Talk:Hille–Yosida theorem

Resolvent formalism

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

$R(z;A)={\frac {1}{A-zI}}$

The residue may be understood to be a projection operator

$\operatorname {res} R(z;A)\vert _{z=\lambda }={\frac {-1}{2\pi i}}\oint _{C_{\lambda }}R(\zeta ;A)d\zeta =P_{\lambda }=\vert \psi _{\lambda }\rangle \langle \psi _{\lambda }\vert$

where λ corresponds to an eigenvalue of A

$A\vert \psi _{\lambda }\rangle =\lambda \vert \psi _{\lambda }\rangle$

and $C_{\lambda }$ 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 $\vert \psi _{\lambda }\rangle$. 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)

Connection to the Laplace transform

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

${\frac {d}{dt}}T_{t}\phi =\lim _{h\to 0}{\frac {T_{t+h}\phi -T_{t}\phi }{h}}=AT_{t}\phi$

If we formally do the Laplace transform of the semigroup

${\mathcal {L}}T_{t}\phi =\int _{0}^{\infty }e^{-\lambda t}T_{t}\phi dt$

we get by integration by parts

${\mathcal {L}}{\frac {d}{dt}}T_{t}\phi =\lambda {\mathcal {L}}T_{t}\phi -\phi$

so when applied to the differential equation above

$\lambda {\mathcal {L}}T_{t}\phi -\phi =A{\mathcal {L}}T_{t}\phi$

or

${\mathcal {L}}T_{t}\phi =(\lambda I-A)^{-1}\phi =R(\lambda ,A)\phi$

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

Re: Is this discussion worthy of inclusion? Yes.--CSTAR 19:39, 12 March 2007 (UTC)

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)