Talk:Hille–Yosida theorem

Latest comment: 17 years ago by CSTAR in topic Connection to the Laplace transform

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