Non-Unitary Dynamics: Is it Allowed?

[dumpling]

I do know that supposedly the time-evolution operator is unitary.
At the same time, I have come across a peculiar case during a calculation.
Suppose that I have a basis that is complete (and not overcomplete), and element of Hilbert-space at t=0.

For some reason, the solution of the Schrödinger-equation is significantly simple, but only if I allow each of the basis-elements to evolve non-unitarily.
The <\Psi|\Psi> norme oscillates in time, but it is always finite, and the states always remain normalisable.

I do know that certain calculations like this exists, for example in nuclear physics, or in certain effective descriptions.
Yet I still would like to know whether this kind of dynamics is mathematically allowed, and whether I can just calculate expectation values in such cases as <\Psi|A|\Psi>/<\Psi|\Psi>.

 

[A. Neumaier]

We just discussed this here: https://www.physicsforums.com/threads/non-standard-models-in-qm.939578

 

[bhobba]

I do know that supposedly the time-evolution operator is unitary.

You might like to acquaint yourself with Wigner's Theorem:
https://arxiv.org/abs/0808.0779

For continuous transformations it can't be anti-unitary because you arrive at a contradiction - there is some point between the points of the transformation - so you have an anti-unitary applied to an anti-unitary which is unitary.

Thanks
Bill