The Asymptotics of Spectral Gaps of a 1D Dirac Operator with Cosine Potential

Letters in Mathematical Physics -     

On Eigenvalues of the One-Dimensional Dirac Operator with Oscillatory Decreasing Potential

In this paper we study the one-dimensional Dirac operator with oscillatory decreasing potential. Using the technique of asymptotic integration together with the ideas of the averaging method, we obtain the formulas for the possible eigenvalues and the eigenfunctions.     

Dirac Operator on Fuzzy Sphere with U（1） Dirac Monopole Background

We treat the Dirac operator on the Fuzzy sphere with the help of the generalized coherent state. It is shown that the derivatives can be constructed by Moyal product with symbol of the operator. We obtain the eigenvalue of the free fermion Dirac operator as same as the result by [Hajime Aoki, Satoshi Iso, and Kelichi Nagao, Phys. Rev.D67 (2003) 065018]. Meanwhile, we also give the eigenvalue of Dirac operator with U(1) Dirac monopole background.     

Dirac Operator on Fuzzy Sphere with U(1) Dirac Monopole Background

We treat the Dirac operator on the Fuzzy sphere with the help of the generalized coherent state. It is shown that the derivatives can be constructed by Moyal product with symbol of the operator. We obtain the eigenvalue of the free fermion Dirac operator as same as the result by [Hajime Aoki, Satoshi Iso, and Kelichi Nagao, Phys. Rev.D67 (2003) 065018]. Meanwhile, we also give the eigenvalue of Dirac operator with U(1) Dirac monopole background.     

Versal Deformations of a Dirac Type Differential Operator

Abstract

If we are given a smooth differential operator in the variable x ∈ R/2πZ, its normal form, as is well known, is the simplest form obtainable by means of the Diff(S ¹)-group action on the space of all such operators. A versal deformation of this operator is a normal form for some parametric infinitesimal family including the operator. Our study is devoted to analysis of versal deformations of a Dirac type differential operator using the theory of induced Diff(S ¹)-actions endowed with centrally extended Lie-Poisson brackets. After constructing a general expression for tranversal deformations of a Dirac type differential operator, we interpret it via the Lie-algebraic theory of induced Diff(S ¹)-actions on a special Poisson manifold and determine its generic moment mapping. Using a Marsden-Weinstein reduction with respect to certain Casimir generated distributions, we describe a wide class of versally deformed Dirac type differential operators depending on complex parameters.     

Nonlinear Dirac Operator and Quaternionic Analysis

Properties of the Cauchy–Riemann–Fueter equation for maps between quaternionic manifolds are studied. Spaces of solutions in case of maps from a K3–surface to the cotangent bundle of a complex projective space are computed. A relationship between harmonic spinors of a generalized nonlinear Dirac operator and solutions of the Cauchy–Riemann–Fueter equation are established.     

Small-Scale Structure of Space-Time and Dirac Operator

Connes' noncommutative geometry is presented and the relevance of the Dirac operator in the elucidation of the structure of space-time at the Planck length is discussed.     

Inverse Problem for the Discrete 1D Schrödinger Operator with Small Periodic Potentials

Consider the discrete 1D Schrödinger operator on ? with an odd 2k periodic potential q. For small potentials we show that the mapping: q→ heights of vertical slits on the quasi-momentum domain (similar to the Marchenko-Ostrovski maping for the Hill operator) is a local isomorphism and the isospectral set consists of 2 ^k distinct potentials. Finally, the asymptotics of the spectrum are determined as q→0.     

Geometric Potential and Dirac Quantization

A fundamental problem regarding the Dirac quantization of a free particle on an () curved hypersurface embedded in N flat space is the impossibility to give the same form of the curvature‐induced quantum potential, the geometric potential as commonly called, as that given by the Schrödinger equation method where the particle moves in a region confined by a thin‐layer sandwiching the surface. This problem is resolved by means of a previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator‐ordering‐free section is identified and is then found sufficient to lead to the expected form of geometric potential.     

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.     

Global Estimates of Resonances for 1D Dirac Operators

We discuss resonances for 1D massless Dirac operators with compactly supported potentials on the real line. We estimate the sum of the negative power of all resonances in terms of the norm of the potential and the diameter of its support.     

Quantum States of a Trapped Dirac Particle in a Pseudoscalar Potential

We obtain the quantum states of a trapped Dirac particle in the presence of a pseudoscalar potential. A change in the geometrical boundary condition can cause an effective electromagnetic field which can act on the trapped object. The nonrelativistic limit is discussed.     

Fermion Particle Production in a Periodic Potential

In this paper we study fermion particle production in the early universe. The present work is motivated to restudy the fermion particle production from the basics and compare the results in the literature through another method developed by one of the present author. One of the authors (SB) has developed a method, known as complex trajectory WKB method, to study particle production in curved as well as flat spacetime. In the present work we have tried to compare the CWKB method with that of other works, current in the literature. In this work we have obtained the particle production amplitude starting from the basics and test our results through both analytical and numerical calculations. For fermion particle production, we first do analytical calculations with a toy example to calculate the production amplitude and verify the same doing fourth order Runge-Kutta calculation. As most problems relevant to early universe are not amenable to analytical calculations, we then take up to study the particle production in periodic potential, generally used in inflationary cosmology. We recheck two recent approaches and obtain almost identical results as that obtained by Greene and Kofman. We also verify the result through CWKB method. Boson particle production has been discussed elsewhere, we discuss it briefly in connection with CWKB. In the present work we generalize the CWKB results of boson production to fermion production. Our works will enable one to understand the various phenomena in early universe related to particle production. Using CWKB we calculate the occupation number and some other results for fermion particle production. The present work will help us clarify the variant results of fermion production current in the literature.     

Langevin Dynamics with a Tilted Periodic Potential

We study a Langevin equation for a particle moving in a periodic potential in the presence of viscosity γ and subject to a further external field α. For a suitable choice of the parameters α and γ the related deterministic dynamics yields heteroclinic orbits. In such a regime, in absence of stochastic noise both confined and unbounded orbits coexist. We prove that, with the inclusion of an arbitrarly small noise only the confined orbits survive in a sub-exponential time scale.     

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.     

Determinants,Grassmanniannians and Elliptic Boundary Problems for the Dirac Operator

We study the relations between different determinants of the Dirac operator over a manifold with boundary considered as sections of a holomorphic line bundle over the Grassmannian of boundary conditions of Atiyah–Patodi–Singer type.     

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves. This is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing-wave solution is stable in and unstable in under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the nonradial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.     

Chaotic Behavior of a Brownian Particle in a Periodic Potential

The classical deterministic dynamics of a Brownian particle with a time-dependent periodic perturbation in a spatially periodic potential is investigated. We have constructed a perturbed chaotic solution near the heteroclinic orbit of the nonlinear dynamics system by using the Constant-Variation method. Theoretical analysis and numerical result show that the motion of the Brownian particle is a kind of chaotic motion. The corresponding chaotic region in parameter space is obtained analytically and numerically.