Quote by A. Neumaier
You may of course regard every 1particle Schroedinger equation as a classical field equation. This doesn't change the fact that he was solving a quantum problem.

I totally agree with you on this. But my point is, thinking of these equations as classical field equations, we can solve for eigenvalues of the classical field equation, and then if we quantize the classical field in to operators, in principle we can solve for oneparticle spectrum from the quantum Hamitonian(for Dirac hydrogen I don't know if anybody can do this explicitly, but I presume this is the logical procedure in QFT frame work).
My examples(free fields and Dirac Hydrogen atom) seem to suggest eigenvalues of field equation are exactly(more or less, if one ignores multiparticle process e.g. lamb shift) oneparticle spectrum, and this doesn't look like a mere coincidence to me, so I was wondering what's the principle behind this.