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\centerline{\Largebf Radiative Decay:}
\vskip .3in
\centerline{\Largebf Nonperturbative Approaches}
\vskip 1in
\centerline{Matthias H\"ubner and Herbert Spohn}
\vskip .3in
\centerline{\it Theoretische Physik, Ludwig-Maximilians-Universit\"at}
\centerline{\it Theresienstra\ss e 37, D-80333 M\"unchen, Germany}
\smallskip
\centerline{\it e-mail: spohn@stat.physik.uni-muenchen.de}
\vskip 2in
{\bf Abstract.} In radiative decay the coupled system, electron bound by
some external potential plus radiation field, relaxes to the ground state
in the long time limit. Our central issue is to prove such a behaviour on
the basis of the corresponding Schr\"odinger equation.
We argue that the spin-boson Hamiltonian is a simple, but physically still
acceptable test case. We relate radiative decay to scattering theory and
prove the existence of wave operators.
Other approaches are reviewed and compared.
Some challenging open problems are listed.
\vfill\eject}
\pageno=1
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\bigskip\noindent
{\largebf 1. Introduction and Hamiltonians}
\medskip\noindent
Radiative decay is {\it the} basic physical mechanism of how excited atoms lose energy
and relax to the radiationless ground state. Within atomic physics one has developed
a very powerful, and successful, perturbational scheme to
handle such processes [1,2].
>From a less computational viewpoint it would be nice to prove radiative decay
as a rigorous consequence of the Schr\"odinger equation. Over the past years
there have been several attempts in this direction. The present paper intends
to put the various approaches in perspective, adds a few novel results, and indicates
open problems. It seems useful to us - and hopefully to the reader - to collect
some background material. However no claims on completeness are made and
we will not provide a grand review on rigorous results for ``Photons and Atoms".
\par
Radiative decay involves a scale of energy small compared to the rest mass of the
electron. Thus we may ignore relativistic effects and base our investigation on
nonrelativistic quantum mechanics of matter coupled to the photon field.
To be concrete we consider a single electron in the force field of an
infinitely heavy nucleus located at the origin. On the level of our undertaking
the spin of the electron plays no role. Thus the Hilbert space for the electron is
${\cal H}_e=L^2(R^3,d^3x)$ and its Hamiltonian is given by
$$H_e={p^2\over 2m}-{e^2\over |x|}.\eqno(1.1)$$
Here $x$ is the position, $p=-i\nabla$ the momentum, $m$ the mass,
and $e$ the charge of the electron. (We set $\hbar=1$.) As standard [1-3], the
electromagnetic field is quantized in the Coulomb gauge, $\nabla\cdot A=0$.
The dynamical variables are then the two transverse components of the vector potential.
Thus the one particle space equals $C^2\otimes L^2(R^3,d^3k)$ and $\cal F$ denotes the
corresponding bosonic Fock space. On $\cal F$ we introduce, in momentum representation,
the two component Bose field $a_j(k),a_j^*(k), j=1,2$, with the commutation relations
$[a_j(k),a_{j'}^*(k')]=\delta_{jj'}\delta(k-k')$.
The Hamiltonian of the photon field is
$$H_p=\sum_{j=1}^2\int d^3k\omega(k)a_j^*(k)a_j(k) \eqno(1.2)$$
with dispersion relation $\omega(k)=|k|$. (We set $c=1$.) $H_p$ is self-adjoint with
domain $D(H_p)\subset {\cal F}$. The electron is minimally coupled to the photon field.
The full Hamiltonian reads then
$$H={1\over 2m}(p-\alpha A(x))^2+V(x)+
\sum_{j=1}^2\int d^3k\omega(k)a_j^*(k)a_j(k). \eqno(1.3)$$
Here $\alpha$ is the coupling constant and $A$ the vector potential,
$$A(x)=\sum_{j=1}^2\int d^3k{1\over\sqrt{2\omega(k)}}e_j(k)
[\hat\rho(k)e^{-ikx}a_j^*(k)+\hat\rho^*(k)e^{ikx}a_j(k)] \eqno(1.4)$$
with $e_1(k),e_2(k),k/|k|$ forming a left-handed dreibein.
We introduced the ultraviolet cutoff function $\hat\rho(k)$ which is
rotational invariant, equal to 1 for $|k|From this point of view we regard rotating wave as a nonadmissible oversimplification.
On the other hand the mathematical difficulties of the more
realistic Hamiltonians as (1.8) or even (1.3) are already present in (1.10).
So along the road the spin-boson Hamiltonian cannot be avoided. For radiative decay
the spin-boson Hamiltonian is the simplest, yet physically still acceptable model.
\par
In the remainder of the paper we study properties of the Hamiltonian (1.10), in
particular spectral properties, long time behaviour, and dilation analyticity. But
before, let us mention a few basic facts. $a(k),a^*(k)$ is a one component Bose
field over $R^3$. Thus $H$ acts on $C^2\otimes{\cal F}$ with $\cal F$ the bosonic Fock
space over $L^2(R^3,d^3k)$. The dispersion relation is $\omega(k)=|k|$.
$\lambda$ is the coupling function.
(For notational convenience we absorbed the coupling constant $\alpha$ into $\lambda(k)$.)
By (1.8), $\lambda(k)=i\alpha\hat\rho(k)\sqrt{\omega(k)}$.
If $\int |\lambda|^2<\infty$, then $a(\lambda),a^*(\lambda)$ are densely defined and closed.
If $\int |\lambda(k)|^2/\omega<\infty$, then the interaction is operator bounded
with relative $H_B$ bound zero, where $H_B=\int d^3k\omega(k)a^*(k)a(k)$.
By the Kato-Rellich theorem, $H$ is bounded from below and self-adjoint on $D(I\otimes H_B)$.
A sufficient condition for $H$ to have a unique
ground state is in addition $\int d^3k|\lambda(k)|^2/\omega(k)^2<\infty$ [11].
In fact, if $\psi_0\in C^2\otimes {\cal F}$ denotes the unique ground state of $H$,
we have $\langle\psi_0|e^{\delta N_B}\psi_0\rangle<\infty$ for any $\delta>0$.
If $\lim_{t\to\infty} t^2\int d^3k|\lambda(k)|^2e^{-\omega(k)t}=\alpha_0>0$, then for
sufficiently strong coupling the true physical ground state acquires an infinite number
of bosons and lies no longer in $C^2\otimes {\cal F}$ [11].
We do not consider such singular cases here.
To summarize: we require
\par
(C1) $\int d^3k |\lambda(k)|^2(1+\omega(k)^{-2})<\infty$.
\par
(C2) $\lambda(k)\not= 0$ a.e. such that all field modes couple. If $\lambda$ would
vanish on the set $\Lambda\subset R^3$ of positive measure, then (1.10) would
split into two commuting pieces by taking the $k$-integration over $\Lambda$
and $R^3\setminus\Lambda$. On $R^3\setminus\Lambda$ our condition is satisfied.
The photons with $k\in\Lambda$ do not couple to the atom.
\par
(C3) $\int dt\;|\int d^3k |\lambda(k)|^2e^{-i\omega(k)t}| (1+|t|)^{\delta}<\infty$
for some $\delta>0$. Implicitely, this is a smoothness condition on $\lambda$.
\par
$H$ commutes with the parity operator
$P=(-1)^{(1+\sigma_z)/2+N_B}$, which induces the symmetry $a(k)\to -a(k),
\sigma_x\to -\sigma_x, \sigma_y\to -\sigma_y,\sigma_z\to \sigma_z$. If $\sigma_x$ is
substituted by a more general coupling operator, this parity is broken.
\par
We are now in a position to formulate with more precision the problem of radiative
decay. The most basic issue is to prove that after a long time the system approaches its
radiationless ground state. Let ${\cal A}_R$ be the algebra of bounded operators
generated by $I,\vec\sigma$, and the Weyl operators
$W(f)=\exp[\int d^3x(f(x)a^*(x)-f(x)^* a(x)]$ with the support of $f$ contained in the
ball $B_R=\{x\mid |x|\le R\}$ in position space. An observable is strictly local if
$A=A^*\in {\cal A}_R$ for some $R$. We should then prove that
$$\lim_{t\to\infty}\langle e^{-iHt}\psi|Ae^{-iHt}\psi\rangle
=\langle\psi_0|A\psi_0\rangle \eqno(1.11)$$
for every $\psi\in{\cal H}$ and at least every strictly local $A$.
The restriction to local $A$'s is essential: e.g. for a bounded function of $H$ (1.11)
does not hold.
\par
More specifically one would like to know how the limit in (1.11) is approached. To pose
this question in generality makes little sense and one considers specific physical
situations. In atomic physics [2] a natural initial condition is to have the two level
atom in the upper state, field in Fock vacuum, i.e. $\psi_u=u\otimes|vac\rangle$ and one
studies the probability to remain in the upper level,
$\langle e^{-iHt}\psi_u|{1+\sigma_z\over 2}e^{-iHt}\psi_u\rangle$.
Its Fourier transform yields the experimentally measured line shape. On the other hand,
in the study of dissipative quantum systems [12,13] one starts with the Hamiltonian (1.9)
and regards $x$ as a quantum mechanical degree of freedom in a symmetric double well
potential. $\psi_1$ is then the ground state, $\psi_2$ the first excited state.
(Indeed, in this case $(\psi_1,x\psi_1)=0=(\psi_2,x\psi_2)$.) A natural initial
condition is then to have the quantum degree of freedom in the right well,
$\psi_+={1\over \sqrt 2}(u+v)\otimes |vac\rangle$, and to investigate the tunneling
probability,
$\langle e^{-iHt}\psi_+|\sigma_x e^{-iHt}\psi_+\rangle$,
which is damped because of the coupling to the boson field.
In fact, in both examples it is arguable that a conditional ground state for
the boson field would be a physically more realistic initial condition.
\par
In the case of an initially excited atom, beyond the decay time, there are a few photons
which carry the energy off to infinity. More generally, one could also have some
incoming photons which scatter off and possibly excite the atom.
>From this point of view we are investigating the wave operators and the $S$-matrix of
the coupled system, radiative decay being a particular initial state.
Thus the problem is to understand the long time behaviour of a general solution
$$e^{-iHt}\psi\qquad {\rm as} \qquad t\to\infty.\eqno(1.12)$$
\par
Clearly (1.11) and (1.12) are closely related. We will come back to them once we have
explained the general scattering theory (Section 2). Section 3 covers perturbation
theory, the standard machinery of atomic physics, and the weak coupling theory as its
rigorous version. Dilation analyticity is discussed in Section 4. We close with some
challenging open problems.
%We would like to emphasize that all these approaches cannot tell us anything about
%the main problem we are interested in: because of the discrete spectrum of the
%models with finitely many field modes (describing systems enclosed in a finite box)
%there occur necessarily recurrences. So there is no chance for such models
%to show a phenomenon like radiative {\it decay}. To establish the desired long time
%relaxation to the ground state or an equilibrium state we have to consider the
%infinitely extended system. This implies that the dispersion relation for the photons
%acts as a self-adjoint operator with purely absolutely continuous spectrum
%on the one-particle wave functions. The Riemann-Lebesgue Lemma, applied to the
%corresponding one-parameter group, expresses then rigorously our intuitive
%feeling that the photons fly away from the atom ``to infinity".
\bigskip\noindent
{\largebf 2. Scattering Theory}
\medskip\noindent
{\bf 2.1 Wave operator, asymptotic completeness}
\smallskip\noindent
After a long time the atom plus surrounding photon field should be in their ground state
and several photons should be travelling outwards with the speed of light.
Except for symmetrization such a state is essentially a product wave function.
If there is a single photon travelling outwards with wave function $\phi$ far away
from the origin, then the state is approximately $\int d^3x\phi(x)a^*(x)\psi_0=
a^*(\phi)\psi_0$, $\psi_0$ being the unique ground state.
More generally for $n$ photons far away from the origin with wave function $\phi^{(n)}$
the appropriate state is given by
$${1\over \sqrt{n!}}\int d^3x_1\ldots d^3x_n\phi^{(n)}(x_1,\ldots,x_n)
a^*(x_n)\ldots a^*(x_1)\psi_0. \eqno(2.1)$$
This defines a map $J$ from $\cal F$ (photon states) to ${\cal H}=C^2\otimes \cal F$.
We write out $J$ more explicitly. Let $\psi_0=(\psi_0^{(0)},\psi_0^{(1)},\ldots)$
be the ground state with norm $1=||\psi_0||^2=$ \hfil\break
$\sum_{\sigma=\pm 1}\sum_{n=0}^{\infty}\int \prod_{j=1}^n d^3x_j
|\psi_0^{(n)}(x_1,\ldots,x_n,\sigma)|^2$ and let $\phi\in\cal F$. Then
$J:{\cal F}\to C^2\otimes{\cal F}$ and $J\phi=\psi_0\otimes_s \phi$ where
$$(\psi_0\otimes_s\phi)^{(n)}(x_1,\ldots,x_n,\sigma)=\sum_{j=0}^n {n\choose j}^{1/2}
S(\psi_0^{(j)}\phi^{(n-j)})(x_1,\ldots,x_n,\sigma). \eqno(2.2)$$
Here $S$ is the symmetrizer,
$$(Sf)(x_1,\ldots,x_n)={1\over n!}\sum_{\pi}f(x_{\pi(1)},\ldots,x_{\pi(n)}),\eqno(2.3)$$
where the sum is over all permutations $\pi$ of $\{1,\ldots,n\}$.
\par
Clearly $J$ is linear, however unbounded. Let
$D_e=\{\phi\in{\cal F}\mid \exists c>0, 00$.
We reserve the linear operator $J$ for the particular choice where $\psi=\psi_0$.
\par
A photon state $\phi$ can be shifted to infinity also by applying the free evolution
\break $\exp(-iH_Bt)$.
\smallskip\noindent
{\bf Lemma 3.} {\it Let $\psi_i\in\cal H$ such that
$\langle\psi_i|e^{\delta N_B}\psi_i\rangle<\infty$ for every $\delta>0$ and let
$\phi\in D_e, i=1,2$. Then
$$\lim_{t\to\infty}
\langle\psi_2\otimes_se^{-iH_Bt}\phi_2|\psi_1\otimes_se^{-iH_Bt}\phi_1\rangle=
\langle\psi_2|\psi_1\rangle\;\langle\phi_2|\phi_1\rangle.\eqno(2.8)$$
\smallskip\noindent
Proof.} The computation is as in Lemma 2. One picks up now at least one factor of the
form \break
$\int d^3k \hat {\psi}(k)^*e^{-i\omega(k)t} \hat\phi(k)
=\int_0^{\infty}d\omega h(\omega)e^{-i\omega t}$ with $h \in L^1(R_+,d\omega)$,
which vanishes as $t\to\infty$ by the Riemann-Lebesgue lemma$.\qquad\squ$
\par
Physically we expect that for every $\psi\in\cal H$ we can find a comparison photon state
$\phi$ such that
$$e^{-iHt}\psi\approx e^{-iE_0t}\psi_0\otimes_se^{-iH_Bt}\phi\eqno(2.9)$$
for $t\to\infty$, where $H\psi_0=E_0\psi_0$ with $E_0$ the ground state energy.
Thus it is natural to define the wave operator
$$\Omega^{\mp}\phi=s-\lim_{t\to\pm\infty}e^{i(H-E_0)t}Je^{-iH_Bt}\phi.\eqno(2.10)$$
We will prove the existence of this limit for all $\phi\in D_e$ by a Cook estimate in
the following subsection.
\par
There is a subtlety here. In the standard two Hilbert space scattering theory $J$ is
bounded. The existence of the limit (2.10) on a dense set implies then also existence on
the closure. In our case we expect $||Je^{-iH_Bt}\phi||$ to be uniformly bounded
for any $\phi\in\cal F$ and sufficiently large $t$. But this remains to be shown
and we can establish the existence of the limit (2.10) only for $\phi\in D_e$.
\par
We prove the standard properties of the wave operator.
\smallskip\noindent
{\bf Proposition 4.} {\it $\Omega^{\mp}$ is an isometry from $\cal F$ to $\cal H$.
\smallskip\noindent
Proof.} For $\phi\in D_e$ also $e^{-iH_Bt}\phi\in D_e$ and
$$\langle\Omega^- \phi|\Omega^- \phi\rangle=\lim_{t\to\infty}
\langle Je^{-iH_Bt}\phi|Je^{-iH_Bt}\phi\rangle=\langle\phi|\phi\rangle\eqno(2.11)$$
by Lemma 3. By continuity we extend $\Omega^-$ to $\cal F.\qquad\squ$
\smallskip\noindent
{\bf Proposition 5.} {\it (Intertwining)
$$e^{-i(H-E_0)t}\Omega^{\mp}=\Omega^{\mp}e^{-iH_Bt}.\eqno(2.12)$$
Proof.} For $\phi\in D_e$ also $e^{-iH_Bt}\phi\in D_e$ and
$$\eqalign{\Omega^-\phi=&\lim_{s\to\infty}e^{i(H-E_0)(t+s)}Je^{-iH_B(t+s)}\phi
= \lim_{s\to\infty}e^{i(H-E_0)t}e^{i(H-E_0)s}Je^{-iH_Bs}e^{-iH_Bt}\phi\cr
=&e^{i(H-E_0)t}\Omega^-e^{-iH_Bt}\phi.\cr}\eqno(2.13)$$
Since $D_e$ is dense in $\cal F$, the result follows$.\qquad\squ$
\par
As a consequence, Ran$\Omega^{\mp}$ are reducing subspaces for $H$, and $H-E_0$
restricted to Ran$\Omega^{\mp}$ is unitarily equivalent to $H_B$ on $\cal F$.
Note that $\Omega^{\mp}|vac\rangle=\psi_0$ and projecting out bound states is not needed.
\smallskip\noindent
{\it Definition 6.} $\Omega^{\mp}$ is called asymptotically complete if
$${\rm Ran}\Omega^{\mp}=C^2\otimes\cal F .\eqno(2.14)$$
\smallskip\noindent
If $\Omega^{\mp}$ is asymptotically complete, then the asymptotics in (2.9) is valid for
all $\psi\in\cal H$ with $\phi=(\Omega^-)^*\psi$. In addition $H-E_0$ is unitarily
equivalent to $H_B$. Thus $H$ has purely absolutely continuous spectrum $[E_0,\infty)$
and a unique bound state at $E_0$. In our perspective, the central issue of
scattering theory is to prove asymptotic completeness.
\par
{\it Remark.} Following [14, Chapter 5] the wave operators are called
weakly asymptotically complete if
$${\rm Ran}\Omega^-={\rm Ran}\Omega^+,$$
complete if
$${\rm Ran}\Omega^-={\rm Ran}\Omega^+= P_{ac}(H)\cal H ,$$
and asymptotically complete if
$${\rm Ran}\Omega^-={\rm Ran}\Omega^+=[P_{pp}(H)\cal H]^{\perp}.$$
Our notion here is stronger, since it requires in addition that
$P_{pp}(H){\cal H}=\{c\psi_0\mid c\in C\}$.
\par
Clearly, the adjoint $(\Omega^{\mp})^*$ is well defined. We would expect then
$$(\Omega^{\mp})^*\psi=\lim_{t\to\infty}e^{iH_Bt}J^*e^{-i(H-E_0)t}\psi.\eqno(2.15)$$
The adjoint $J^*:C^2\otimes{\cal F}\to{\cal F}$ and is given by
$$\eqalign{(J^*\psi)^{(n)}(x_1,\ldots,x_n)=&\sum_{m=0}^{\infty}{m+n \choose n}^{1/2}
\sum_{\sigma}\int d^3x_{n+1}\ldots d^3x_{n+m}\cr
&\psi_0^{(n+m)}(x_1,\ldots,x_{n+m},\sigma)^*\psi^{(m)}(x_{n+1},\ldots,x_{n+m},\sigma).\cr}\eqno(2.16)$$
As in Lemma 1, $J^*$ is well defined for $\psi\in C^2\otimes D_e$. The difficulty with
(2.15) is that even if $\psi\in C^2\otimes D_e$ we do not know how to guarantee that
also $e^{-iHt}\psi\in C^2\otimes D_e$. By purely energetic considerations it is not
forbidden to have very many infrared photons. A control on the number of photons
uniformly in time is one of the deep problems of our subject.
\par
As standard, we define the $S$-matrix by
$$S=(\Omega^-)^*\Omega^+.\eqno(2.17)$$
If the wave operators are weakly asymptotically complete, then $S:{\cal F}\to\cal F$ is
unitary. By Proposition 5
$$e^{iH_Bt}Se^{-iH_Bt}=S, \eqno(2.18)$$
which reflects the energy conservation for in- and outgoing states. In constructive
quantum field theory the analytic structure of the kernel of the $S$-matrix has been
studied in fair detail. It would be of interest to understand how much of this survives
in our nonrelativistic model.
\medskip\noindent
{\bf 2.2 Cook estimate}
\smallskip\noindent
In this subsection we prove the existence of the wave operator $\Omega^-$. We first
establish two identities for $\phi\in D_e$,
$$a (\lambda)[\psi_0\otimes_s\phi]=a(\lambda)\psi_0\otimes_s\phi+
\psi_0\otimes_sa(\lambda)\phi, \eqno(2.19)$$
and
$$a^*(\lambda)[\psi_0\otimes_s\phi]=a^*(\lambda)\psi_0\otimes_s\phi.\eqno(2.20)$$
The cumbersome way is to use directly the definition (2.1).
A short cut goes as follows. Let
$$\phi=(0,\ldots,0,\phi^{(n)},0,\ldots).$$
Then $\psi_0\otimes_s\phi$ is equal to
$${1\over \sqrt{n!}}\int d^3x_1\ldots d^3x_n
\phi^{(n)}(x_1,\ldots,x_n)a^*(x_n)\ldots a^*(x_1)\psi_0,\eqno(2.21)$$
and
$$\eqalign{&a(\lambda)[\psi_0\otimes_s\phi]={1\over \sqrt{n!}}\int d^3x_1\ldots d^3x_n
\phi^{(n)}(x_1,\ldots,x_n) a(\lambda) a^*(x_n)\ldots a^*(x_1)\psi_0 \cr
=&a(\lambda)\psi_0\otimes_s\phi\cr
&+{1\over \sqrt{(n-1)!}}\sqrt n \int d^3x_1\ldots d^3x_n
\phi^{(n)}(x_1,\ldots,x_n){\lambda}^*(x_n)a^*(x_{n-1})\ldots a^*(x_1)\psi_0, \cr}
\eqno(2.22) $$
and
$$\eqalign{a^*(\lambda)[\psi_0\otimes_s\phi]=&{1\over \sqrt{n!}}\int d^3x_1\ldots d^3x_n
\phi^{(n)}(x_1,\ldots,x_n)a^*(x_n)\ldots a^*(x_1)a^*(\lambda)\psi_0 \cr
=&a^*(\lambda)\psi_0\otimes_s\phi.\cr}\eqno(2.23)$$
{\bf Proposition 7.} {\it Let $\int_0^{\infty} dt
|(\lambda,e^{-i\omega t}\lambda)|<\infty$. For all $\phi\in D_e$ the limit
$$s-\lim_{t\to\infty}e^{i(H-E_0)t}Je^{-iH_Bt}\phi=\Omega^-\phi\eqno(2.24)$$
exists.
\smallskip\noindent
Proof.} If $\phi=|vac\rangle$ then $J\phi=\psi_0$ and the limit exists. Let then
$\langle\phi|vac\rangle=0$ and $\phi\in D_e\cap D(H_B)$. Then
$Je^{-iH_Bt}\phi\in D(H)$ and, using (2.19) and (2.20),
$$\eqalign{{d\over dt}e^{i(H-E_0)t}Je^{-iH_Bt}\phi=&
ie^{i(H-E_0)t}(HJ-E_0J-JH_B)e^{-iH_Bt}\phi \cr
=&ie^{i(H-E_0)t}\sigma_x\psi_0\otimes_s (a(\lambda)e^{-iH_Bt}\phi)).\cr}\eqno(2.25)$$
Thus
$$e^{i(H-E_0)t}Je^{-iH_Bt}\phi=J\phi+ i\int_0^t ds
e^{i(H-E_0)t}\sigma_x\psi_0\otimes_s (a(\lambda)e^{-iH_Bt}\phi) \eqno(2.26)$$
and we have to show that $t\to||\sigma_x\psi_0\otimes_S(a(\lambda)e^{-iH_Bt}\phi)||$
is integrable for a dense set of $\phi$'s.
For this purpose, we define ${\cal M}_{\lambda}\subset L^2(R^3)$ to be the linear subspace
spanned by the set $\{e^{-i\omega t}\lambda\mid t\in R\}$.
We choose the $\phi$'s of product form, $\phi=(0,\ldots,0,\phi^{(n)},0,\ldots)$,
$\phi^{(n)}(x_1,\ldots,x_n)=\prod_{j=1}^n\phi_j(x_j)$ with
$\phi_j\in {\cal M}_{\lambda}\oplus {\cal M}_{\lambda}^{\perp}$, i.e.
$$\phi_j=\sum_{l=1}^{l_j} \alpha_{jl}e^{-i\omega t_{jl}}\lambda+\phi_j^{\perp},
\quad j=1,\ldots,n,\qquad {\rm with}
\quad \alpha_{jl}\in C,\quad\phi_j^{\perp}\in {\cal M}_{\lambda}^{\perp}.\eqno (2.27)$$
Then
$$||(\sigma_x\psi_0\otimes_s(a(\lambda)e^{-iH_Bt}\phi)||\le const.
\sum_{j=1}^n|\int d^3k\lambda(k)^*e^{-i\omega(k)t}\hat\phi_j(k)|,\eqno(2.28) $$
Integrating over $t$ gives a finite sum of integrals of the form
$$\int_0^{\infty}dt \;|\int d^3k\lambda(k)^*e^{-i\omega(k)(t+t_{jl})}\lambda(k)|,
\eqno(2.29)$$
which are bounded by assumption.
\par
The existence of the limit (2.24) for all finite particle number states follows
by equicontinuity in the one particle functions, and for states in $D_e$
similarly$.\qquad\squ$
\par
Since the operators $\Omega^{\mp}$ are bounded by Proposition 4, they can be
uniquely extended to the whole Fock space by taking closures. The finiteness of the
time integral in Proposition 7 is implied by condition (C3).
\medskip\noindent
{\bf 2.3 Long time decay}
\smallskip\noindent
Let $\cal W$ be the Weyl
algebra defined as the uniform closure of the Weyl operators
$W(f)=\exp(a^*(f)-a(f)), f\in L^2(R^3)$ on $\cal F$. $\cal W$ is also the uniform
closure of the local algebras ${\cal A}_R$, ${\cal W}=\overline{\cup_R{\cal A}_R}$ [15].
\smallskip\noindent
{\bf Proposition 8.} {\it Let $A$ be local in the sense that $A\in B(C^2)\otimes\cal W$.
Then for every $\psi\in$Ran$\Omega^-$ with $||\psi||=1$
$$\lim_{t\to\infty}\langle e^{-iHt}\psi |Ae^{-iHt}\psi \rangle=
\langle \psi_0|A \psi_0\rangle. \eqno(2.30)$$
\smallskip\noindent
Remark.} If $\Omega^-$ is asymptotically complete, then the limit (2.30) holds for all
$\psi\in\cal H$.
\smallskip\noindent
{\it Proof.} By Proposition 7 we have for all $\psi$ such that $\psi=\Omega^-\phi$
with $\phi\in D_e$
$$\lim_{t\to\infty}\langle e^{-iH t}\psi |A e^{-iH t}\psi \rangle=
\lim_{t\to\infty}\langle Je^{-iH_Bt}\phi |AJe^{-iH_Bt}\phi \rangle .\eqno(2.31)$$
We now choose $A=M\otimes W(f)\in B(C^2)\otimes\cal W$ and an $n$-particle product
state $\phi={1\over \sqrt{n!}}\prod_{j=1}^na^*(\phi_j)\psi_0$. Then
$Je^{-iH_Bt}\phi={1\over \sqrt{n!}}\prod_{j=1}^na^*(\phi_j(t))\psi_0$ with
$\phi_j(t)=e^{-i\omega t}\phi_j, \phi_j\in L^2(R^3)$. Since
$W(f)a^*(\phi)=a^*(\phi)W(f)-(f,\phi)W(f)$, we have
$$\eqalign{&\langle Je^{-iH_Bt}\phi |AJe^{-iH_Bt}\phi \rangle={1\over n!}
\langle\prod_{j=1}^na^*(\phi_j(t))\psi_0|M\otimes W(f)
\prod_{j=1}^na^*(\phi_j(t))\psi_0 \rangle \cr
=&\langle\prod_{j=1}^na^*(\phi_j(t))\psi_0|
\prod_{j=1}^na^*(\phi_j(t))M\otimes W(f)\psi_0\rangle \cr
&+{1\over n!}\sum_{b\subset\{1,\ldots,n\},b\not=\emptyset} sign(b)
\prod_{j\in b }^n(f,\phi_j(t))\langle\prod_{j=1}^na^*(\phi_j(t))\psi_0|
\prod_{j\in b^c}^na^*(\phi_j(t))M\psi_0\rangle \cr}\eqno(2.32)$$
with $sign(b)$ depending on the number of contractions.
Taking the limit $t\to\infty$, the second term vanishes because
$\lim_{t\to\infty}(f,\phi_j(t))=0$ by Riemann-Lebesgue. The first term converges
to $||\phi||^2\langle\psi_0|M\otimes W(f)\psi_0\rangle$ using Lemma 3 and the estimate
$$\langle W(f)\psi_0|e^{ \delta N_B}W(f)\psi_0\rangle<\infty,\eqno(2.33)$$
which follows from [11]. Thus the limit (2.30) holds for a particular choice of $A$ and
$\psi$. By taking linear combinations and strong limits for $\psi$, uniform limits for
$A$, one reaches all $A\in B(C^2)\otimes\cal W$ and all $\psi\in$Ran$\Omega^-.\qquad\squ$
\bigskip\noindent
{\largebf 3. Perturbation and weak coupling theories}
\medskip\noindent
{\bf 3.1 Level shift and lifetime}
\smallskip\noindent
Physically the coupling $\alpha$ to the photon field is weak and it is natural to
gain some understanding through perturbation theory, which, however, is singular, since
we perturb around the $\alpha=0$ Hamiltonian, which exhibits no radiative decay at all.
\par
The basic problem is to consider the excited state $\psi_u=u\otimes |vac\rangle$,
and to investigate the survival amplitude $\langle\psi_u|e^{-iHt}\psi_u\rangle$.
We do this through the resolvent expansion
$$\langle\psi_u|(H-z)^{-1}\psi_u\rangle=\sum_{n=0}^{\infty}\langle\psi_u|
(H_0-z)^{-1}[-\alpha V(H_0-z)^{-1}]^n\psi_u\rangle,\eqno(3.1)$$
with $V=\sigma_x\otimes(a^*(\lambda)+a(\lambda))$.
In this expansion we sum now only those terms where $a^*(\lambda)$ and $a(\lambda)$
alternate. Then
$$\eqalign{\langle\psi_u|(H-z)^{-1}\psi_u\rangle&\approx\sum_{n=0}^{\infty}
(\mu-z)^{-1}[\alpha^2\int d^3k|\lambda(k)|^2(\omega(k)-z)^{-1}(\mu-z)^{-1}]^n\cr
&=(\mu-\alpha^2R(z)-z)^{-1}.\cr} \eqno(3.2)$$
For the limit $t\to+\infty$ we need $R(z)$ just above the real axis. We have
$$\lim_{\eta\downarrow 0}R(E+i\eta)
=\lim_{\eta\downarrow 0}\int d^3k{|\lambda(k)|^2\over{\omega(k)-E-i\eta}}
=\lim_{\eta\downarrow 0}\int_0^{\infty}{\rho(\omega)d\omega\over {\omega-E-i\eta}}
={i\over 2}\gamma(E)-\beta(E).\eqno(3.3)$$
We adopted here the conventions
$$\rho(\omega)=\int d^3k |\lambda(k)|^2\delta(\omega-\omega(k)),\eqno(3.4)$$
where $\rho(\omega)>0$ for $\omega>0$, and
$$\eqalign{\int_0^{\infty}dt e^{iEt}\int_0^{\infty}d\omega\rho(\omega)e^{-i\omega t}
=&{1\over 2}\gamma(E)+i\beta(E) \cr
=&\pi\rho(E)-i{\cal P}\int d\omega{\rho(\omega)\over \omega-E}. \cr}\eqno(3.5)$$
Therefore
$$\lim_{\eta\downarrow 0}\langle\psi_u|(H-E-i\eta)^{-1}\psi_u\rangle\approx
[\mu-E+\alpha^2\beta(E)-i\alpha^2\gamma(E)/2]^{-1}.\eqno(3.6)$$
Since $\alpha$ is small and $\beta,\gamma$ are smooth, the dominant part in (3.6) comes
from $E=\mu$. Setting $\beta=\beta(\mu),\gamma=\gamma(\mu)$ we have
$$\lim_{\eta\downarrow 0}\langle\psi_u|(H-E-i\eta)^{-1}\psi_u\rangle\approx
[\mu-E+\alpha^2\beta-i\alpha^2\gamma/2]^{-1}.\eqno(3.7)$$
In this approximation, the resolvent has a pole on the second Riemann sheet. The
eigenvalue at $\mu$ for $\alpha=0$ is shifted to $\mu+\alpha^2(\beta-i\gamma/2)$.
Such a resonance can be unfolded through complex scaling, cf. Section 4.
\par
Translated to real time, (3.7) yields an exponential decay as
$$\langle\psi_u|e^{-iHt}\psi_u\rangle\approx\exp
[-i(\mu+\alpha^2\beta)t-\alpha^2(\gamma/2) t].\eqno(3.8)$$
$\alpha^2\gamma$ is the decay rate for the excited state of the atom.
\medskip\noindent
{\bf 3.2 Reduced dynamics in the weak coupling limit}
\smallskip\noindent
In Section 3.1 we considered a particular atomic state and observable.
More generally, we could consider an arbitrary atomic observable $M\in B(C^2)$
and its expectation. We write $M\mapsto\langle e^{-iHt}\psi|M\otimes Ie^{-iHt}\psi\rangle
={\rm tr}M\rho_t$ as a linear functional on $B(C^2)$, defining thereby
the reduced (atomic) density matrix $\rho_t$. Since $\rho_t$ is a statistical
state, we might as well take the initial state to be a density matrix, i.e.
$\rho\ge 0, {\rm tr}\rho=1$.
Usually, one assumes that the atom and photon field are initially uncorrelated and that
the field is in the vacuum. Then the reduced dynamics, $T_t^{\alpha}$, is defined by
$${\rm tr}[M\;T_t^{\alpha}\rho]=
{\rm tr}[M\otimes I\;e^{-iHt}\rho\otimes P_{vac}e^{iHt}]\eqno(3.9)$$
with $P_{vac}$ the projection onto $|vac\rangle$.
The reduced dynamics carries all information about atomic observables.
$T_t^{\alpha}$ is a linear map on $B(C^2)$, which is (completely) positive and
preserves normalization, i.e.
$T_t^{\alpha}\rho\ge 0, {\rm tr}T_t^{\alpha}\rho={\rm tr}\rho$.
However, $T_t^{\alpha}$ does not have the semigroup property. Only at small coupling
the loss of energy to the photon field can be approximated by a dissipative generator.
\smallskip\noindent
{\bf Theorem 9} (Davies [16]). {\it Let $[\sigma_z,\cdot], K$ be
linear operators on $B(C^2)$ defined by
$$[\sigma_z,\cdot]\rho=[\sigma_z,\rho],\qquad
K\rho={1\over 2}\gamma([\sigma^-\rho,\sigma^+]+[\sigma^-,\rho\sigma^+])\eqno(3.10)$$
with the spin raising and lowering operators $\sigma^+,\sigma^-$,
$\sigma^+v=u,\sigma^+u=0,\sigma^-=(\sigma^+)^*$. Then for every $\tau>0$
$$\lim_{\alpha\to 0}\sup_{0\le t\le\alpha^{-2}\tau}||T_t^{\alpha}\rho-\exp
[-i(\mu/2+\alpha^2\beta)[\sigma_z,\cdot]t+\alpha^2 Kt]
\rho||=0.\eqno(3.11)$$
Remark.} Since dissipation is in effect on the time scale $\alpha^{-2}\tau$, the limit
in (3.11) is meaningful only if the comparison is on that time scale.
$-i[\sigma_z,\cdot]$ and $\alpha^2K$ differ in order $\alpha^2$.
As a consequence $K$ is not uniquely fixed by the limit (3.11) [17,18].
\smallskip\noindent
{\it Proof.} The only condition to be checked is that
$$\int_0^{\infty}dt\;|\int_0^{\infty}d\omega\rho(\omega)e^{-i\omega t}|
(1+t)^{\delta}<\infty \eqno(3.12)$$
which is implied by condition (C3)$.\qquad\squ$
\par
Davies also investigates the structure of the photon field at times of order
$\alpha^{-2}$. As before the initial state is $\psi_u=u\otimes|vac\rangle$.
At weak coupling only one photon is emitted. Its wave function is described in
\smallskip\noindent
{\bf Theorem 10.} {\it Let $\phi_{\lambda}(t)=
(0,e^{-i\omega(k)t}\lambda(k),0,\ldots)\in{\cal F}$. Then for every $\tau>0$
$$\eqalign{\lim_{\alpha\to 0}&\sup_{0\le t\le\alpha^{-2}\tau}||e^{-iHt}\psi_u
- \Bigl( e^{-i(\mu+\alpha^2\beta)t-\alpha^2(\gamma/2)t}\psi_u \cr
&- i\int_0^t ds e^{ i(\mu(t-s)}v\otimes\phi_{\lambda}(t-s)
e^{-i(\mu+\alpha^2\beta)s-\alpha^2(\gamma/2)s}\Bigr)||=0.\cr}\eqno(3.13)$$
\smallskip\noindent
Proof.} The result is a consequence of [17, Theorem 3.1] with
${\cal H}=C^2\otimes{\cal F}$ and $P_0$ projection onto $\psi_u.\qquad\squ$
\par
(3.13) expresses that the excited atom acts as a source with exponentially
decaying amplitude and source function $\lambda$.
\medskip\noindent
{\bf 3.3 Scattering}
\smallskip\noindent
The $S$-matrix relates in- and outgoing photon states, cf. (2.9), the atom being in the
ground state in the remote past and future. To investigate the structure of the
$S$-matrix for small coupling we follow a while formal scattering theory
[19]. One defines the $T$-matrix through $S=I-2\pi iT$.
Let $\psi,\phi\in{\cal F}$. Then
$$\eqalign{-2\pi i\langle\phi|T\psi\rangle=&\langle\phi|(S-I)\psi\rangle
= \langle\phi|(\Omega^--\Omega^+)^*\Omega^+\psi\rangle
= \langle(\Omega^--\Omega^+)\phi|\Omega^+\psi\rangle \cr
=&\lim_{\eta\downarrow 0}(-i)\int_0^{\infty}dt e^{-\eta t}
\langle e^{i(H-E_0)t}\sigma_x\psi_0\otimes_sa(\lambda) e^{-iH_Bt}\phi\cr
&+e^{-i(H-E_0)t}\sigma_x\psi_0\otimes_sa(\lambda) e^{ iH_Bt}\phi|
\Omega^+\psi\rangle \cr
=&\lim_{\eta\downarrow 0}(-i)\int_0^{\infty}dt e^{-\eta t}\Bigl(
\langle\sigma_x\psi_0\otimes_sa(\lambda) e^{-iH_Bt}\phi|
\Omega^+e^{-iH_Bt}\psi\rangle \cr
&+\langle\sigma_x\psi_0\otimes_sa(\lambda) e^{iH_Bt}\phi|
\Omega^+e^{iH_Bt}\psi\rangle \Bigr). \cr}\eqno(3.14)$$
We used (2.25) to write $(\Omega^--\Omega^+)\phi$ as an integral,
and the intertwining relation (2.12).
\par
It is convenient to work in the momentum representation. To avoid too many indices we set
$k\in\bigcup_{n\ge 0}(R^3)^n, \psi(k)=(\psi^{(0)},\psi^{(1)}(k_1),\ldots)$ and
$dk$ stands for $\sum_{n=0}^{\infty}\prod_{j=1}^nd^3k_j$. $H_B$ becomes
multiplication by $E(k)=(0,\omega(k_1),\ldots,\sum_{j=1}^n\omega(k_j),\ldots)$.
In (3.14) we have then the factor
$$\lim_{\eta\downarrow 0}-i\int_0^{\infty}dt e^{-\eta t}
(e^{i[E(k')-E(k)]t}+e^{-i[E(k')-E(k)]t})=-2\pi i\delta(E(k')-E(k)) \eqno(3.15)$$
as a distribution. If the kernel of the $T$-matrix in momentum space is sufficiently
smooth, then
$$\langle\phi|T\psi\rangle=\int dkdk'\phi(k)^*\langle k|{\cal T}|k'\rangle
\delta(E(k')-E(k))\psi(k') \eqno(3.16)$$
with the off energy shell $\cal T$ matrix
$$\langle\phi|{\cal T}\psi\rangle
=\int dkdk'\phi(k)^*\langle k|{\cal T}|k'\rangle \psi(k')
=\langle\sigma_x\psi_0\otimes_sa(\lambda)\phi|\Omega^+\psi\rangle. \eqno(3.17)$$
If we use again equation (2.25) and (2.19), we obtain
$$\eqalign{\sqrt{n!}&\sqrt{m!}\langle k_1,\ldots,k_m|{\cal T}|k'_1,\ldots,k'_n\rangle
= \langle\sigma_x[a(\lambda),\prod_{j=1}^ma^*(k_j)]\psi_0|
\prod_{j=1}^na^*(k'_j)\psi_0\rangle \cr
+&\lim_{\eta\downarrow 0}
\langle\sigma_x[a(\lambda),\prod_{j=1}^ma^*(k _j)]\psi_0|(E_0+E(k')-H+i\eta)^{-1}
\sigma_x[a(\lambda),\prod_{j=1}^na^*(k'_j)]\psi_0\rangle.\cr}\eqno(3.18)$$
The rate of scattering from $k$ into $k'+dk'$ equals
$$\Gamma(k\to k')dk'=2\pi\delta(E(k)-E(k'))|\langle k|{\cal T}|k'\rangle|^2dk'.
\eqno(3.19)$$
The $\delta$-function reflects energy conservation in the sense that the kinetic
energies of in- and outgoing states are the same.
\par
Of particular interest is the one photon scattering for small coupling. Then (3.18)
simplifies to
$$\langle k|{\cal T}|k'\rangle
=\alpha\lambda(k)\langle\sigma_x\psi_0|a^*(k')\psi_0\rangle
+\lim_{\eta\downarrow 0} \alpha\lambda(k)\alpha\lambda(k')^*
\langle\sigma_x\psi_0|(E_0+\omega(k')-H+i\eta)^{-1}
\sigma_x\psi_0\rangle,\eqno(3.20)$$
where we introduced explicitly the coupling $\alpha$.
For the first term the perturbed vacuum $\psi_0$ is expanded to first order in
$\alpha$ yielding
$$-\alpha^2\lambda(k)\lambda(k')^*(\omega(k')+\mu)^{-1},\eqno(3.21)$$
while the second term equals
$$\alpha^2\lambda(k)\lambda(k')^*\lim_{\eta\downarrow 0} \langle\psi_u|
(E_0+\omega(k')-H+i\eta)^{-1}\psi_u\rangle=\alpha^2\lambda(k)\lambda(k')^*
(\omega(k')-\mu-\alpha^2\beta+i\alpha^2\gamma/2)^{-1} \eqno(3.22)$$
in the approximation (3.7). If $k$ is nonresonant, i.e. $\omega(k)\not= \mu$, then
$$\langle k|{\cal T}|k'\rangle=\alpha^2\lambda(k)\lambda(k')^*2\mu
(\omega(k')^2-\mu^2)^{-1} \eqno(3.23)$$
to second order in $\alpha$ and the scattering rate $\Gamma(k\to k')$ results as
$$\Gamma(k\to k')d^3k'
=2\pi\delta(\omega(k)-\omega(k'))\alpha^4|\lambda(k)|^2|\lambda(k')|^24\mu^2
(\omega(k')^2-\mu^2)^{-2}d^3k'.\eqno(3.24)$$
It would be of interest to establish rigorously the limit $\alpha\to 0$ of the
$\cal T$ matrix. By (3.21) and (3.23) the lowest nonvanishing contribution is of
order $\alpha^2$. Physically, it is more informative to take $\omega(k)$ close to the
resonance as $\omega(k)-\mu=O(\alpha^2)$. Then, as can be seen from (3.22), the limit
is of order one and depends on $\beta,\gamma$, whereas the contribution (3.21) is
negligible. This is the situation envisioned in Theorem 10.
\par
It is only a small step to compute the full second order $T$-matrix. Using the
energy conservation, $\delta(E(k)-E(k'))$ in (3.16), one obtains
$$\eqalign{\langle k_1,\ldots,k_m|& T|k'_1,\ldots,k'_n\rangle
=\alpha^2\delta_{nm}{1\over n!}\sum_{j=1}^n
\sum_{i=1}^m\lambda(k_j)\lambda(k'_i)^*\delta(\omega(k_j)-\omega(k'_i))\cr
& [(\omega(k_j)-\mu)^{-1}-(\omega(k_j)+\mu)^{-1}]
\langle vac|\prod_{l=1,l\not= j}^na(k_l)\prod_{l=1,l\not= i}^na^*(k'_l)]vac\rangle\cr}
\eqno(3.25)$$
provided $\omega(k_j)\not=\mu$ for $j=1,\ldots,n$.
As expected, except for symmetrization, photons scatter independently. For resonant
scattering $\mu$ has to be shifted to $\mu+\alpha^2\beta-i\alpha^2\gamma/2$, as in (3.22).
\bigskip\noindent
{\largebf 4. Complex scaling}
\medskip\noindent
Decaying states also occur on the level of a one particle Schr\"odinger equation. The
standard examples are a metastable local minimum as in the potential $V(x)=x(1-x^2)$
locally with $V(x)\to 0$ as $x\to\infty$ or a potential with term
$E\cdot x$ added (Stark effect).
A very powerful technique to analyze the time-dependent behaviour is complex scaling.
As the spectrum is rotated one uncovers eigenvalues of the dilated Schr\"odinger
operator just below the real axis.
These resonances govern the bulk part of the dynamics.
\par
We explain how the dilation technique applies to the spin-boson Hamiltonian [20,21].
For this purpose we first have to define the analytic dilation $H_{\theta}$ of $H$.
For real $\theta$ let
$$U(\theta)\phi(k)=e^{-3\theta/2}\phi(e^{-\theta}k) \eqno(4.1)$$
be the unitary dilation acting on the one-particle functions. Lifting $U(\theta)$ to
Fock space yields
$$\eqalign{U(\theta)\psi^{(n)}( k_1,\ldots, k_n)&=e^{-3n\theta/2}
\psi^{(n)}(e^{-\theta}k_1,\ldots,e^{-\theta}k_n),\cr
U(\theta)|vac\rangle&=|vac\rangle,\cr}\eqno(4.2)$$
and
$$H_{B,\theta}=U(\theta)H_BU(-\theta)=\int d^3k e^{-\theta}|k| a^*(k)a(k).\eqno(4.3)$$
We extend now the definition (4.3) to complex $\theta$ with $|{\rm Im}\theta|<\pi/2$.
\par
Let $C_a$ be the open strip $\{z\in C\mid |{\rm Im}z|0$, its
essential (purely absolutely continuous) spectrum is rotated downwards to the two
halflines $\{z\mid z=re^{-i{\rm Im}\theta}, z=\mu+re^{-i{\rm Im}\theta}, r>0\}$
compare with Figure 1.
If $H_B$ would have a spectral gap, e.g. by considering massive photons
$\omega(k)=\sqrt{k^2+M_0^2}$ or by merly shifting as $\omega(k)=|k|+M_0$, then the
resonant eigenvalue at $\mu$ would become isolated for the dilated Hamiltonian.
One can then use analytic perturbation theory in the coupling $\alpha$.
$H_{\theta}(\alpha)$ has the discrete eigenvalue $E(\alpha)$ such that
$\lim_{\alpha\to 0} E(\alpha)=\mu$.
Furthermore for $|\alpha|<\alpha_0$, $E(\alpha)$ is given through the convergent power series
$$E(\alpha)=\mu+a_1\alpha+a_2\alpha^2+\cdots. \eqno(4.6)$$
The coefficients can be computed through standard perturbation theory.
$a_1$ vanishes since $\langle\psi_u|\sigma_x(a^*(\lambda)+a(\lambda))\psi_u\rangle=0$.
The second coefficient is given by
$$a_2=\lim_{\eta\downarrow 0}\langle\psi_u|\sigma_x(a^*(\lambda)+a(\lambda))
(\mu-H_B+i\eta)^{-1}\sigma_x(a^*(\lambda)+a(\lambda))\psi_u\rangle
=\beta-i\gamma/2,\eqno(4.7)$$
compare with (3.7).
\par
For the true dispersion relation $\omega(k)=|k|$ the eigenvalues $0,\mu$ are no longer
isolated. Still Bach, Fr\"ohlich and Sigal succeed to provide a sufficiently precise
estimate on the resolvent set of $H_{\theta}$ for small $\alpha$.
It is convenient to discuss their result at the hand of Figure 1. By turning on
the coupling the spectrum of $H_{\theta}$ shifts downward to order $\alpha^2$ at $\mu$
and to the left by the same order at $0$. In addition the sharp line for $\alpha=0$
opens to a cone with angle $O(\alpha^2)$. Since the eigenvalues are not isolated, one has
to allow for an additional layer of $O(\alpha^3)$. But then every point away from the shaded
region is guaranteed to be in the resolvent set of $H_{\theta}$.
\par
The resonant decay can now be understood as coming mostly from the spectral part
of the dilated Hamiltonian located at distance $\alpha^2$ below $\mu$. One has no
information on the fate of the eigenvalue inside the cone. But the time behaviour has
the appearance as if there sits an eigenvalue computed from perturbation theory.
\par
In fact [21] provides estimates on the resolvent. Thereby one can obtain information on the
time dependence of physical observables . As one typical result we quote
\smallskip\noindent
{\bf Theorem 12.} [21] {\it Let $\lambda$ satisfy the condition (4.4). Then for
$|\alpha|<\alpha_0$
$$\langle\psi_u|e^{-itH}\psi_u\rangle=\exp[-i(\mu+\alpha^2\beta)t-\alpha^2(\gamma/2)t]
+O(\alpha^2) \eqno(4.8)$$
uniformly in $t$.
\smallskip\noindent
Remark.} Under more stringent conditions on $\lambda$ the same result is proved by King [20].
\par
The error in (4.8) comes from the contributions $O(\alpha^2)$ close to the ground state
energy, on which analytic dilation provides no information. If in (4.8) we would localize
somewhat in energy as $\langle\psi_u|g(H)e^{-itH}\psi_u\rangle$ with some smooth $g$
supported, say, on $[\mu/2,3\mu/2]$, then this amplitude has only a uniform error
which decays as $C_n(1+t)^{-n}$, arbitrary $n>0$ [22]. Similar arguments [23] imply that
$H$ has purely absolutely continuous spectrum in $[E_0+O(\alpha^2),\infty)$.
We expect that estimates of the form (4.8) could be established also for local observables.
\par
%To conclude our discussion of dilation analyticity, we mention that Weder [24] studied
%by this method the spectrum of the Lee model describing
%strong interactions of static baryons.
%In [25] complex scaling is used to prove the weak
%coupling limit, $\alpha\to 0, t\to\infty, \alpha^2t=\tau$, for the dipole approximation
%with massive photons. This improves [16] where the coupling operator, i.e. $x$ in the
%case of (1.9), is assumed to be a bounded operator.
If applicable, analytic dilation provides an explicit dynamical information.
One further case is [24], where the weak coupling limit $\alpha\to 0, t\to\infty,
\alpha^2 t=\tau$, is proved for the dipole approximation with massive photons and
without cut-off in the interaction. On the other hand it seems difficult to go
beyond small coupling and to extract information on infrared photons.
E.g. if one just wants to prove that the survival amplitude (4.8)
decays as $t\to\infty$, other methods, such as scattering theory, are required.
\bigskip\noindent
{\largebf 5. Cut-offs}
\medskip\noindent
{\bf 5.1 Finite photon approximation}
\smallskip\noindent
Considering the foregoing it seems unlikely that the long time asymptotics (2.30)
could be proved, so to speak, in one blow. One step towards a partial solution is to
restrict by fiat the number of photons to be less than or equal to $N$ with $N$
arbitrary. I.e. if $P_{\le N}$ projects onto the subspace of $\cal F$ spanned by
$(\psi^{(0)},\ldots,\psi^{(N)},0,\ldots)$ then one studies
$P_{\le N}HP_{\le N}$. To have a structure close to the standard $N$-body Schr\"odinger
operator it is convenient to regard in addition the photons as distinguishable [25].
The states of a single photon are of the form $(c,\psi)\in C\oplus L^2(R^3)$, where
$c$ is the amplitude for the photon being dead and $\psi$ is the wave function for the
alive photon. The Hilbert space for $N$ distinguishable photons is then ${\cal H}_N=
\otimes_{j=1}^N {\cal H}_j$ with each ${\cal H}_j$ a copy of $C\oplus L^2(R^3)$ and the
Hilbert space for the coupled system is $C^2\otimes {\cal H}_N$. The Hamiltonian, $H_N$,
is fixed by the requirement that $H_N$ restricted to the symmetric subspace
$C^2\otimes P_{\le N}{\cal F}$ is unitarily equivalent to $P_{\le N}HP_{\le N}$.
Clearly, the kinetic energy of the photons is
$$T=\sum_{j=1}^N T_j \eqno(5.1)$$
with
$$T_j=I\otimes\cdots\otimes\pmatrix{0&0\cr0&\omega\cr}_j\otimes\cdots\otimes I,\eqno(5.2)$$
which reflects that photons move independently. The energy of the atom is
$\mu(1+\sigma_z)/2$, as before. Somewhat surprisingly, the interaction between photons
and the atom is not strictly of two-body type. Let
$Q_j=I\otimes\cdots\otimes\pmatrix{0&0\cr0&I\cr}_j\otimes\cdots\otimes I$ be the
projection onto the $j$-th photon being alive and let
$Y_j=\sum_{i=1,i\not= j}^N(1-Q_j)$ be the number of dead photons except for photon $j$.
Then the interaction reads
$$\alpha\sigma_x\otimes V=\sum_{j=1}^N\alpha(1+Y_j)^{-1/2}\sigma_x\otimes V_j,\eqno(5.3)$$
where $V_j=I\otimes\cdots\otimes V_j\otimes\cdots\otimes I$ and
$$V_j\pmatrix{c\cr\psi\cr}=\pmatrix{0&(\lambda,\cdot)\cr\lambda&0\cr}
\pmatrix{c\cr\psi\cr}=\pmatrix{(\lambda,\psi)\cr c\lambda}\eqno(5.4)$$
on factor $j$. From (5.3) we conclude that the interaction strength depends on the number
of dead photons around. A dead photon becomes alive with wave function $\lambda$,
whereas an alive photon at $x$ dies with amplitude $\lambda(x)$. Since $\lambda$
decays as $|x|\to\infty$, photons which have escaped from the atom can no longer
disappear. Combining the three pieces, the full Hamiltonian is given by
$$H_N=\mu{1+\sigma_z\over 2}\otimes I+I\otimes T+\alpha\sigma_x\otimes V.\eqno(5.5)$$
For $N=1$ we decompose our Hilbert space into ${\cal H}_+\oplus{\cal H}_-$ according to
$(-1)^{(1+\sigma_z)/2+Q}=\mp 1$. ${\cal H}_+$ and ${\cal H}_-$ are reducing subspaces
for $H_1$. $H_1$ restricted to ${\cal H}_+$ reads
$$\pmatrix{0&(\lambda,\cdot)\cr\lambda&\omega\cr}\eqno(5.6)$$
and $H_1$ restricted to ${\cal H}_-$ reads
$$\pmatrix{\mu&(\lambda,\cdot)\cr\lambda&\omega\cr}.\eqno(5.7)$$
Thus the one-photon problem reduces to the Friedrichs model [26].
In the context of the theory of elementary particles related versions are known as Lee
model. The Friedrichs model has been studied very extensively, amongst other
approaches also through dilation analyticity [27]. The decay amplitude of the state
$(1,0)$ can be estimated in the form of Eq. (4.8) [28,29].
The case $N=2$ corresponds to the three-body problem, for which
asymptotic completeness is proved for small coupling [30].
\par
For general $N$ we establish [25] the analogue of the HVZ theorem
\smallskip\noindent
{\bf Theorem 13.} {\it Let $E_N$ be the ground state energy and $\Sigma_N$ be the edge
of the essential spectrum of $H_N$. Then}
$$E_{N+1}0,\qquad C \;{\rm compact}.\eqno(5.12)$$
\smallskip\noindent
Remark.} The theorem uses the extension of Mourre estimates to isometric semigroups [33].
\par
If $\psi_n$ is an eigenvector of $H_N$, then by the virial theorem
$\langle\psi_n|[I\otimes\tilde A,H_N]\psi_n\rangle=
0\ge\kappa-\langle\psi_n|C\psi_n\rangle$.
Thus through control over $\kappa$ and $C$ we can count the number of eigenvalues.
With the $I\otimes\tilde A$ as given in (5.9) the estimates never fall below 2.
>From an improvement of the conjugate operator we infer
\smallskip\noindent
{\bf Theorem 15.} [33] {\it Let $\lambda$ be in the domain of $D^2$. Then for every
$\mu>0$ there exists an $\alpha_0$ such that for $0<\alpha<\alpha_0$ the spectrum
of $H_N$ decomposes as
$\sigma_{pp}=\{E_N\}, \sigma_{sc}=\emptyset, \sigma_{ac}=[E_{N-1},\infty)$.}
\medskip\noindent
{\bf 5.2 Finite mode approximation}
\smallskip\noindent
A widely studied problem [34] is the finite mode approximation to (1.10) which in the
extreme case of a single mode becomes
$$H=\mu{1+\sigma_z\over 2}\otimes I+I\otimes\omega a^*a
+\alpha\sigma_x\otimes(a^*+a).\eqno(5.13)$$
This Hamiltonian describes the physics of an atom placed in a cavity of suitable
geometry so that the interaction is dominated by a single mode of the electromagnetic
field. In particular the radiation emitted by the atom will be reflected at the walls
of the cavity and subsequently reexcite the atom. If one ignores dissipative losses,
the back and forth bouncing goes on indefinitely and there is no radiative decay.
Correspondingly, the spectrum of $H$ is purely discrete and we have a quasiperiodic
time evolution. Since a nonlinear oscillator
($\sigma_z$) is coupled to a linear one, questions of nonlinear dynamics as irregular
recurrences and level statistics come into focus.
\par
If one writes in (5.13) the interaction as $(\sigma^-+\sigma^+)\otimes(a^*+a)$ and
neglects the rotating terms $\sigma^-\otimes a,\sigma^+\otimes a^*$, then one obtains
the Jaynes-Cummings model. It can be solved exactly, with the amplitude to be in the
upper state exhibiting quasi-random revivals [35]. Three level models and their
experimental realizations are surveyed in [36]. The hydrogen atom coupled to a single
radiation mode was studied through analytic dilation in [37]. [38-40]
discuss the quality of the rotating wave approximation.
\bigskip\noindent
{\largebf 6. Perspectives}
\medskip\noindent
We indicate some open problems.
\par\noindent
{\it (i) Asymptotic completeness}
\par
As discussed in Section 2, a central problem is asymptotic completeness of the wave
operators. Even with a cut-off in the photon number $N$ this property remains to
be established.
\par\noindent
{\it (ii) Absence of embedded eigenvalues}
\par
Even if (i) holds, there could be bound states
embedded in the continuum (by the Mourre estimate there cannot be too many). It would be
of interest to understand whether methods as in [14] would allow one to exclude embedded
eigenvalues. For finite $N$ we can exclude embedded eigenvalues only for small
coupling [33].
\par\noindent
{\it (iii) Bounds on the number of photons}
\par
For the sequence of ground states one knows that $\psi_0^{(N)}\to \psi_0$ as $N\to\infty$.
In fact, with $p_0(n)$ denoting the probability to have $n$ photons in the state
$\psi_0^{(N)}$, the bound $\sum_{n\ge 0}p_0(n)e^{\delta n}<\infty$ for all $\delta>0$
holds uniformly in $N$. No such bound is known to us for time-dependent states.
If $p(t,n)$ is the probability to have $n$ photons in the state $e^{-iHt}\psi$ and if
$p(0,n)$ has compact support initially, then we would expect an exponential bound of
$p(t,n)$ uniformly in time. The need for such bounds reappears at various stages.
E.g. if possible eigenvectors of $H$ (with dispersion $\omega(k)=|k|$) would satisfy
such a bound, we could get complete spectral information for small coupling [33].
\par\noindent
{\it (iv) More realistic atoms}
\par
Already in the dipole approximation with no cut-off in the coupling operator $x$,
cf. (1.8),(1.9), very little is known on a rigorous level. The uniqueness of the
ground state is established only under restrictive conditions.
>From a Mourre estimate with the conjugate operator $I\otimes\tilde A$, $\tilde A$
given by the second quantization of $D$, cf. (5.11),
one cannot even conclude that dim$P_{pp}(H)<\infty$ for small $\alpha$.
The technical difficulty results here from the unboundedness of the coupling
operator $x$.
\par
A further step towards realistic atoms is the minimally coupled Hamiltonian (1.3) with a
smeared out Coulomb potential. In such a model the atom can ionize. This opens a
new scattering channel, where the electron travels off to infinity.
\par\noindent
{\it (v) A closed theory of point charges interacting with the radiation field}
\par
Point charges interacting through the electrostatic Coulomb potential give rise to a
self-adjoint Hamiltonian, thus to a mathematically well defined theory. If one includes
the radiation field as in the Pauli-Fierz Hamiltonian (1.3), then the point charge limit
can be reached only through a suitable renormalization which removes the ultraviolet
cut-off in (1.3). For a particle coupled to a scalar Bose field such a program is carried
through by Fr\"ohlich [41]. At present, it is not clear, at least to us, whether a
mathematically well defined theory of point charges plus radiation field can be achieved.
Constructive quantum field theory is heading for much more ambitious goals.
>From this perspective mathematical physics seems to have completely overlooked that the
Pauli-Fierz Hamiltonian (generalized to many electrons and nuclei) covers
essentially all of low energy physics.
\bigskip\noindent
{\it Acknowledgements.} We are grateful to V.Bach for explaining us the results in
Ref. [21] and for comments on the paper. This work is supported by a grant of
Deutsche Forschungsgemeinschaft.
\vskip 4in\noindent
{\largebf Figure captions}
\bigskip\noindent
{\bf Figure 1.} The resolvent set of $H_{\theta}$.
The dashed lines are the spectrum of $H_{\theta}$ for $\alpha=0$. $\bullet$ are
eigenvalues. For small $\alpha$ the resonant sector shifts downwards and the ground
state sector to the left by $O(\alpha^2)$. The half lines open up to cones of
width $O(\alpha^2)$ and an error zone of $O(\alpha^3)$ must be allowed.
Points away from the shaded regions are in the resolvent set of $H_{\theta}$.
\vfill\eject
\parindent=0pt
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