  The transfer operator due to Bogomolny provides a convenient method for
obtaining a semiclassical approximation to the energy eigenvalues of a quantum
system, no matter what the nature of the analogous classical system. In this
paper, the method is applied to integrable systems which are rotationally
invariant, in two and three dimensions.
  In two dimensions, the transfer operator is expanded in a Fourier series in
the angle variable, while in three dimensions it is expanded in spherical
harmonics. In both cases, when the Fourier coefficients are evaluated using the
stationary phase approximation, we arrive at the Einstein-Brillouin-Keller
(EBK) quantization conditions.
  The associated Maslov indices are shown to agree with the results calculated
by well-known simple rules. The theory is applied to several rotationally
invariant systems, including the hydrogen atom and the isotropic harmonic
oscillator in two and three dimensions, the circle billiard, a billiard inside
a spherical cavity, and a harmonic potential with a singular magnetic flux
line.
