  The generalized pseudospectral Legendre method is used to carry out accurate
calculations of eigenvalues of the spherically confined isotropic harmonic
oscillator with impenetrable boundaries. The energy of the confined state is
found to be equal to that of the unconfined state when the radius of
confinement is suitably chosen as the location of the radial nodes in the
unconfined state. This incidental degeneracy condition is numerically shown to
be valid in general. Further, the full set of pairs of confined states defined
by the quantum numbers [(n+1, \ell) ; (n, \ell+2)], n = 1,2,.., and with the
radius of confinement {(2 \ell +3)/2}^{1/2} a.u., which represents the single
node in the unconfined (1, \ell) state, is found to display a constant energy
level separation exactly given by twice the oscillator frequency. The results
of similar numerical studies on the confined Davidson oscillator with
impenetrable boundary as well as the confined isotropic harmonic oscillator
with finite potential barrier are also reported .The significance of the
numerical results are discussed.
