Quantum monodromy and its generalizations and molecular manifestations

Abstract

Quantum monodromy is a non-trivial qualitative characteristic of certain non-regular lattices formed by the joint eigenvalue spectrum of mutually commuting operators. The latter are typically the Hamiltonian (energy) and the momentum operator(s) which label the eigenstates of the system. We give a brief review of known quantum systems with monodromy, which include such fundamental systems as the hydrogen atom in external fields, Fermi resonant vibrations of the CO₂ molecule, and non-rigid triatomic molecules. We emphasize the correspondence between the classical Hamiltonian monodromy and its quantum analogue and discuss possible generalizations of this characteristic in classical integrable Hamiltonian dynamical systems and their quantum counterparts.

Notes

†Recall that the point ξ of the initial phase space is a regular point of EM, and the corresponding value f = EM(ξ) is a regular value if the differentials (dF ₁, dF ₂, … , dF _N ) are linearly independent at ξ. Alternatively, if (dF ₁, dF ₂, … , dF_N ) are linearly dependent at some point ξ _c, the latter is a critical point of EM, and the corresponding value f _c = EM(ξ _c) is a critical value.

‡Notice that by fibre over f we imply a connected component of EM⁻¹(f), the latter can therefore be a union of fibres.

§Codimension equals the difference between the dimension of the space and the dimension of the subspace.

†Matrices M in the group SL(N, ) are N × N matrices with integer entries and det M = 1.

†Generalized monodromy matrices M belong to , i.e. det M = 1 and entries are rational numbers.

†Notice that hereafter we will imply atomic units in which ℏ = 1, and neglect semiclassical corrections μ.

† is an N-dimensional cubic lattice formed by all integers in the space which in our case is the space of the values of local actions.

†Spectroscopists often use the abbreviation 1:2 to designate 3D Fermi systems with doubly degenerate bending mode. This might, however, be confusing, as it does not allow one to distinguish the 3D systems from 2D systems with 1:2 resonance which do not show the presence of monodromy.