Planar Dirac electron in Coulomb and magnetic fields: A Bethe ansatz approach

Chiang, C. H.; Ho, Choon‐Lin

doi:10.1063/1.1418426

Cited by 23 publications

(15 citation statements)

References 19 publications

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“…Consider the (2+1)-dimensional relativistic system of a Dirac electron (with mass m e ) in the presence of an external electromagnetic field A µ . This system was also examined in [25] via a similar Bethe ansatz approach. The covariant Dirac equation (in the unit = c = 1) has the form…”

Section: Planar Dirac Electron In Coulomb and Magnetic Fieldsmentioning

confidence: 99%

Exact polynomial solutions of second order differential equations and their applications

Zhang¹

2012

J. Phys. A: Math. Theor.

130

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We find all polynomials Z(z) such that the differential equationwhere X(z), Y (z), Z(z) are polynomials of degree at most 4, 3, 2 respectively, has polynomial solutions S(z) = n i=1 (z − z i ) of degree n with distinct roots z i . We derive a set of n algebraic equations which determine these roots. We also find all polynomials Z(z) which give polynomial solutions to the differential equation when the coefficients of X(z) and Y (z) are algebraically dependent. As applications to our general results, we obtain the exact (closed-form) solutions of the Schrödinger type differential equations describing: 1) Two Coulombically repelling electrons on a sphere; 2) Schrödinger equation from kink stability analysis of φ 6 -type field theory; 3) Static perturbations for the non-extremal Reissner-Nordström solution; 4) Planar Dirac electron in Coulomb and magnetic fields; and 5) O(N ) invariant decatic anharmonic oscillator.2000 Mathematics Subject Classification. 34A05, 30C15, 81U15, 81Q05, 82B23.

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Section: Planar Dirac Electron In Coulomb and Magnetic Fieldsmentioning

confidence: 99%

Exact polynomial solutions of second order differential equations and their applications

Zhang¹

2012

J. Phys. A: Math. Theor.

130

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“…This situation is alluded to in Refs. [3,4]. On the other hand, we are always able to find well behaved numerical solutions in a soft-core potential.…”

Section: E Zeeman Effectmentioning

confidence: 78%

Relativistic Photoionization Computations with the Time Dependent Dirac Equation

Gordon¹,

Hafizi²

2016

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Standard Form 298 (Rev. 8-98)Prescribed by ANSI Std. Z39.18 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Ionization of inner shell electrons by laser fields often occurs in the relativistic regime. A complete description of this phenomenon requires both relativistic and quantum mechanical treatment. The Dirac equation describes the motion of a spin one-half particle in an external electromagnetic field. The stationary states in both Coulomb and soft-core potentials are solved either analytically or numerically. These are used as initial conditions in time dependent calculations. It is found that, at least in one case, two-dimensional simulations predict a greater ionization rate than the one predicted by the Coulomb corrected strong field approximation. Computational performance is important due to the rapid oscillations in a relativistic wavefunction. Multiple General Purpose Graphical Processing Units are utilized in parallel to speed up calculations. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) SPONSOR / MONITOR'S ACRONYM(S) 9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) SPONSOR / MONITOR'S REPORT NUMBER(S)

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“…The consistency of the linear homogeneous system generated by the three-term recurrence relation (16), for j = 0, 1, • • • , m + 1, enforces the vanishing of the determinant, denoted by ∆ (m+1) , that establish the sufficient condition…”

Section: The Inverse Square-root Potentialmentioning

confidence: 99%

“…Chiang et al [16] considered an electron moving in a Coulombic field, A 0 = Z e/r, with a constant homogeneous magnetic field described by A 1 = −B y/2 and A 2 = B x/2. With this choice of potential and for r > 0, (64) becomes…”

Section: Two-dimensional Dirac Equationmentioning

confidence: 99%

On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application

et al. 2020

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The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree n, n−1, and n−2 respectively. We show that for n≥3 the necessary condition is not enough to ensure the existence of the polynomial solutions. Applying Scheffé’s criteria to this differential equation we have extracted n generic equations that are analytically solvable by two-term recurrence formulas. We give the closed-form solutions of these generic equations in terms of the generalized hypergeometric functions. For arbitrary n, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly along with the necessary and sufficient conditions. We demonstrate the validity of the algorithm by constructing the polynomial solutions for the case of n=4. We also demonstrate the simplicity and applicability of our constructive approach through applications to several important equations in theoretical physics such as Heun and Dirac equations.

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Planar Dirac electron in Coulomb and magnetic fields: A Bethe ansatz approach

Cited by 23 publications

References 19 publications

Exact polynomial solutions of second order differential equations and their applications

Exact polynomial solutions of second order differential equations and their applications

Relativistic Photoionization Computations with the Time Dependent Dirac Equation

On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application

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