Most typical 1 ratio 2 resonant perturbation of the hydrogen atom by weak electric and magnetic fields.

We study a perturbation of the hydrogen atom by small homogeneous static electric and magnetic fields in a specific mutual alignment with angle approximately pi/3 which results in the 1 ratio 2 resonance of the linearized Keplerian n-shell approximation. The bifurcation diagram of the classical integrable approximation has for most such field configurations the same typical structure that we describe. The structure of the corresponding quantum energy spectrum, which we describe in detail, is in certain ways an analogue of the well-known degeneracy found by Herrick [Phys. Rev. A 26, 323 (1982)] for the quadratic Zeeman effect.

Assigning vibrational polyads using relative equilibria: application to ozone.

We demonstrate how relative equilibria of a vibrating molecule, which are families of principal periodic orbits otherwise known as nonlinear normal modes, can be used to describe the global polyad structure of vibrational energy levels. The classical action integral n(E) computed along these orbits at different energies E corresponds to the polyad quantum number n so that the energy En of different relative equilibria describes the splitting of n-polyads. Further information on the internal polyad structure can be driven from the stability analysis of relative equilibria. We use the ozone molecule as a concrete example where n-polyads or "hyperpolyads" should be distinguished from the well-known polyads of the 1:1 stretching mode resonance; the stretching polyads are structural elements of hyperpolyads. We give dynamical interpretation of the relation between relative equilibria and n-polyads based on the normal form reduction in the limit of small vibrations near the equilibrium.

CO2 molecule as a quantum realization of the 1:1:2 resonant swing-spring with monodromy.

We consider the wide class of systems modeled by an integrable approximation to the 3 degrees of freedom elastic pendulum with 1:1:2 resonance, or the swing-spring. This approximation has monodromy which prohibits the existence of global action-angle variables and complicates the dynamics. We study the quantum swing-spring formed by bending and symmetric stretching vibrations of the CO2 molecule. We uncover quantum monodromy of CO2 as a nontrivial codimension 2 defect of the three dimensional energy-momentum lattice of its quantum states.