  In 1956 $\AA$. Pleijel gave his celebrated theorem showing that the
inequality in Courant's theorem on the number of nodal domains is strict for
large eigenvalues of the Laplacian. This was a consequence of a stronger result
giving an asymptotic upper bound for the number of nodal domains of the
eigenfunctions as the eigenvalues tend to $+\infty$. A similar question occurs
naturally for Schr"\odinger operators. The first significant result has been
obtained recently by the first author for the harmonic oscillator. The purpose
of this paper is to consider more general potentials which are radial. We will
analyze the case when the potential tends to $+\infty$ and the case when the
potential is negative and tends to zero, where the considered eigenfucntion are
associated to the eigenvalues below the essential spectrum.
