View source for Lee-Friedrichs model

What is known today (1998) as the Lee–Friedrichs model [[#References|[a1]]], [[#References|[a2]]] is characterized by a [[Self-adjoint operator|self-adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l1200601.png" /> on a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l1200602.png" />, which is the sum of two self-adjoint operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l1200603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l1200604.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l1200605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l1200606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l1200607.png" /> have a common domain, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l1200608.png" /> has absolutely continuous spectrum (of uniform multiplicity) except for the end-point of the semi-bounded from below spectrum, and one or more eigenvalues which may or may not be embedded in the continuum (cf. also [[Spectrum of an operator|Spectrum of an operator]]; [[Absolute continuity|Absolute continuity]]). The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l1200609.png" /> is compact and of finite rank (cf. also [[Compact operator|Compact operator]]; [[Rank|Rank]]), and induces a mapping from the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006010.png" /> spanned by the eigenvectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006011.png" /> to the subspace corresponding to the continuous spectrum (and the reverse). The central idea of the model is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006012.png" /> does not map the subspace corresponding to the continuous spectrum into itself, and, as a consequence, the model becomes solvable in the sense described below.

In the physical applications of the model, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006013.png" /> corresponds to the Hamiltonian operator, the self-adjoint operator (often the self-adjoint completion of an essentially self-adjoint operator) that generates the unitary evolution (through the [[Schrödinger equation|Schrödinger equation]]) of the vector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006014.png" /> representing the state of the physical system in time (cf. also [[Hamilton operator|Hamilton operator]]).

The resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006015.png" /> generated by the [[Laplace transform|Laplace transform]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006016.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006017.png" /> on the Schrödinger evolution operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006018.png" /> (both acting on some suitable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006019.png" />) is analytic in the upper half <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006020.png" />-plane. Denoting by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006021.png" /> (with [[Lebesgue measure|Lebesgue measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006022.png" />) the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006023.png" /> on the continuous spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006026.png" /> and by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006027.png" /> the eigenvector with eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006028.png" /> (assuming for the sake of argument just one discrete eigenvector), one sees that the second resolvent equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006030.png" />, can be exactly solved by the pair of equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006032.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

Substituting (a3) into (a2), one sees that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006035.png" /></td> </tr></table>

If the discrete spectral value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006036.png" /> is separated from the continuum (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006037.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006038.png" /> has a pole on the real axis at the point

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006040.png" /> is embedded in the continuum that lies on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006041.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006042.png" />), one can avoid the generation of a real pole on the negative half-line by the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006043.png" /></td> </tr></table>

The projection of the time evolution of the quantum-mechanical state represented by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006044.png" /> back onto the initial state is given by (using units in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006045.png" />, the Planck constant divided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006046.png" />, is unity):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>

where the contour goes from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006048.png" /> in the negative direction of the real axis and a small distance above it, around the branch point counter-clockwise at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006049.png" />, and back to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006050.png" /> below the real axis. The construction defined on the left-hand side of (a6) was used by E.P. Wigner and V.F. Weisskopf [[#References|[a3]]], [[#References|[a28]]] in 1930 as a model for the description of unstable systems; they used it to calculate the line-width of a radiating atom.

The contour of the integral in (a6) can be deformed so that the integration below the real line is shifted to the negative imaginary axis where, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006051.png" /> sufficiently positive, this contribution can be considered as negligible (except near the branch cut). The integral path above the real axis can be similarly deformed into the second Riemann sheet of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006052.png" /> to the negative real axis, but there is a possibility that the second sheet extension of this function has a pole in the lower half-plane. One observes this in the Lee–Friedrichs model [[#References|[a1]]], [[#References|[a2]]] by studying (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006053.png" /> real)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006054.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006055.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006056.png" /></td> </tr></table>

Choosing a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006057.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006058.png" /> is the boundary value on the real axis of an analytic function in some (sufficiently large) domain in the lower half-plane, one sees that the second sheet continuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006059.png" /> is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006060.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>

Now,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006061.png" /></td> </tr></table>

if the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006062.png" /> looked for is sufficiently close to the real axis, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006063.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006064.png" /> may have a zero for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006065.png" />. If the real part vanishes as well, one has a pole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006066.png" /> at, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006067.png" />, which implies a decay law of the time evolution of the so-called survival amplitude (a6), of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006068.png" />, an exponential decay. The imaginary part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006069.png" /> is the semi-decay width computed in lowest-order perturbation theory by Wigner and Weisskopf [[#References|[a3]]], [[#References|[a28]]].

In quantum-mechanical scattering theory [[#References|[a4]]], [[#References|[a11]]], the scattered wave is expressed in terms of an operator-valued function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006070.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006071.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>

analytic in the same domain as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006072.png" />. The transition amplitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006073.png" /> contains, by the hypotheses of the Lee–Friedrichs model, the reduced resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006074.png" />, and the second sheet pole discussed above dominates the behaviour of the scattering for an interval of energies near the real part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006075.png" />, appearing as a scattering resonance. Hence the Lee–Friedrichs model offers an opportunity to describe scattering and resonance phenomena, along with the behaviour of an unstable system, in the framework of a single mathematical model [[#References|[a5]]].

The pole in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006076.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006077.png" /> suggests that, in some sense, there may be an eigenvalue equation of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006078.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>

This equation is exactly solvable in the Lee–Friedrichs model, with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006079.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>

but the eigenvalue equation (a9) is satisfied only after [[Analytic continuation|analytic continuation]] to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006080.png" /> in the same way as described above. This analytic continuation can be done in terms of the sesquilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006081.png" /> for a suitable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006082.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006083.png" /> is the boundary value of an analytic function on an adequate domain in the lower half-plane (including the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006084.png" /> within its boundary). (The eigenfunction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006085.png" /> must lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006086.png" /> as well.) The [[Banach space|Banach space]] functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006087.png" /> defined in this way lies in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006088.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006089.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006090.png" />, i.e., an element of a Gel'fand triple (cf. [[Rigged Hilbert space|Rigged Hilbert space]]) [[#References|[a6]]], [[#References|[a7]]], [[#References|[a12]]], [[#References|[a13]]], [[#References|[a14]]], [[#References|[a15]]], [[#References|[a16]]], [[#References|[a17]]]. This construction has provided the basis for useful physical applications [[#References|[a8]]], [[#References|[a27]]], [[#References|[a18]]], [[#References|[a19]]], [[#References|[a20]]], [[#References|[a21]]], [[#References|[a22]]].

Note that the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006091.png" /> studied in the Wigner–Weisskopf theory [[#References|[a3]]], [[#References|[a28]]] can never be precisely exponential in form (i.e., more generally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006092.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006093.png" /> is a projection, cannot be a [[Semi-group|semi-group]]) [[#References|[a9]]], [[#References|[a23]]] although for sufficiently large (but not too large) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006094.png" />, it may well approximate an exponential. For example, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006095.png" />-derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006096.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006097.png" /> vanishes if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006098.png" /> is defined. The time dependence of the Gel'fand triple function may, however, be exactly exponential (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l12006099.png" /> is sufficiently stable).

Lee's original model [[#References|[a1]]], formulated in the framework of non-relativistic [[Quantum field theory|quantum field theory]], was motivated by an interest in the process of [[Renormalization|renormalization]]; it can be seen from (a5) that the interaction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120060/l120060100.png" /> induces a shift in the point spectrum. There is a conserved quantum number in Lee's field theory which enables the model to be written in one sector as a quantum-mechanical model equivalent to the structure used by K.O. Friedrichs [[#References|[a2]]], whose motivation was to study the general framework of the perturbation of continuous spectra.

A relativistically covariant form of the Lee–Friedrichs model has been developed in [[#References|[a10]]] (see also [[#References|[a24]]], [[#References|[a25]]], [[#References|[a26]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.D. Lee,   "Some special exampls in renormalizable field theory"  ''Phys. Rev'' , '''95'''  (1954)  pp. 1329–1334</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K.O. Friedrichs,   "On the perturbation of continuous spectra"  ''Commun. Pure Appl. Math.'' , '''1'''  (1948)  pp. 361–406</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.F. Weisskopf,   E.P. Wigner,   "Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie"  ''Z. Phys.'' , '''63'''  (1930)  pp. 54</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.R. Taylor,   "Scattering theory" , Wiley  (1972)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.P. Horwitz,   J.-P. Marchand,   "The decay scattering system"  ''Rocky Mtn. J. Math.'' , '''1'''  (1971)  pp. 225–253</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I.M. Gel'fand,   G.E. Shilov,   "Generalized functions" , '''4''' , Acad. 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