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

Restriction of some unitary representations of O(1,N) to symmetric subgroups