Monodromy of the quantum 1:1:2 resonant swing spring

Abstract

We describe the qualitative features of the joint spectrum of the quantum 1:1:2 resonant swing spring. The monodromy of the classical analogue of this problem is studied in Dullin et al. [Physica D 190, 15–37 (2004)]. Using symmetry arguments and numerical calculations we compute its three-dimensional (3D) lattice of quantum states and show that it possesses a codimension 2 defect characterized by a nontrivial 3D-monodromy matrix. The form of the monodromy matrix is obtained from the lattice of quantum states and depends on the choice of an elementary cell of the lattice. We compute the quantum monodromy matrix, that is the inverse transpose of the classical monodromy matrix. Finally we show that the lattice of quantum states for the 1:1:2 quantum swing spring can be obtained—preserving the symmetries—from the regular 3D-cubic lattice by means of three “elementary monodromy cuts”

Fractional Hamiltonian monodromy from a Gauss–Manin monodromy

“…Hamiltonian monodromy has profound implications both in classical and quantum mechanics [Ngoc99] since it is the simplest topological obstruction to the existence of global action-angle variables [Dui80] and thus of global good quantum numbers [Ngoc99]. The phenomenon of Hamiltonian monodromy has been exhibited in a large variety of physical systems both in classical and quantum mechanics [AKE04, SC00, KR03, CWT99, SZ99, WJD03, GCSZ04,EJS04].…”

Classical and quantum-mechanical plane switching in CO2

“…In this Appendix, we give relations between various sets of dynamical variables used here and in related work [3,5,6] to describe a two-dimensional harmonic oscillator in resonance 1:1 and its perturbations. In CO 2 these variables represent bending vibrations (mode 2), in the swing-spring they describe pendular oscillations.…”

“…We then comment on our treatment of the whole system with three degrees of freedom and compare it to Refs. [3,5,6]. …”

“…We now briefly survey the way to reduce the symmetries of the 1:1:2 system while preserving its geometry [3,5,6] in order to show that our Eqs. (2.4) and (2.10) are indeed direct higher order analogs of the ones studied previously for the resonant swing-spring [3].…”

“…The resonant swing spring -a system which consists of a spring with one end fixed, a mass attached at the other end, and which is submitted to a constant vertical gravitation fieldhas recently received much attention from mathematicians and theoretical physicists [1][2][3][4][5][6].…”