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Register a new userMatrix continued fraction solution to the relativistic spin-$0$ Feshbach-Villars equations
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The Feshbach-Villars equations, like the Klein-Gordon equation, are relativistic quantum mechanical equations for spin-$0$ particles. We write the Feshbach-Villars equations into an integral equation form and solve them by applying the Coulomb-Sturmian potential separable expansion method. We consider bound-state problems in a Coulomb plus short range potential. The corresponding Feshbach-Villars Coulomb Greens operator is represented by a matrix continued fraction.
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We propose a solution method for studying relativistic spin-$0$ particles. We adopt the Feshbach-Villars formalism of the Klein-Gordon equation and express the formalism in an integral equation form. The integral equation is represented in the Coulomb-Sturmian basis. The corresponding Greens operator with Coulomb and linear confinement potential can be calculated as a matrix continued fraction. We consider Coulomb plus short range vector potential for bound and resonant states and linear confining scalar potentials for bound states. The continued fraction is naturally divergent at resonant state energies, but we made it convergent by an appropriate analytic continuation.
In this paper we treat the time evolution of unitary elements in the N level system and consider the reduced dynamics from the unitary group U(N) to flag manifolds of the second type (in our terminology). Then we derive a set of differential equations of matrix Riccati types interacting with one another and present an important problem on a nonlinear superposition formula that the Riccati equation satisfies. Our result is a natural generalization of the paper {bf Chaturvedi et al} (arXiv : 0706.0964 [quant-ph]).
The formalism of SUSYQM (SUperSYmmetric Quantum Mechanics) is properly modified in such a way to be suitable for the description and the solution of a classical maximally superintegrable Hamiltonian System, the so-called Taub-Nut system, associated with the Hamiltonian: $$ mathcal{H}_eta ({mathbf{q}}, {mathbf{p}}) = mathcal{T}_eta ({mathbf{q}}, {mathbf{p}}) + mathcal{U}_eta({mathbf{q}}) = frac{|{mathbf{q}}| {mathbf{p}}^2}{2m(eta + |{mathbf{q}}|)} - frac{k}{eta + |{mathbf{q}}|} quad (k>0, eta>0) , .$$ In full agreement with the results recently derived by A. Ballesteros et al. for the quantum case, we show that the classical Taub-Nut system shares a number of essential features with the Kepler system, that is just its Euclidean version arising in the limit $eta to 0$, and for which a SUSYQM approach has been recently introduced by S. Kuru and J. Negro. In particular, for positive $eta$ and negative energy the motion is always periodic; it turns out that the period depends upon $ eta$ and goes to the Euclidean value as $eta to 0$. Moreover, the maximal superintegrability is preserved by the $eta$-deformation, due to the existence of a larger symmetry group related to an $eta$-deformed Runge-Lenz vector, which ensures that in $mathbb{R}^3$ closed orbits are again ellipses. In this context, a deformed version of the third Keplers law is also recovered. The closing section is devoted to a discussion of the $eta<0$ case, where new and partly unexpected features arise.
We introduce an application of the Quasi-Gasdynamic method for a solution of ideal magnetohydrodynamic equations in the modeling of compressible conductive gas flows. A time-averaging procedure is applied for all physical parameters in order to obtain the quasi-gas-dynamic system of equations for magnetohydrodynamics. Evolution of all physical variables is presented in an unsplit divergence form. Divergence-free evolution of the magnetic field is provided by using a constrained transport method based on Stokes theorem. Accuracy and convergence of this method are verified on a large set of standard 1D and 2D test cases.
The iterative method of Sinkhorn allows, starting from an arbitrary real matrix with non-negative entries, to find a so-called scaled matrix which is doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and with all line sums equal to 1. We conjecture that a similar procedure exists, which allows, starting from an arbitrary unitary matrix, to find a scaled matrix which is unitary and has all line sums equal to 1. The existence of such algorithm guarantees a powerful decomposition of an arbitrary quantum circuit.
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