  We establish an explicit expression for the smallest non-zero eigenvalue of
the Laplace--Beltrami operator on every homogeneous metric on the 3-sphere, or
equivalently, on SU(2) endowed with left-invariant metric. For the subfamily of
3-dimensional Berger spheres, we obtain a full description of their spectra. We
also give several consequences of the mentioned expression. One of them
improves known estimates for the smallest non-zero eigenvalue in terms of the
diameter for homogeneous 3-spheres. Another application shows that the spectrum
of the Laplace--Beltrami operator distinguishes up to isometry any
left-invariant metric on SU(2). It is also proved the non-existence of constant
scalar curvature metrics conformal and arbitrarily close to any non-round
homogeneous metric on the 3-sphere. All of the above results are extended to
left-invariant metrics on SO(3), that is, homogeneous metrics on the
3-dimensional real projective space.
