which, together, carry an N=2 superSchrödinger symmetry. Part of the latter has its origin in the universal enveloping algebra of the superPoincaré algebra.
The Dirac equation for an electron in two spatial dimensions in the Coulomb and homogeneous magnetic fields is a physical example of quasi-exactly solvable systems. This model, however, does not belong to the classes based on the algebra $sl(2)$ which underlies most one-dimensional and effectively one-dimensional quasi-exactly solvable systems. In this paper we demonstrate that the quasi-exactly solvable differential equation possesses a hidden $osp(2,2)$ superalgebra.
The Dirac equation for an electron in two spatial dimensions in the Coulomb and homogeneous magnetic fields is an example of the so-called quasi-exactly solvable models. The solvable parts of its spectrum was previously solved from the recursion relations. In this work we present a purely algebraic solution based on the Bethe ansatz equations. It is realised that, unlike the corresponding problems in the Schrödinger and the Klein-Gordon case, here the unknown parameters to be solved for in the Bethe ansatz equations include not only the roots of wave function assumed, but also a parameter from…
The Dirac equation for an electron in two spatial dimensions in the Coulomb and homogeneous magnetic fields is discussed. For weak magnetic fields, the approximate energy values are obtained by semiclassical method. In the case with strong magnetic fields, we present the exact recursion relations that determine the coefficients of the series expansion of wave functions, the possible energies and the magnetic fields. It is found that analytic solutions are possible for a denumerably infinite set of magnetic field strengths. This system thus furnishes an example of the so-called quasi-exactly so…
The inverse problem of recovering a smooth simply connected multisheet planar domain from its Steklov spectrum is equivalent to the problem of determination, up to a gauge transform, of a smooth positive function $a$ on the unit circle from the spectrum of the operator $aΛ$, where $Λ$ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by $Z_m(a)={\rm Tr}[(aΛ)^{2m}-(aD)^{2m}]$ for every smooth function $a$. In the case of a positive $a$, zeta-invariants are determined by the Steklov spectrum. We obtain some estimate from below for $Z_m(a)$ in the case of a real f…
We consider Dirac operators defined on planar domains. For a large class of boundary conditions, we give a direct proof of their self-adjointness in the Sobolev space $H^1$.
A detailed Hamiltonian analysis for a five-dimensional St{ü}eckelberg theory with a compact dimension is performed. First, we develop a pure Dirac's analysis of the theory, we show that after performing the compactification, the theory is reduced to four-dimensional St{ü}eckelberg theory plus a tower of Kaluza-Klein modes. We develop a complete analysis of the constraints, we fix the gauge and we show that there are present pseudo-Goldstone bosons. Then we quantize the theory by constructing the Dirac brackets. As complementary work, we perform the Faddeev-Jackiw quantization for the theory un…
Let $D$ be a (generalized) Dirac operator on a non-compact complete Riemannian manifold $M$ acted on by a compact Lie group $G$. Let $v:M --> Lie(G)$ be an equivariant map, such that the corresponding vector field on $M$ does not vanish outside of a compact subset. These data define an element of $K$-theory of the transversal cotangent bundle to $M$. Hence a topological index of the pair $(D,v)$ is defined as an element of the completed ring of characters of $G$. We define an analytic index of $(D,v)$ as an index space of certain deformation of $D$ and we prove that the analytic and topologica…
Exact analytic solutions are found for the Dirac equation in 2+1 dimensions for a spin-one-half particle in a combination of the Lorentz 3-vector and scalar Coulomb as well as Aharonov--Bohm potentials. We employ the two-component Dirac equation which contains a new parameter introduced by Hagen to describe the spin of the spin-1/2 particle. We derive a transcendental equations that implicitly determine the energy spectrum of an electron near the negative-energy continuum boundary and the critical charges for some electron states. Fermion pair production from a vacuum by a strong Coulomb field…
Symmetry generators and quantum numbers for fermionic circularly symmetric motion
São Paulo State University (UNESP), School of Engineering and Sciences（圣保罗州立大学（UNESP）工程与科学学院）；CFisUC, Physics Department, University of Coimbra（科英布拉大学物理系CFisUC）
New extrinsic lower bounds are given for the classical Dirac operator on the boundary of a compact domain of a spin manifold. The main tool is to solve some boundary problems for the Dirac operator of the domain under boundary conditions of Atiyah-Patodi-Singer type. Spinorial techniques are used to give simple proofs of classical results for compact embedded hypersurfaces.
In this paper, Dirac`s equation in anisotropic Bianchi-type-I background spacetimes is treated w.r.t. orthonormal frames. By specializing to the massless spinor case and metrics with power-law scale factors and planar symmetry, an analytical expression of the approximate time evolution operator is derived. By use of this operator, all approximate spinor solutions can be generated. By construction, these solutions agree asymptotically and at early times with the exact solutions. The operator approach is also well adapted for studying the case of small deviations from conformally flat background…
In this paper, we get the Kastler-Kalau-Walze theorem associated to Dirac operators with torsion on compact manifolds with boundary. We give two kinds of operator-theoretic explanations of the gravitational action in the case of 4-dimensional compact manifolds with flat boundary. Furthermore, we get the Kastler-Kalau-Walze type theorem for four dimensional complex manifolds associated with nonminimal operators.
In this paper, we prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-form perturbations on $4$-dimensional compact manifolds.
We consider the massive Dirac equation in the exterior region of the 5-dimensional Myers-Perry black hole. Using the resulting ODEs obtained from the separation of variables of the Dirac equation, we construct an integral spectral representation for the solution of the Cauchy problem with compactly supported smooth initial data. We then prove that the probability of presence of a Dirac particle to be in any compact region of space decays to zero as $t\to\infty$, in analogy with the case of the Dirac operator in the Kerr-Newman geometry.
We prove regularity for a class of boundary value problems for first order elliptic systems, with boundary conditions determined by spectral decompositions, under coefficient differentiability conditions weaker than previously known. We establish Fredholm properties for Dirac-type equations with these boundary conditions. Our results include sharp solvability criteria, over both compact and non-compact manifolds; weighted Poincare and Schroedinger-Lichnerowicz inequalities provide asymptotic control in the non-compact case. One application yields existence of solutions for the Witten equation …
The spectrum and degeneracies of the Dirac operator are analysed on compact coset spaces when there is a non-zero homogeneous background gauge field which is compatible with the symmetries of the space, in particular when the gauge field is derived from the spin-connection. It is shown how the degeneracy of the lowest Landau level in the recently proposed higher dimensional quantum Hall effect is related to the Atiyah-Singer index theorem for the Dirac operator on a compact coset space.
We address the question of the existence of bound states for a suitably projected two-dimensional massless Dirac operator in the presence of a Bessel-Macdonald potential (also known as $K_0$-potential potential), raised by De Lima, Del Cima and Miranda, in Eur.Phys.J. B (2020) 93, 187. Based on Relativistic Hardy Inequality, we prove that this operator has no bound states if $γ\leqslant γ_{\rm crit}$ (subcritical region), where $γ$ is a coupling constant.
