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Talk:Hille–Yosida theorem

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Resolvent formalism

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Hi CSTAR, fantastic article that I'm still trying to grok. I would like to sugges a low-brow section, or maybe a distinct article, called, for example, resolvent formalism, that uses the standard notation commonly seen in books on quantum mechanics. (even though its somewhat poorly defined) Viz:

The resolvent captures the spectral properties of an operator in the analytic structure of the resolvent. Given an operator A, the resolvent may be defined as

The residue may be understood to be a projection operator

where λ corresponds to an eigenvalue of A

and is a contour in the positive dirction around the eigenvalue λ.

The above is more-or-less a textbook defintion of the resolvant as used in QM. Unfortunately, I've never seen anything "better" than this, in particlar, don't know how to qualify . Must this be an element of a Hilbert space? Something more general? Frechet space? Banach space? Does the operator A have to be Hermitian? Nuclear? Trace-class? or maybe not?

Since the above is occasionally seen in the lit. I'd like to get a good solid article for it, with the questions at least partly addressed (and it seems the Hille-Yosida theorem adresses these, in part). Let me know what you think. linas 21:30, 22 November 2005 (UTC)[reply]

Connection to the Laplace transform

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By definitions of the infinitisimal operator and the semigroup we have that

If we formally do the Laplace transform of the semigroup

we get by integration by parts

so when applied to the differential equation above

or


Is this discussion worthy of inclusion? (Igny 19:27, 12 March 2007 (UTC))[reply]

Re: Is this discussion worthy of inclusion? Yes.--CSTAR 19:39, 12 March 2007 (UTC)[reply]
You might be able to shorten it, since the Hille-Ypsida theorem already is stated (implicitly) in terms of the Laplacc transform.--CSTAR 00:05, 13 March 2007 (UTC)[reply]