Simple Waves Do Not Avoid Eigenvalue Crossings

Molecular propagation through crossings and avoided crossings of electron energy levels

The time–dependent Born–Oppenheimer approximation describes the quantum mechanical motion of molecular systems. This approximation fails if a wavepacket propagates through an electron energy level crossing or “avoided crossing.” We discuss the various types of crossings and avoided crossings and describe what happens when molecular systems propagate through them. It is not practical to solve th...

Waves and instabilities in a differentially rotating disc containing a poloidal magnetic field

The theory of waves and instabilities in a differentially rotating disc containing a poloidal magnetic field is developed within the framework of ideal magnetohydrodynamics. A continuous spectrum, for which the eigenfunctions are localized on individual magnetic surfaces, is identified but is found not to contain any instabilities associated with differential rotation. The normal modes of a wea...

D ec 1 99 8 Adiabatic Evolution for Systems with Infinitely many Eigenvalue Crossings

We formulate an adiabatic theorem adapted to models that present an instantaneous eigenvalue experiencing an infinite number of crossings with the rest of the spectrum. We give an upper bound on the leading correction terms with respect to the adiabatic limit. The result requires only differentiability of the considered spectral projector, and some geometric hypothesis on the local behaviour of...

A Simple Criterion of Transverse Linear Instability for Solitary Waves

We prove an abstract instability result for an eigenvalue problem with parameter. We apply this criterion to show the transverse linear instability of solitary waves on various examples from mathematical physics.

Adiabatic evolution for systems with infinitely many eigenvalue crossings