PT-symmetric operators and metastable states of the 1D relativistic oscillators

We consider the one-dimensional Dirac equation for the harmonic oscillator and the associated second order separated operators giving the resonances of the problem by complex dilation. The same operators have unique extensions as closed PT-symmetric operators defining infinite positive energy levels converging to the Schroedinger ones as c tends to infinity. Such energy levels and their eigenfunctions give directly a definite choice of metastable states of the problem. Precise numerical computations shows that these levels coincide with the positions of the resonances up to the order of the width. Similar results are found for the Klein-Gordon oscillators, and in this case there is an infinite number of dynamics and the eigenvalues and eigenvectors of the PT-symmetric operators give metastable states for each dynamics.Comment: 13 pages, 2 figure

Perturbation Theory for Metastable States of the Dirac Equation with Quadratic Vector Interaction

The spectral problem of the Dirac equation in an external quadratic vector potential is considered using the methods of the perturbation theory. The problem is singular and the perturbation series is asymptotic, so that the methods for dealing with divergent series must be used. Among these, the Distributional Borel Sum appears to be the most well suited tool to give answers and to describe the spectral properties of the system. A detailed investigation is made in one and in three space dimensions with a central potential. We present numerical results for the Dirac equation in one space dimension: these are obtained by determining the perturbation expansion and using the Pad\'e approximants for calculating the distributional Borel transform. A complete agreement is found with previous non-perturbative results obtained by the numerical solution of the singular boundary value problem and the determination of the density of the states from the continuous spectrum.Comment: 10 pages, 1 figur

Wannier-Bloch oscillators

. We consider a Wannier-Stark problem for small field f in the oneladder case. We prove that a generical first band state is a metastable state (Wannier-Bloch oscillator) with the lifetime determined by the imaginary part of the Wannier-Stark ladder. The infinite resonances of the ladder cause Bloch oscillations as a global beating effect. For an adiabatic time ø = ft large enough, but much smaller than the resonance lifetime, we have a new version of the acceleration theorem and well specified Bloch oscillators. In the x representation and in the adiabatic scale: x ! x(f) = ¸=f + y the state vanishes externally to a pulsating region of j¸j defined by j¸j ! ¸ + (ø ) where ¸ + (n) = 0 and ¸ + (n + 1=2) is the maximum value equal to the first band width. For ¸ and ø such that j¸j is in this region and for y in a fixed domain, the state approaches a finite combination of oscillating Bloch states. 1. Introduction Let us consider the dynamics of an electron driven by a constant el..

Stability of the molecular structure

We prove the stability of the molecular structure given by a bidimensional nonlinear model for a stochastic perturbation. In particular, for a small stochastic perturbation, the racemisation effect doesn't happen when during the validity timew of the mode

The top resonances of the cubic oscillator

We study the top resonance states of the cubic anharmonic oscillator H(\beta)= p^2+x^2+i\sqrt{\beta}x^3 for \beta on the complexplane cut along the negative semiaxis. In particular, by the semiclassical scaling and semiclassical methods, we prove that the top resonance states do not belong to to L^2(R)