[time 922] Re: [time 921] Re: [time 919] Re: [time 914] Re: [time 909] About your proof ofunitarity

Hitoshi Kitada (hitoshi@kitada.com)
Sat, 9 Oct 1999 21:52:43 +0900

Dear Matti,

Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:

Subject: [time 921] Re: [time 919] Re: [time 914] Re: [time 909] About your
proof of unitarity


> > I am speaking of general context without such a condition. If the limit
> > exists, then it follows from it the unitarity.
> >
> You argue that you can avoid somehow the assumption about the
> existence of the time development operator and get unitarity from
> algebraic structure alone. Or that you have unitary time development
> operator in case that you have only E=0 states
> of Hamiltonian?

This is not my argument, but it is an old theory of T. Kato and S. T. Kuroda:

Theory of simple scattering and eigenfunction expansions, Functional Analysis
and Related Topics, Springer-Verlag, 1970, pp. 99-131,


The abstract theory of scattering, Rocky Mount. J. Math., Vol. 1 (1971),

I am not sure if your interaction term satisfies their assumptions. If it
works with your case, their argument treats the Hamiltonian without assuming
conditions like Virasoro conditions. They get a unitarity (completeness in
their terminology) for general spectra. The spectral projection onto the space
corresponding to E=0 would then give your unitarity.

The problem may be in the interaction term if their method does not work.

A question related with this is if the E=0 states are genuine eigenvectors
or generalized ones. Maybe to see if this is the case or not is included in
your problem?

Best wishes,

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