13,806 research outputs found
Metrics and isospectral partners for the most generic cubic PT-symmetric non-Hermitian Hamiltonian
We investigate properties of the most general PT-symmetric non-Hermitian
Hamiltonian of cubic order in the annihilation and creation operators as a ten
parameter family. For various choices of the parameters we systematically
construct an exact expression for a metric operator and an isospectral
Hermitian counterpart in the same similarity class by exploiting the
isomorphism between operator and Moyal products. We elaborate on the subtleties
of this approach. For special choices of the ten parameters the Hamiltonian
reduces to various models previously studied, such as to the complex cubic
potential, the so-called Swanson Hamiltonian or the transformed version of the
from below unbounded quartic -x^4-potential. In addition, it also reduces to
various models not considered in the present context, namely the single site
lattice Reggeon model and a transformed version of the massive sextic
x^6-potential, which plays an important role as a toy modelto identify theories
with vanishing cosmological constant.Comment: 21 page
Finite volume schemes on Lorentzian manifolds
We investigate the numerical approximation of (discontinuous) entropy
solutions to nonlinear hyperbolic conservation laws posed on a Lorentzian
manifold. Our main result establishes the convergence of monotone and
first-order finite volume schemes for a large class of (space and time)
triangulations. The proof relies on a discrete version of entropy inequalities
and an entropy dissipation bound, which take into account the manifold geometry
accurately and generalize techniques and estimates that were known in the
(flat) Euclidian setting, only. The strong convergence of the scheme then is
then a consequence of the well-posed theory recently developed by Ben-Artzi and
LeFloch for conservation laws on manifolds.Comment: 24 page
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