Various Quantum Mechanical Aspects of Quadratic Forms

Two of the oldest known classical integrable systems are: (i) n-decoupled harmonic oscillators, constrained to the surface of the n-sphere (i.e., the classical C. Neumann system), and (ii) geodesic flow on an n-axial ellipsoid. We integrate both these systems at the quantum level. That is, in both cases we construct n−1 pairwise-commuting partial differential operators which, in turn, commute with the respective quantum Hamiltonians. Moreover, the joint eigenvalues of the commuting partial differential operators appear to be encoded linearly as parameters in an n-parameter eigenvalue equation given by a second-order complex ordinary differential equation with automorphic boundary conditions. We then focus on the particular case of two degrees of freedom (i.e., R3). Via the microlocal wave-averaging ansatz combined with explicit WKB constructions, we give a spectral analysis (both classical and semiclassical) of the above systems, and subsequently exhibit connections between the various spectral asymptotics. In the course of this analysis, we show that the above two systems are intimately related to yet another quantum integrable system, the reduced quantum asymmetric rigid body in a vacuum.