Lee-Friedrichs model

What is known today (1998) as the Lee–Friedrichs model [a1], [a2] is characterized by a self-adjoint operator $H$ on a Hilbert space $\mathcal{H}$, which is the sum of two self-adjoint operators $H _ { 0 }$ and $V$ such that $H$, $H _ { 0 }$ and $V$ have a common domain, $H _ { 0 }$ 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; Absolute continuity). The operator $V$ is compact and of finite rank (cf. also Compact operator; Rank), and induces a mapping from the subspace of $\mathcal{H}$ spanned by the eigenvectors of $H _ { 0 }$ to the subspace corresponding to the continuous spectrum (and the reverse). The central idea of the model is that $V$ 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, $H$ 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) of the vector in $\mathcal{H}$ representing the state of the physical system in time (cf. also Hamilton operator).

The resolvent $G ( z ) = ( z - H ) ^ { - 1 }$ generated by the Laplace transform on $[ 0 , \infty )$ by $e ^ { i z t }$ on the Schrödinger evolution operator $e ^ { - i H t }$ (both acting on some suitable $f \in \mathcal{H}$) is analytic in the upper half $z$-plane. Denoting by $\langle \lambda | f )$ (with Lebesgue measure $d \lambda$) the representation of $f \in \mathcal{H}$ on the continuous spectrum $\lambda$ of $H _ { 0 }$ on $[ 0 , \infty )$ and by $\phi \in \mathcal{H}$ the eigenvector with eigenvalue $E _ { 0 }$ (assuming for the sake of argument just one discrete eigenvector), one sees that the second resolvent equation

\begin{equation} \tag{a1} G ( z ) = G _ { 0 } ( z ) + G _ { 0 } ( z ) V G ( z ), \end{equation}

where $G _ { 0 } ( z ) = ( z - H _ { 0 } ) ^ { - 1 }$, can be exactly solved by the pair of equations

\begin{equation} \tag{a2} ( \phi , G ( z ) \phi ) = \end{equation}

\begin{equation*} = \frac { 1 } { z - E _ { 0 } } + \frac { 1 } { z - E _ { 0 } } \int _ { 0 } ^ { \infty } d \lambda ( V \phi | \lambda \rangle \langle \lambda | G ( z ) \phi ) \end{equation*}

and

\begin{equation} \tag{a3} \langle \lambda | G ( z ) \phi ) = \frac { 1 } { z - \lambda } \langle \lambda | V \phi ) ( \phi , G ( z ) \phi ). \end{equation}

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

\begin{equation} \tag{a4} h ( z ) ( \phi , G ( z ) \phi ) \equiv \end{equation}

\begin{equation*} \equiv \left( z - E _ { 0 } - \int _ { 0 } ^ { \infty } \frac { | ( V \phi | \lambda \rangle | ^ { 2 } } { z - \lambda } d \lambda \right) ( \phi , G ( z ) \phi ) = 1. \end{equation*}

If the discrete spectral value $E _ { 0 }$ is separated from the continuum ($E _ { 0 } < 0$), then $( \phi , G ( z ) \phi )$ has a pole on the real axis at the point

\begin{equation} \tag{a5} E _ { 1 } = E _ { 0 } + \int _ { 0 } ^ { \infty } \frac { | ( V \phi | \lambda \rangle | ^ { 2 } } { E _ { 1 } - \lambda } d \lambda < 0. \end{equation}

If $E _ { 0 }$ is embedded in the continuum that lies on $[ 0 , \infty )$ ($E _ { 0 } > 0$), one can avoid the generation of a real pole on the negative half-line by the inequality

\begin{equation*} \int _ { 0 } ^ { \infty } \frac { | ( V \phi | \lambda \rangle |^ { 2 } } { \lambda } d \lambda < E _ { 0 }. \end{equation*}

The projection of the time evolution of the quantum-mechanical state represented by $\phi$ back onto the initial state is given by (using units in which $\hbar$, the Planck constant divided by $2 \pi$, is unity):

\begin{equation} \tag{a6} ( \phi , e ^ { - i H t } \phi ) = \frac { 1 } { 2 \pi i } \int _ { C } e ^ { - i z t } ( \phi , G ( z ) \phi ) d z, \end{equation}

where the contour goes from $+ \infty$ in the negative direction of the real axis and a small distance above it, around the branch point counter-clockwise at $0$, and back to $+ \infty$ below the real axis. The construction defined on the left-hand side of (a6) was used by E.P. Wigner and V.F. Weisskopf [a3], [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 $t$ 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 $h ( z ) ^ { - 1 }$ 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 [a1], [a2] by studying (for $\zeta$ real)

\begin{equation*} h ( \zeta + i \epsilon ) - h ( \zeta - i \epsilon ) = \end{equation*}

\begin{equation*} = \int _ { 0 } ^ { \infty } | ( V \phi | \lambda ) | ^ { 2 } \left( \frac { 1 } { \zeta - \lambda - i \epsilon } - \frac { 1 } { \zeta - \lambda + i \epsilon } \right) d \lambda = \end{equation*}

\begin{equation*} = 2 \pi i | ( V \phi | \zeta \rangle | ^ { 2 }. \end{equation*}

Choosing a $V$ so that $W ( \zeta ) = | ( V \phi \ | \ \zeta \rangle | ^ { 2 }$ 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 $h ( z )$ is

\begin{equation} \tag{a7} h ^ { I I } ( z ) = h ( z ) + 2 \pi i W ( z ). \end{equation}

Now,

\begin{equation*} \operatorname { Im } h ^ { I I } ( z ) = \operatorname { Im } z \left( \int _ { 0 } ^ { \infty } \frac { | ( V \phi | \lambda \rangle | ^ { 2 } } { | z - \lambda | ^ { 2 } } d \lambda \right) + 2 \pi \operatorname { Re } W ( z ); \end{equation*}

if the value of $z$ looked for is sufficiently close to the real axis, $\operatorname { Re } W ( z ) > 0$, and $\operatorname { Im } h ^ { I I } ( z )$ may have a zero for $\operatorname{Im} z < 0$. If the real part vanishes as well, one has a pole of $h ^ { I I } ( z ) ^ { - 1 }$ at, say $\tilde{z}$, which implies a decay law of the time evolution of the so-called survival amplitude (a6), of the form $e ^ { - i z t }$, an exponential decay. The imaginary part of $\tilde{z}$ is the semi-decay width computed in lowest-order perturbation theory by Wigner and Weisskopf [a3], [a28].

In quantum-mechanical scattering theory [a4], [a11], the scattered wave is expressed in terms of an operator-valued function of $z$,

\begin{equation} \tag{a8} T ( z ) = V + V G ( z ) V, \end{equation}

analytic in the same domain as $G ( z )$. The transition amplitude $\langle \lambda | T ( z ) | \lambda ^ { \prime } \rangle$ contains, by the hypotheses of the Lee–Friedrichs model, the reduced resolvent $( \phi , G ( z ) \phi )$, and the second sheet pole discussed above dominates the behaviour of the scattering for an interval of energies near the real part of $\tilde{z}$, 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 [a5].

The pole in $h ^ { I I } ( z )$ at $\tilde{z}$ suggests that, in some sense, there may be an eigenvalue equation of the form

\begin{equation} \tag{a9} z f ( z ) = H f ( z ). \end{equation}

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

\begin{equation} \tag{a10} \langle \lambda | f ( z ) ) = \frac { 1 } { \lambda - z } \langle \lambda | V \phi ) ( \phi , f ( z ) ), \end{equation}

but the eigenvalue equation (a9) is satisfied only after analytic continuation to $\tilde{z}$ in the same way as described above. This analytic continuation can be done in terms of the sesquilinear form $( g, f ( z ) )$ for a suitable $g \in \mathcal{D} \subset \mathcal{H}$, such that $( \lambda | g )$ is the boundary value of an analytic function on an adequate domain in the lower half-plane (including the point $\tilde{z}$ within its boundary). (The eigenfunction $\phi$ must lie in $\mathcal{D}$ as well.) The Banach space functional $f$ defined in this way lies in the space $\overline{\mathcal{H}}$ dual to $\mathcal{D}$, for which $\overline { \mathcal{H} } \supset \mathcal{H} \supset \mathcal{D}$, i.e., an element of a Gel'fand triple (cf. Rigged Hilbert space) [a6], [a7], [a12], [a13], [a14], [a15], [a16], [a17]. This construction has provided the basis for useful physical applications [a8], [a27], [a18], [a19], [a20], [a21], [a22].

Note that the quantity $( \phi , e ^ { - i H t } \phi )$ studied in the Wigner–Weisskopf theory [a3], [a28] can never be precisely exponential in form (i.e., more generally, $P e ^ { - i H t } P$, where $P$ is a projection, cannot be a semi-group) [a9], [a23] although for sufficiently large (but not too large) $t$, it may well approximate an exponential. For example, the $t$-derivative of $| ( \phi , e ^ { - i H t } \phi ) | ^ { 2 }$ at $t = 0$ vanishes if $H \phi$ is defined. The time dependence of the Gel'fand triple function may, however, be exactly exponential (if $\mathcal{D}$ is sufficiently stable).

Lee's original model [a1], formulated in the framework of non-relativistic quantum field theory, was motivated by an interest in the process of renormalization; it can be seen from (a5) that the interaction $V$ 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 [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 [a10] (see also [a24], [a25], [a26]).

References

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