  We find the complete branching law for the restriction of certain unitary
representations of $O(1,n+1)$ to the subgroups $O(1,m+1)\times O(n-m)$, $0\leq
m\leq n$. The unitary representations we consider belong either to the unitary
spherical principal series, the spherical complementary series or are
unitarizable subquotients of the spherical principal series.
  In the crucial case $0<m<n$ the decomposition consists of a continuous part
and a discrete part. The continuous part is given by a direct integral of
unitary principal series representations whereas the discrete part consists of
finitely many representations which either belong to the complementary series
or are unitarizable subquotients of the principal series. The explicit
Plancherel formula is computed on the Fourier transformed side of the
non-compact realization of the representations by using the spectral
decomposition of a certain hypergeometric type ordinary differential operator.
The main tool connecting this differential operator with the representations
are second order Bessel operators which describe the Lie algebra action in this
realization.
  To derive the spectral decomposition of the ordinary differential operator we
use Kodaira's formula for the spectral decomposition of Schr\"odinger type
operators.
