19 research outputs found
Bender-Wu singularities
We consider a family of quantum Hamiltonians , where is an imaginary
double well potential. We prove the existence of infinite eigenvalue crossings
with the selection rules of the eigenvalue pairs taking part in a crossing.
This is a semiclassical localization effect. The eigenvalues at the crossings
accumulate at a critical energy for some of the Stokes lines
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)