Semicircle form factor and spectral measure

This explicit calculation specializes the regular Friedrichs spectral method. Broader spectral, resonance, and bound-state issues are treated in FacchiLigaboLonigro,Lonigro.

The exact survival law is not exponential:

It need not be monotone. An autonomous trajectory is therefore not identified with all prefixes of a monotone collision schedule.

Here $|f_T({\omega }_0+x){|}^2$ is even in $x$. The symmetry gives

These are expectation identities on the prepared trajectory, not the operator identity $[H,H_0]=0$.