ON OPERATORS COMMUTATIVE WITH ALL INVARIANTS FOR A HARMONIC-OSCILLATOR WITH COMMENSURABLE FREQUENCIES

For the d-dimensional quantum mechanical harmonic oscillator with r commensurability relation between frequencies, d independent operators commutative with the oscillator Hamiltonian and all other commutative with Hamiltonian operators are constructed. These operators form a basis in the centre of the algebra of invariants (integrals of motion) for the quantum mechanical oscillator with commensurable frequencies. Not all of these d operators have classical analogues. A classical harmonic oscillator only has d - r commutative with all other invariants. These classical integrals first appeared in the Gustavson work on the Birkhoff normalization. Operators with such properties are of interest for the perturbation theory, since any of them may be (at least formally) continued to become the invariant of the perturbed Hamiltonian.