We study the two-dimensional massless Dirac equation for a potential that is allowed to depend on the energy and on one of the spatial variables. After determining a modified orthogonality relation and norm for such systems, we present an application involving an energy-dependent version of the hyperbolic Scarf potential. We construct closed-form bound state solutions of the associated Dirac equation.
The effective mass Klein-Gordon equation in one dimension for the Woods-Saxon potential is solved by using the Nikiforov-Uvarov method. Energy eigenvalues and the corresponding eigenfunctions are computed. Results are also given for the constant mass case.
We present exact analytical solutions of the Dirac equation in $(1+1)$-dimensions for the generalized Kratzer potential by taking the pseudoscalar interaction term as an attractive Coulomb potential. We study the problem for a particular (spin) symmetry of the Dirac Hamiltonian. After a qualitative analyse, we study the results for some special cases such as Dirac-Coulomb problem in the existence of the pseudoscalar interaction, and the pure Coulomb problem by discussing some points about pseudospin and spin symmetries in one dimension. We also plot some figures representing the dependence of the energy on quantum number, and potential parameters.
Two-dimensional Scarf~II quantum model is considered in the framework of Supersymmetrical Quantum Mechanics (SUSY QM). Previously obtained results for this integrable system are systematized, and some new properties are derived. In particular, it is shown that the model is exactly or quasi-exactly solvable in different regions of parameter of the system. The degeneracy of the spectrum is detected for some specific values of parameters. The action of the symmetry operators of fourth order in momenta is calculated for the arbitrary wave functions, obtained by means of double shape invariance.
We utilize the relation between soliton solutions of the mKdV and the combined mKdV-KdV equation and the Dirac equation to construct electrostatic fields which yield exact zero energy states of graphene.
The equation of state of a weakly interacting two-dimensional Bose gas is studied at zero temperature by means of quantum Monte Carlo methods. Going down to as low densities as na^2 ~ 10^{-100} permits us for the first time to obtain agreement on beyond mean-field level between predictions of perturbative methods and direct many-body numerical simulation, thus providing an answer to the fundamental question of the equation of state of a two-dimensional dilute Bose gas in the universal regime (i.e. entirely described by the gas parameter na^2). We also show that the measure of the frequency of a breathing collective oscillation in a trap at very low densities can be used to test the universal equation of state of a two-dimensional Bose gas.
A. Schulze-Halberg
,P. Roy
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(2017)
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"Bound states of the two-dimensional Dirac equation for an energy-dependent hyperbolic Scarf potential"
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Pinaki Roy
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