Reviews in Mathematical Physics

The Ghirardi–Rimini–Weber (GRW) theory is a physical theory that, when combined with a suitable ontology, provides an explanation of quantum mechanics. The so-called collapse of the wave function is problematic in conventional quantum theory but not in the GRW theory, in which it is governed by a stochastic law. A possible ontology is the flash ontology, according to which matter consists of random points in space-time, called flashes. The joint distribution of these points, a point process in space-time, is the topic of this work. The mathematical results concern mainly the existence and uniqueness of this distribution for several variants of the theory. Particular attention is paid to the relativistic version of the GRW theory that was developed in 2004.

Our aim is to analyze the various energy functionals appearing in the physics literature and describing the behavior of a Bose–Einstein condensate in an optical lattice. We want to justify the use of some reduced models and control the error of approximation. For that purpose, we will use the semi-classical analysis developed for linear problems related to the Schrödinger operator with periodic potential or multiple wells potentials. We justify, in some asymptotic regimes, the reduction to low dimensional problems and analyze the reduced problems.

We propose an alternative to the usual time-independent Born–Oppenheimer approximation that is specifically designed to describe molecules with non-symmetrical hydrogen bonds. In our approach, the masses of the hydrogen nuclei are scaled differently from those of the heavier nuclei, and we employ a specialized form for the electron energy level surface. As a result, the different vibrational modes appear at different orders of approximation.

Although we develop a general theory, our analysis is motivated by an examination of the FHCl^- ion. We describe our results for it in detail.

We prove the existence of quasimodes and quasienergies for the nuclear vibrational and rotational motion to arbitrary order in the Born–Oppenheimer parameter ∊. When the electronic motion is also included, we provide simple formulas for the quasienergies up to order ∊³ that compare well with experiment and numerical results.