# Eigenstates of the scattering matrix?

Consider a single-particle non-relativistic problem.

Consider a 3D spherically symmetric potential.

What are the eigenstates of the $$S$$-matrix? Are they spherically symmetric?

And what are the corresponding eigenvalues? It is stated that the S-matrix is unitary, so the eigenvalues should be of modulus one.

• They exist because the operator is normal (unitary in this case). However, barring a little number of them, they are not spherically symmetric. But, in general, spherical symmetry transforms eigenvectors to (different) eigenvectors. This is an elementary case of spontaneous symmetry breaking. – Valter Moretti Jun 8 at 8:20