Newton equivalent Hamiltonians for the harmonic oscillator

We consider a one-parameter family of Hamilton functions yielding the Newto n equation of the harmonic oscillator, (x(double overdot)) + omega (2)x = 0 . The parameter may be viewed as the speed of light c, the nonrelativistic limit c --> infinity yielding the usual Hamiltonian. For c < infinity, the classical Hamiltonians are the product of a function of x and a function of p. In the quantum case, with a suitable ordering, we explicitly find the s pectrum and the eigenfunctions of the Hamiltonian. (C) 2001 Academic Press.