2+1 Dimensional Noncommutative Dirac Oscillator and (Anti)-Jaynes-Cummings Models

We study Dirac oscillator in 2+1 dimensional noncommutative space. The model is solved exactly and the relationship with Jaynes-Cummings (JC) or anti-Jaynes-Cummings (AJC) models are investigated. We find that for a positive noncommutative parameter, there is an exact map from the 2+1 dimensional noncommutative Dirac oscillator to AJC model. However, for a negative noncommutative parameter, the noncommutative planar Dirac oscillator contains both AJC and JC terms simultaneously. Our investigation may afford a new way to study relativistic quantum mechanics models in noncommutative space by means of quantum optics method, and vice verse.     

Eigenvalue Boundary Problems for the Dirac Operator

 On a compact Riemannian spin manifold with mean-convex boundary, we analyse the ellipticity and the symmetry of four boundary conditions for the fundamental Dirac operator including the (global) APS condition and a Riemannian version of the (local) MIT bag condition. We show that Friedrich's inequality for the eigenvalues of the Dirac operator on closed spin manifolds holds for the corresponding four eigenvalue boundary problems. More precisely, we prove that, for both the APS and the MIT conditions, the equality cannot be achieved, and for the other two conditions, the equality characterizes respectively half-spheres and domains bounded by minimal hypersurfaces in manifolds carrying non-trivial real Killing spinors. Received: 12 November 2001 / Accepted: 25 June 2002 Published online: 21 October 2002 RID="*" ID="*" Research of S. Montiel is partially supported by a Spanish MCyT grant No. BFM2001-2967 and by European Union FEDER funds     

Eigenvalue density of the non-Hermitian Wilson Dirac operator

We find the lattice spacing dependence of the eigenvalue density of the non-Hermitian Wilson Dirac operator in the ? domain. The starting point is the joint probability density of the corresponding random matrix theory. In addition to the density of the complex eigenvalues we also obtain the density of the real eigenvalues separately for positive and negative chiralities as well as an explicit analytical expression for the number of additional real modes.     

The First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces

We give a formula for the first eigenvalue of the Dirac operator acting on spinor fields of a spin compact irreducible symmetric space G/K.     

On the Chirality Quantum Phase Transition of the Dirac Oscillator

Dirac oscillator subjects to an external magnetic field is re-examined. We show that this model can be mapped onto different quantum optics models if one insists to introduce two kinds of phonons which associate with the excitations of Dirac oscillator and magnetic field respectively. The conclusion about chirality quantum phase transition in the paper “Chirality quantum phase transition in the Dirac oscillator” (Bermudez et al. Phys. Rev. A, 77, 063815 2008) is only valid for a specific mapped quantum optics models rather than the Dirac oscillator itself. Thus, the conclusions about chirality quantum phase transitions in this paper are not universal.     

The Q-deformed Oscillator Algebra with an Integer Number Eigenvalue and a Half Odd Integer Number Eigenvalue

In this paper a new q-deformed oscillator algebra with an integer number eigenvalue and a half odd integer number eigenvalue is proposed. For this algebra, the associated energy spectrum and thermodynamic behavior is discussed.     

The Three-Dimensional Dirac Oscillator in the Presence of Aharonov-Bohm and Magnetic Monopole Potentials

No Heading We study the Dirac equation in 3+1 dimensions with non-minimal coupling to an isotropic radial three-vector potential and in the presence of a static electromagnetic potential. The space component of the electromagnetic potential has angular (non-central) dependence such that the Dirac equation separates completely in spherical coordinates. We obtain solutions for the case where the three-vector potential is linear in the radial coordinate (Dirac oscillator) and the time component of the electro-magnetic potential vanishes. The relativistic energy spectrum and spinor eigenfunctions are obtained.     

An Algebraic Approach to the Kemmer Equation for Dirac Oscillator

We present energy eigenvalues and corresponding eigenfunction of the relativistic Kemmer equation for Dirac oscillator. The associated ladder operators being established, it is shown that these operators satisfy the commutation relation of the su(1,1) group. A suitable representation of this group is presented and analytical matrix elements of some functions are obtained.     

Eigenvalue estimates for the Dirac operator with generalized APS boundary condition

Under two boundary conditions, the generalized Atiyah–Patodi–Singer boundary condition and the modified generalized Atiyah–Patodi–Singer boundary condition, we get the lower bounds for the eigenvalues of the fundamental Dirac operator on compact spin manifolds with nonempty boundary.     

Analytic Continuation of Eigenvalues of a Quartic Oscillator

We consider the Schrödinger operator on the real line with even quartic potential x ⁴ + α x ² and study analytic continuation of eigenvalues, as functions of parameter α. We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues are given by branches of two multi-valued analytic functions, one for even eigenfunctions and one for odd ones. 2. The only singularities of these multi-valued functions in the complex α-plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the α-plane.     

The (2+1) Dirac Equation with a Delta Potential

In this Letter the bound states of (2+1) Dirac equation with the cylindrically symmetric (r–r ₀) potential are discussed. It is surprisingly found that the relation between the radial functions at two sides of r ₀ can be established by an SO(2) transformation. We obtain a transcendental equation for calculating the energy of the bound state from the matching condition in the configuration space. The condition for existence of bound states is determined by the Sturm-Liouville theorem.     

Dirac equation in (1+1)-dimensional curved space-time

The Dirac equation in (1+1)-dimensional curved space-time is solved explicitly for the spatially flat Robertson-Walker space-time and the cigar metric considered by Witten.     

Analytical Expressions of Matrix Elements of Physical Quantities for Dirac Oscillator

The analytical expressions of the matrix elements for physicalquantities are obtained for the Dirac oscillator in two andthree spatial dimensions. Their behaviour for the case ofoperator's square is discussed in details. The two-dimensionalDirac oscillator has similar behavior to that forthree-dimensional one.     

The quantum mechanical two-Coulomb-centre problem in the Dirac equation framework in 2+1 dimensions

With the help of perturbation theory the asymptotic expansions (at small and large internuclear distances R) of the eigenvalues (potential curves) E(R) of the two-Coulomb-centre problem in 2+1 dimensions are obtained. We compare the results obtained with the data from similar approximation for two-Coulomb-centre problem in 3+1 dimensions.     

Eigenvalue Spectrum of a Dirac Particle in Static and Spherical Complex Potential

It has been observed that a quantum theory need not be Hermitian to have a real spectrum. We study the non-Hermitian relativistic quantum theories for many complex potentials, and obtain the real relativistic energy eigenvalues and corresponding eigenfunctions of a Dirac-charged particle in complex statically and spherically symmetric potentials. Complex Dirac–Eckart, complex Dirac–Rosen–Morse II, complex Dirac–Scarf and complex Dirac–Poschl–Teller potential are investigated.     

The (1+1) Dimensional Dirac Equation With Pseudoscalar Potentials: Path Integral Treatment

The supersymetric path integrals in solving the problem of relativistic spinning particle interacting with pseudoscalar potentials is examined. The relative propagator is presented by means of path integral, where the spin degrees of freedom are described by odd Grassmannian variables and the gauge invariant part of the effective action has a form similar to the standard pseudoclassical action given by Berezin and Marinov. After integrating over fermionic variables (Grassmannian variables), the problem is reduced to a nonrelativistic one with an effective supersymetric potential. Some explicit examples are considered, where we have extracted the energy spectrum of the electron and the wave functions. PACS numbers: 03.65. Ca-Formalism, 03.65. Db-Functional analytical methods, 03.65. Pm-Relativistic wave equations.