Search results for "math-ph"
showing 10 items of 525 documents
A computational method for the Helmholtz equation in unbounded domains based on the minimization of an integral functional
2012
Abstract We study a new approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains. Our approach is based on the minimization of an integral functional arising from a volume integral formulation of the radiation condition. The index of refraction does not need to be constant at infinity and may have some angular dependency as well as perturbations. We prove analytical results on the convergence of the approximate solution. Numerical examples for different shapes of the artificial boundary and for non-constant indexes of refraction will be presented.
Quantum waveguides with magnetic fields
2019
International audience; We study generalised quantum waveguides in the presence of moderate and strong external magnetic fields. Applying recent results on the adiabatic limit of the connection Laplacian we show how to construct and compute effective Hamiltonians that allow, in particular, for a detailed spectral analysis of magnetic waveguide Hamiltonians. We apply our general construction to a number of explicit examples, most of which are not covered by previous results.
First-Principles Computed Rate Constant for the O + O 2 Isotopic Exchange Reaction Now Matches Experiment
2018
We show, by performing exact time-independent quantum molecular scattering calculations, that the quality of the ground electronic state global potential energy surface appears to be of utmost importance in accurately obtaining even as strongly averaged quantities as kinetic rate constants. The oxygen isotope exchange reaction, 18O + 32O2, motivated by the understanding of a complex long-standing problem of isotopic ozone anomalies in the stratosphere and laboratory experiments, is explored in this context. The thermal rate constant for this key reaction is now in quantitative agreement with all experimental data available to date. A significant recent progress at the frontier of three rese…
An algorithm for computing geometric relative velocities through Fermi and observational coordinates
2013
We present a numerical method for computing the \textit{Fermi} and \textit{observational coordinates} of a distant test particle with respect to an observer. We apply this method for computing some previously introduced concepts of relative velocity: \textit{kinematic}, \textit{Fermi}, \textit{spectroscopic} and \textit{astrometric} relative velocities. We also extend these concepts to non-convex normal neighborhoods and we make some convergence tests, studying some fundamental examples in Schwarzschild and Kerr spacetimes. Finally, we show an alternative method for computing the Fermi and astrometric relative velocities.
Discrete-Gauss states and the generation of focussing dark beams
2014
Discrete-Gauss states are a new class of gaussian solutions of the free Schr\"odinger equation owning discrete rotational symmetry. They are obtained by acting with a discrete deformation operator onto Laguerre-Gauss modes. We present a general analytical construction of these states and show the necessary and sufficient condition for them to host embedded dark beams structures. We unveil the intimate connection between discrete rotational symmetry, orbital angular momentum, and the generation of focussing dark beams. The distinguishing features of focussing dark beams are discussed. The potential applications of Discrete-Gauss states in advanced optical trapping and quantum information pro…
Spherical symmetric parabolic dust collapse: ${\cal C}^{1}$ matching metric with zero intrinsic energy
2016
The collapse of marginally bound, inhomogeneous, pressureless (dust) matter, in spherical symmetry, is considered. The starting point is not, in this case, the integration of the Einstein equations from some suitable initial conditions. Instead, starting from the corresponding general exact solution of these equations, depending on two arbitrary functions of the radial coordinate, the fulfillment of the Lichnerowicz matching conditions of the interior collapsing metric and the exterior Schwarzschild one is tentatively assumed (the continuity of the metric and its first derivatives on the time-like hypersurface describing the evolution of the spherical 2-surface boundary of the collapsing cl…
Ferromagnetism of the Hubbard Model at Strong Coupling in the Hartree-Fock Approximation
2005
As a contribution to the study of Hartree-Fock theory we prove rigorously that the Hartree-Fock approximation to the ground state of the d-dimensional Hubbard model leads to saturated ferromagnetism when the particle density (more precisely, the chemical potential mu) is small and the coupling constant U is large, but finite. This ferromagnetism contradicts the known fact that there is no magnetization at low density, for any U, and thus shows that HF theory is wrong in this case. As in the usual Hartree-Fock theory we restrict attention to Slater determinants that are eigenvectors of the z-component of the total spin, {S}_z = sum_x n_{x,\uparrow} - n_{x,\downarrow}, and we find that the ch…
A non self-adjoint model on a two dimensional noncommutative space with unbound metric
2013
We demonstrate that a non self-adjoint Hamiltonian of harmonic oscillator type defined on a two-dimensional noncommutative space can be diagonalized exactly by making use of pseudo-bosonic operators. The model admits an antilinear symmetry and is of the type studied in the context of PT-symmetric quantum mechanics. Its eigenvalues are computed to be real for the entire range of the coupling constants and the biorthogonal sets of eigenstates for the Hamiltonian and its adjoint are explicitly constructed. We show that despite the fact that these sets are complete and biorthogonal, they involve an unbounded metric operator and therefore do not constitute (Riesz) bases for the Hilbert space $\L…
Non-Hermitian Hamiltonian for a Modulated Jaynes-Cummings Model with PT Symmetry
2015
We consider a two-level system such as a two-level atom, interacting with a cavity field mode in the rotating wave approximation, when the atomic transition frequency or the field mode frequency is periodically driven in time. We show that in both cases, for an appropriate choice of the modulation parameters, the state amplitudes in a generic $n${-}excitation subspace obey the same equations of motion that can be obtained from a \emph{static} non-Hermitian Jaynes-Cummings Hamiltonian with ${\mathcal PT}$ symmetry, that is with an imaginary coupling constant. This gives further support to recent results showing the possible physical interest of ${\mathcal PT}$ symmetric non-Hermitian Hamilto…
Pseudobosons, Riesz bases, and coherent states
2010
In a recent paper, Trifonov suggested a possible explicit model of a PT-symmetric system based on a modification of the canonical commutation relation. Although being rather intriguing, in his treatment many mathematical aspects of the model have just been neglected, making most of the results of that paper purely formal. For this reason we are re-considering the same model and we repeat and extend the same construction paying particular attention to all the subtle mathematical points. From our analysis the crucial role of Riesz bases clearly emerges. We also consider coherent states associated to the model.