Unstable States in a Model of Nonrelativistic Quantum Electrodynamics: Corrections to the Lorentzian Distribution

Abstract

We revisit the Lee-Friedrichs model as a model of atomic resonances in the hydrogen atom, using the dipole-moment matrix-element functions which have been exactly computed by Nussenzveig. The Hamiltonian H of the model is positive and has absolutely continuous spectrum. Although the return probability amplitude $R_{\Psi }(t) = (\Psi , \exp (-iHt) \Psi )$ of the initial state $\Psi$, taken as the so-called Weisskopf–Wigner (W.W.) state, cannot be computed exactly, we show that it equals the sum of an exponentially decaying term and a universal correction $O(\beta ^{2}\frac{1}{t})$, for large positive times t and small coupling constants $\beta$, improving on some results of King (Lett Math Phys 23:215–222, 1991). The remaining, non-universal, part of the correction is also shown to be of the same qualitative type. The method consists in approximating the matrix element of the resolvent operator operator in the W.W. state by a Lorentzian distribution. No use is made of complex energies associated to analytic continuations of the resolvent operator to ”physical” Riemann sheets. Other new results are presented, in particular a physical interpretation of the corrections, and the characterization of the so-called sojourn time $\tau _{H}(\Psi )= \int _{0}^{\infty } |R_{\Psi }(t)|^{2} dt$ as the average lifetime of the decaying state, a standard quantity in (quantum) probability.

Appendix A: Completion of the Proof of Theorem 2.1

In this appendix we prove that (74) of Theorem 2.1 holds. Together with (71), this proves (70), and thereby completes the proof of Theorem 2.1.

We first write (72) as the limit, as $\delta \downarrow 0$, of the corresponding integral from $\delta >0$ to $\infty$. By integration by parts on the latter, we find

$$\begin{aligned}&D_{L}^{1}(t) = \lim _{\delta \downarrow 0}\left[ -\frac{w(\delta )}{it\alpha (\delta )\beta (\delta )}\right. \nonumber \\&\quad \left. + \frac{\int _{\delta }^{\infty }d\lambda \exp (-it\lambda )\frac{d}{d\lambda }(\frac{w(\lambda )}{\alpha (\lambda )\beta (\lambda )})}{it}\right] \end{aligned}$$

(A.1)

where, for $\lambda >0$, $\alpha (\lambda )$ and $\beta (\lambda )$ are given by (51) and (73) of the main text, but we repeat them here for clarity:

$$\begin{aligned} \alpha (\lambda ) \equiv E_{0}-\lambda -\beta ^{2}F(\lambda )-i\pi \beta ^{2}G(\lambda ) \end{aligned}$$

(A.2)

and

$$\begin{aligned} \beta (\lambda ) \equiv \kappa (\lambda -\lambda _{0})-i\pi \beta ^{2}G(\lambda _{0}) \end{aligned}$$

(A.3)

We have, the prime denoting, as usual, the first derivative,

$$\begin{aligned} \alpha ^{'}(\lambda ) = -1-\beta ^{2}F^{'}(\lambda )-i\pi \beta ^{2}G^{'}(\lambda ) \end{aligned}$$

(A.4)

and

$$\begin{aligned} \beta ^{'}(\lambda ) = \kappa \end{aligned}$$

(A.5)

From (52),

$$\begin{aligned}&w^{'}(\lambda )=-\beta ^{2}(F^{'}(\lambda )-F^{'}(\lambda _{0}))-i\pi \beta ^{2}(G^{'}(\lambda )-G^{'}(\lambda _{0})) \end{aligned}$$

(A.6)

$$\begin{aligned}&\frac{d}{d\lambda }\left( \frac{w(\lambda )}{\alpha (\lambda )\beta (\lambda )}\right) \nonumber \\&\quad = \frac{w^{'}(\lambda )}{\alpha (\lambda )\beta (\lambda )} - \frac{w(\lambda )\alpha ^{'}(\lambda )}{\alpha (\lambda )^{2}\beta (\lambda )} \nonumber \\&\quad \quad - \frac{w(\lambda )\beta ^{'}(\lambda )}{\alpha (\lambda )\beta (\lambda )^{2}} \end{aligned}$$

(A.7)

From (21), (22), (36) and (37) we have

$$\begin{aligned}&G(\lambda ) = \lambda (\lambda ^{2}+a^{2})^{-4} \text{ for } \lambda \ge 0 \end{aligned}$$

(A.8.1)

$$\begin{aligned}&G^{'}(\lambda ) = (\lambda ^{2}+a^{2})^{-4}-8\lambda ^{2}(\lambda ^{2}+a^{2})^{-5} \end{aligned}$$

(A.8.2)

When writing f(0) in the following, for some function f, it will be meant the limit $\lim _{\delta \downarrow 0} f(\delta )$. The finiteneness of the resulting limits, for all the functions which follow, will result from (38), which will be proved later as part of the forthcoming property b.) of the function F. We have, then:

$$\begin{aligned}&F(0) = \int _{0}^{\infty }(k^{2}+a^{2})^{-4}dk \end{aligned}$$

(A.8.3)

$$\begin{aligned}&G(0) = 0 \end{aligned}$$

(A.8.4)

$$\begin{aligned}&w(0)= -\beta ^{2}[F(0)-F(\lambda _{0})+\lambda _{0}F^{'}(\lambda _{0})]-i\pi \beta ^{2}\lambda _{0}G^{'}(\lambda _{0}) \end{aligned}$$

(A.8.5)

$$\begin{aligned}&\alpha (0) = E_{0}-\beta ^{2}F(0) \end{aligned}$$

(A.8.6)

$$\begin{aligned}&\beta (0) = -\kappa \lambda _{0}-i\pi \beta ^{2}G(\lambda _{0}) \end{aligned}$$

(A.8.7)

The first term in (A.1) satisfies, in the limit $\delta \downarrow 0$, the bound on the r.h.s. of (75), by (A.8.5), (A.8.6) and (A.8.7). Therefore, by (A.1) and (A.7), in order to conclude the proof of (74), we need only prove that

$$\begin{aligned}&\left| \int _{0}^{\infty } \frac{\alpha ^{'}(\lambda )}{\alpha (\lambda )^{2}\beta (\lambda )} w(\lambda ) d\lambda \right| <\infty \end{aligned}$$

(A.9.1)

$$\begin{aligned}&\left| \int _{0}^{\infty } \frac{\beta ^{'}(\lambda )}{\alpha (\lambda )\beta (\lambda )^{2}} w(\lambda ) d\lambda \right| <\infty \end{aligned}$$

(A.9.2)

$$\begin{aligned}&\left| \int _{0}^{\infty } \frac{1}{\alpha (\lambda )\beta (\lambda )} w^{'}(\lambda ) d\lambda \right| <\infty \end{aligned}$$

(A.9.3)

It follows from (A.2), (A.3), (A.4), (A.5), (A.8.1) and (A.8.2) and (52) that (A.9.1)–(A.9.3) hold if the two following assertions are true:

(a):

For $\lambda$ sufficiently large, $F(\lambda )$ and $F^{'}(\lambda )$ are uniformly bounded in $\lambda$;

(b):

For $\lambda$ in a sufficiently small right-neighbourhood of zero, $F(\lambda )$ is uniformly bounded, (38) holds and

$$\begin{aligned} F^{'}(\lambda ) = -\log \lambda + D \end{aligned}$$

where $0<D<\infty$ is independent of $\lambda$.

Indeed, b.) implies that $\alpha ^{'}$, as well as $w^{'}$, are integrable in a neighbourhood of zero, which suffice to prove integrability of $\frac{\alpha ^{'}(\lambda )}{\alpha (\lambda )^{2}\beta (\lambda )} w(\lambda )$ and of $\frac{1}{\alpha (\lambda )\beta (\lambda )}w^{'}(\lambda )$, in a neighbourhood of zero, which are elements in the proof of (A.9.1) and (A.9.3). Convergence at infinity of the integrals on the left hand sides of (A.9.1)–(A.9.3) is an immediate consequence of the explicit formulae for $\alpha$, $\beta$ and w, together with a.).

In order to prove a.) and b.), we come back to (37), whereby, for any $\lambda >0$,

$$\begin{aligned} F(\lambda ) = \lim _{r\rightarrow 0}\int _{|k-\lambda |\ge r} \frac{G(k)}{k-\lambda }dk \end{aligned}$$

We write

$$\begin{aligned}&\int _{|k-\lambda |\ge r} \frac{G(k)}{k-\lambda } \\&\quad =\int _{0}^{\lambda -r} \frac{G(k)}{k-\lambda }dk + \int _{\lambda +r}^{2\lambda } \frac{G(k)}{k-\lambda }\\&\quad \quad +\int _{2\lambda }^{\infty } \frac{G(k)}{k-\lambda }dk \end{aligned}$$

but

$$\begin{aligned}&\int _{0}^{\lambda -r} \frac{G(k)}{k-\lambda }dk + \int _{\lambda +r}^{2\lambda } \frac{G(k)}{k-\lambda } \\&\quad = \int _{r}^{\lambda } \frac{1}{k}[G(k+\lambda )-G(k-\lambda )]dk \end{aligned}$$

Write

$$\begin{aligned} G(k+\lambda )-G(k-\lambda ) = k\int _{-1}^{1}dt G^{'}(\lambda +kt) \end{aligned}$$

Thus,

$$\begin{aligned}&F(\lambda ) = \int _{0}^{\lambda }dk\int _{-1}^{1}dt\{[(\lambda +kt)^{2}+a^{2}]^{-4}\nonumber \\&\quad -8(\lambda +kt)^{2}[(\lambda +kt)^{2}+a^{2}]^{-5}\}\nonumber \\&\quad + \int _{2\lambda }^{\infty } \frac{G(k)}{k-\lambda }dk \end{aligned}$$

(A.9.4)

We write

$$\begin{aligned}&F(\lambda ) = -7\int _{0}^{\lambda }dk\int _{-1}^{1}dt[(\lambda +kt)^{2}+a^{2}]^{-4}\\&\quad +8a^{2} \int _{0}^{\lambda }dk\int _{-1}^{1}dt [(\lambda +kt)^{2}+a^{2}]^{-5}\\&\quad + \int _{2\lambda }^{\infty } \frac{G(k)}{k-\lambda }dk \end{aligned}$$

from which

$$\begin{aligned}&F^{'}(\lambda ) = -7\int _{-1}^{1}dt[\lambda ^{2}(1+t)^{2}+a^{2}]^{-4} \nonumber \\&\quad + 8a^{2} \int _{-1}^{1} dt [\lambda ^{2}(1+t)^{2}+a^{2}]^{-5}\nonumber \\&\quad + 28 \int _{0}^{\lambda }dk\int _{-1}^{1} [(\lambda +kt)^{2}+a^{2}]^{-5}2(\lambda +kt)\nonumber \\&\quad -40a^{2} \int _{0}^{\lambda }dk\int _{-1}^{1}dt [(\lambda +kt)^{2}+a^{2}]^{-6}2(\lambda +kt)\nonumber \\&\quad -2 \int _{2\lambda }^{\infty } \frac{G(k)}{k-\lambda }dk - \int _{2\lambda }^{\infty } \frac{G(k)}{(k-\lambda )^{2}}dk \end{aligned}$$

(A.9.5)

By (A.9.4), we obtain directly a.) for $F(\lambda )$, as well as the statements in b.) which concern $F(\lambda )$. Statement b.) for $F^{'}(\lambda )$ follows from (A.8.1) and the last term in (A.9.5). Statement a.) for $F^{'}(\lambda )$ is not entirely obvious from (A.9.5), but we use

$$\begin{aligned} b(\lambda +kt) \le (\lambda +kt)^{2} +a^{2} \end{aligned}$$

which is true for b sufficiently small, to bound the third and fourth terms in (A.9.5) in absolute value by

$$\begin{aligned} \text{ const. } \int _{0}^{\lambda } dk((\lambda -k)^{2}+a^{2})^{-4} \text{ resp. } \text{ const. } \int _{0}^{\lambda }dk ((\lambda -k)^{2}+a^{2})^{-5} \end{aligned}$$

which are trivially seen to be uniformly bounded in $\lambda$ by a change of variable. This completes the proof of (74). q.e.d.