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.

  • $\begingroup$ 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. $\endgroup$ – Valter Moretti Jun 8 at 8:20

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