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\begin{document}
\title{Van Hove Hamiltonians---exactly solvable models\\of the infrared and ultraviolet problem}
%
\author{Jan Derezi\'{n}ski\footnote{{\bf Acknowledgments.} The author would like to acknowledge useful
discussions with
C. G\'{e}rard and S. DeBi\`evre. He is also grateful to the referee for
the remarks, which helped him to make some improvements in the
paper. His research was partly supported by the
Postdoctoral Training Program HPRN-CT-2002-0277 and the grant SPUB127
funded by KBN.
A part of this work was done during his visit
to the Aarhus University supported by MaPhySto funded
by the Danish National Research Foundation.
}
\\ \\
Department of Mathematical Methods in Physics \\
Warsaw University \\ Ho\.{z}a 74, 00-682, Warszawa, Poland}
\maketitle
\noindent{\bf Abstract.}
Quadratic bosonic Hamiltonians with a linear perturbation are studied.
Depending on the infrared and ultraviolet behavior of the
perturbation, their properties are described from the point of view of
spectral and scattering theory.
\section{Introduction}
\label{s1}
Some of the simplest exactly solvable models
in classical and quantum physics are quadratic Hamiltonians with a linear
perturbation. Following \cite{Sch} we will call them van Hove
Hamiltonians. They arise in classical and
quantum field theory in many contexts. For
example, the Hamiltonian of electrodynamics with
prescribed external charges is a van Hove Hamiltonian.
In the case of finite degrees of freedom,
an appropriate translation in phase space transforms
a van Hove Hamiltonian into a purely quadratic Hamiltonian. In
the quantum case this translation can be implemented by a unitary
Bogolubov transformation.
If the phase space is infinite dimensional,
the situation becomes more complicated, because the
above mentioned Bogolubov transformation can become ``improper'' (not
implementable by a unitary operator). Still, as
it is well known, one can fully
analyze the properties of van Hove Hamiltonians even in the case of
the infinite dimensional phase space.
These properties are quite interesting, both
physically and mathematically. In fact, some types of
van Hove Hamiltonians can be viewed as
exactly solvable toy models of renormalization, both in the
infrared and ultraviolet regime.
Depending on the assumptions on the perturbation,
one can distinguish several types
of these Hamiltonians with distinct properties.
Let us describe some of the results of our paper
in a somewhat informal language.
We will restrict ourselves to the quantum case, since the classical
case is very similar. (In the main part of the paper a
different, more compact notation is used; moreover,
both the classical and
quantum case is treated).
Let $(K,\d k)$ be a space with a measure. (We use the notation $\d k$
to denote an arbitrary measure, not necessarily the Lebesgue measure).
Let $K\ni k\mapsto a^*(k)$
and $
K\ni k\mapsto a(k)$ be the corresponding bosonic creation and annihilation
operators. Let $K\ni k\mapsto h(k)$ be an almost everywhere positive measurable
function, describing the dispersion relation. Finally, suppose
that $K\ni k\mapsto z(k)$ is a complex function describing the interaction.
By a van Hove Hamiltonian we will mean a self-adjoint operator given
by the formal expression
\beq H:=\int \bigl(h(k)
a^*(k)a(k)\d k+ z(k)a^*(k)\d k+ \bar
z(k)a(k)\bigr)\d k+c,\label{hov0}\end{equation}
where $c$ is an arbitrary (maybe infinite) constant.
It turns out that if \beq
\begin{array}{l}
\int_{h(k)<1}|z(k)|^2\d k+\int_{h(k)\geq 1}\frac{|z(k)|^2}{h(k)^2}\d
k<\infty.\end{array}
\label{con0}\end{equation}
then properly interpreted (\ref{hov0}) defines a family of
self-adjoint operators.
There are two natural choices of $c$ in (\ref{hov0}).
The first choice is $c=0$. It is possible if we assume that
\beq
\begin{array}{l}
\int_{h(k)<1}|z(k)|^2\d k+\int_{h(k)\geq 1}\frac{|z(k)|^2}{h(k)}\d
k<\infty,\end{array}
\label{con01}\end{equation}
Under this condition
we can define the van Hove Hamiltonian of the first kind:
\beq
H_\rI:=\int \bigl(h(k)
a^*(k)a(k)\d k+ z(k)a^*(k)\d k+ \bar
z(k)a(k)\bigr)\d k.\label{hov}\end{equation}
The second choice is $c=\int \frac{|z(k)|^2}{h(k)}\d
k$. It is possible if
\beq
\begin{array}{l}
\int_{h(k)<1}\frac{|z(k)|^2}{h(k)}\d k+\int_{h(k)\geq 1}
\frac{|z(k)|^2}{h(k)^2}\d
k<\infty.\end{array}
\label{con02}\end{equation} Under this condition we can define
the van Hove Hamiltonians of the second kind:
\beq
H_{\II}:=\int h(k) \Big(a^*(k)+\frac{z(k)}{h(k)}\Big)
\Big(a(k)+\frac{\bar z(k)}{h(k)}\Big)
\d k.\label{hova}\end{equation}
The conditions (\ref{con01}) and (\ref{con02}) are true at the same time
iff
\beq
\int
\frac{|z(k)|^2}{h(k)}\d
k<\infty,\end{equation}
and then the two types of van Hove Hamiltonians are well defined and
differ by a constant:
\beq H_\rII=H_\rI+\int
\frac{|z(k)|^2}{h(k)}\d
k.\label{cou}\end{equation}
%(Note that the condition (\ref{con0}) can be satisfied and neither
%(\ref{con01}) nor (\ref{con02}))
Next we would like to describe various types of behavior of van Hove
Hamiltonians.
We will consider separately
the case when the dispersion relation
is separated away from zero and the case of a bounded dispersion relation.
In the former case the infrared problem is absent and we can study the
ultraviolet problem in its pure form.
In the latter case the ultraviolet problem is absent and all the
difficulties are due to the infrared problem.
We will see that one can distinguish $3 \times 3=9$ distinct
classes of van Hove Hamiltonians.
\subsection{Ultraviolet problem}
Assume that for some $00$ such that
$\{w_0+t w\ :\ |t|<\epsilon\}\subset\Dom\,G$ and there exists
\[\frac\d{\d t}G(w_0+tw)\Big|_{t=0}=:w\nabla G(w_0).\]
We will say that $G$ is differentiable at $w_0$ iff
\[\cD:=\{w\in\cW\ :\
w\nabla G(w_0)\ \hbox{ exists }\}\]
is a dense linear subspace of
$\cW$ and
$\cD\ni w\mapsto w\nabla G(w_0)$
is a bounded linear functional. If this is the case, then
the gradient
of $G$ at $w_0$ is denoted by $\nabla G(w_0)\in\cW$ and defined by
\[\Re(w|\nabla G(w_0))=w\nabla G(w_0),\ \ w\in\cW.\]
We will say that a map $\alpha:\cW\to\cW$ preserves the scalar product iff
\[(\alpha(w_1)-\alpha(w_2)|\alpha(w_3)-\alpha(w_4))=
(w_1-w_2|w_3-w_4),\ \ \ w_1,\dots,w_4\in\cW.\]
%If $A$, $B$ are subsets of a fixed vector space, then, as usual, we set
%$A+B:=\{a+b\ :\ a\in A,\ b\in B\}$
\subsection{Unbounded perators}
Let $h$ be a self-adjoint operator on
$\cW$ with $h\geq0$. $\sp h$ will denote the spectrum of $h$.
Let $\Dom\,h$ denote its domain and $( \Dom\,h)^*$ the space of
bounded antilinear functionals on $\Dom\,h$.
If $w_1\in\Dom\, h$ and $w_2\in(\Dom\, h)^*$, then we will write
$(w_1|w_2)$ for the action of $w_2$ on $w_1$.
Clearly, $\cW$ can be
embedded in a natural way in $(\Dom\,h)^*$.
The operators $h$ and $h^{1/2}$ extend to bounded maps from $\cW$ to
$(\Dom\, h)^*$. We denote by $h\cW$ and $h^{1/2}\cW$ respectively the
images of these extensions.
Clearly, the algebraic sums
$\cW+h^{1/2}\cW$, $h^{1/2}\cW+h\cW$ and $\cW+h\cW$ are well defined
subspaces of $(\Dom\, h)^*$. (Note, in particular, that $\cW+h\cW$
coincides with $(\Dom\, h)^*$).
%(Note in particular that if $h\geq c_0>0$, then $h\cW=(\Dom\,h)^*$).
\subsection{Fock spaces}
\cite{RS2,BR} If $\cX$ is a vector space, then $\Gammal_\s(\cX)$
will denote the
algebraic symmetric Fock space over $\cX$, that means the space of
finite linear combinations of symmetric tensor products of elements of
$\cX$. $\Omega$ will denote the vacuum.
If $\cW$ is a Hilbert space, then $\Gamma_\s(\cW)$ will denote the
(complete) Fock space, that is the completion of $\Gammal_\s(\cW)$.
If $u$ is a contraction on $\cW$, then $\Gamma(u)$ denotes the
contraction on $\Gamma_\s(\cW)$ that on the $n$ particle sector
equals $u^{\otimes n}$.
If $h$ is a self-adjoint operator, then $\d\Gamma(h)$ denotes the
self-adjoint operator that on the $n$ particle sector
equals $h\otimes 1^{\otimes (n-1)}+\cdots+ 1^{\otimes (n-1)}\otimes h$.
Let $b$ be a sesquilinear form on $\cW$ with the domain
$\widetilde\cW$. Abusing the notation, we will use the symbol
$\d\Gamma(b)$ to denote the sequilinear form on
$\Gamma_\s(\cW)$ with the domain $\Gammal_\s(\widetilde\cW)$
that on the $n$ particle sector
equals $b\otimes 1^{\otimes (n-1)}+\cdots+ 1^{\otimes (n-1)}\otimes b$.
\subsection{Creation and annihilation operators}
\label{s2.5}
The notion of creation and annihillation operators is standard in the
context of Fock spaces and can be found eg. in
\cite{BR,RS2}. Nevertheless, we will need slight generalizations of
these concepts.
Let $w$ be an antilinear form on $\cW$ with the domain $\Dom
w=\widetilde\cW
\subset\cW$. We define the annihilation
operator
$\bar w(a)$ as an operator with the domain
$\Gammal_\s(\widetilde\cW)$ satisfying
\[ \bar w(a)\,z^{\otimes n}:=\sqrt n(w|z)
z^{\otimes(n-1)},\ \ \ z\in\widetilde\cW.\]
(Note that vectors of the form $z^{\otimes n}$ span
$\Gammal_\s(\widetilde\cW)$).
The operator $\bar w(a)$ is closable iff $w\in\cW$. If this is the
case, we denote its closure by the same symbol. Its
adjoint is called the creation operator and denoted
\[w(a^*):=\bar w (a)^*.\]
If $f\in\Gammal_\s(\cZ)$, then $f(a^*)$ and $\bar f(a)$ have the
obvious meaning as polynomials in $a^*$ and $a$. For instance
$z^{\otimes n}(a^*)=(z(a^*))^n$ and $\bar z^{\otimes n}(a)=(\bar z(a))^n$.
We also introduce the field operator
\beq\phi(w):=\frac{1}{\sqrt2}(w(a^*)+\bar w(a)),\label{phi}\end{equation}
which
is self-adjoint on $\Dom\,\bar w(a)$. We
also introduce the Weyl operators
\beq W(w):=\e^{\i\phi(w)}.\label{phi1}\end{equation}
If $w\not\in\cW$, then the annihilation operator $\bar w(a)$ is not
closable. In this case
the creation operator $w(a^*)$ is not even densely
defined and is not of much use. On the other hand, we can define the
``creation form'', denoted $w(a^{*\f})$:
\[\Gammal_\s(\widetilde\cW)\times\Gamma_\s(\cW)\ni
(\Phi,\Psi)\mapsto(\Phi|w(a^{*\f})\Psi):=
(\bar w (a)\Phi|\Psi).\]
Note that $w(a^{*\f})$ is a different object from $w(a^*)$. In what
follows, however, we will sometimes abuse the notation and we will write
$w(a^*)$ instead of
$w(a^{*\f})$.
\section{Van Hove Hamiltonians}
\label{s3}
In Subsections \ref{s3.1} and \ref{s3.1a} we consider the classical
dynamics $\alpha$ generated by a Hamiltonian $G$ equal to a
quadratic polynomial. In
the remaining subsections we discuss their quantum counterparts---the
dynamics $\beta$ and its Hamiltonian $H$.
\subsection{Classical dynamics}
\label{s3.1}
%If $G:\cW\to\rr$ is a $C^1$ function, then its derivative can be
%viewed as a map\[\d G:\cW\to\cW.\]
%(note that we use the identification of $\cW$ with its real dual).
%If we treat $G$ as a ``classical Hamiltonian'', then the Hamilton
%equation for the dynamics generated by $G$ read
%\[\begin{array}{l}
%\Im(w_1|\frac{\d}{\d t})=\Re(\d G(w)|w_1),\ \ \ w_1\in\cW,\end{array}\]
%which can be rewitten as
%\[\frac{\d}{\d t}w=\i\d G(w).\]
Suppose that $h$ is a positive
operator on $\cW$. We will assume that $\Ker h=\{0\}$.
Let \beq
z\in\cW+h\cW\label{wqw}\end{equation}
For $w\in\cW$, $t\in\rr$, we define
\[\alpha^t(w):=\e^{\i th}w+(\e^{\i th}-1)h^{-1} z.\]
It is easy to see that $\rr\ni t\mapsto \alpha^t$ is a 1-parameter
group of transformations preserving the scalar product.
%Therefore, it preserves the real scalar product $\Re(\cdot|\cdot)$
%and the symplectic form $\Im(\cdot|\cdot)$.
Let us note the following property of the dynamics $\alpha$.
Let $p_1,p_2$ be two complementary orthogonal projections commuting
with $h$. For $i=1,2$ let $\cW_i:=\Ran p_i$. For $w\in\cW$, set
$w_i:=p_i w$. Set $h_i:=p_ih$, treated as a self-adjoint operator on
$\cW_i$. Let $\alpha_i$ be the
dynamics on $\cW_i$ defined by $h_i$, $z_i$. Then the dynamics
$\alpha$ splits:
\[\alpha^t(w_1,w_2)=(\alpha_1^t(w_1),\alpha_2^t(w_2)).\]
In particular, we can take \beq p_1:=1_{[0,1]}(h),\ \
p_2:=1_{]1,\infty[}(h).\label{facto1}\eeq
Then $h_1$
is
bounded and $h_2$ is bounded from below by a positive constant.
In the case of $h_1$, the ultraviolet problem is absent, but the
infrared problem can show up. In the case of $h_2$ we have the
opposite situation:
the infrared problem is absent, but we can face the ultraviolet problem.
\subsection{Classical Hamiltonian}
\label{s3.1a}
The dynamics $\alpha$ preserves the symplectic form $\Im(\cdot|\cdot)$.
Therefore, we should expect that $\alpha$ possesses a Hamiltonian
$G$, that is a function satisfying
\beq\begin{array}{l}
\Im(w_1|\frac{\d}{\d t}\alpha^t(w))=\Re(w_1|\nabla
G(\alpha^t(w))),\ \ w_1\in\cW,\end{array}
\label{hami}\end{equation}
or equivalently,
\beq\frac{\d}{\d t}\alpha^t(w)=\i\nabla G(\alpha^t(w)).\label{hamio}\eeq
In this subsection we will construct such Hamiltonians.
(Note, however, that in general
these Hamiltonians will not be everywhere defined).
First we define Hamiltonians of $\alpha$ in two special cases.
If $z\in \cW+h^{1/2}\cW$, then we define $\Dom\,G_\rI:=\Dom\,h^{1/2}$ and
for $w\in\Dom\,G_\rI$ we set
\[G_\rI(w):=\frac{1}{2}
\left((w|hw)+(z|w)+(w|z)\right).\]
We will say that $G_\rI$ is the Hamiltonian of $\alpha$ of the first kind.
If $z\in h^{1/2}\cW+h\cW$, then we define
$\Dom\,G_\rII:=\{w\in\cW\ :\ h^{1/2}w+h^{-1/2}z\in\cW\}$, and for
$w\in\Dom\,G_\rII$ we set
\[G_\rII(w):=\frac{1}{2}
\|h^{1/2}w+h^{-1/2}z\|^2.\]
We will say that $G_\rII$ is the Hamiltonian of $\alpha$ of the second kind.
Clearly, both $G_\rI$ and $G_\rII$ are well defined iff $z\in
h^{1/2}\cW$, and then
\[G_\rII=G_\rI+\frac12(z|h^{-1}z).\]
If $z$ is any functional satisfying (\ref{wqw}) we split the dynamics
$\alpha$ into two parts $\alpha_1$ and $\alpha_2$, as explained at
the end of Subsection \ref{s3.1}. We can then define the Hamiltonian
$G_{1,\rI}$ for the dynamics $\alpha_1$ and the Hamiltonian
$G_{2,\rII}$ for the dynamics $\alpha_2$. We will say that a function
$G$ is a Hamiltonian of $\alpha$ iff its domain equals $\Dom\,
G_{1,\rI}\oplus
\Dom\, G_{2,\rII}$ and it is equal to
\[G(w_1,w_2):=G_{1,\rI}(w_1)+G_{2,\rII}(w_2)+c,\]
where $c\in\rr$ is arbitrary.
%The name ``a Hamiltonian for $\alpha$''
% will be justified by the theorem below.
Note that, in general, $(\cW+h^{1/2}\cW)\cup( h^{1/2}\cW+h\cW)$ is
strictly smaller than $\cW+h\cW$. Therefore, for some $z$,
the dynamics $\alpha$ is well
defined but neither the Hamiltonian of the first kind $G_\rI$
nor the Hamiltonian of the second
kind $G_\rII$ are well defined.
\bet Let $z\in
\cW+h\cW$ and let $G$ be a Hamiltonian for $\alpha^t$. Then the
following statements are true:
\ben\item The function $G$ is differentiable at $w\in\cW$ iff $h w
+z$ belongs to $\cW$. We then have
\[\nabla G(w)=hw+z.\]
The dynamics $t\mapsto \alpha^t(w)$
is differentiable wrt $t$ iff $h\alpha^t(w)+z\in\cW$. We
have
\[
\frac{\d}{\d t}\alpha^t(w)=\i(h\alpha^t(w)+z),\]
which can be written in the form (\ref{hamio}).
Note also that $\alpha^t$ leaves $\Dom\,G$ invariant and $G$ is
constant along the trajectories.
\item $0$ belongs to $\Dom\,G$ iff
$z\in\cW+h^{1/2}\cW$. We have $G=G_\rI$ iff $G_\rI(0)=0$.
\item
$G$ is bounded from below iff
$z\in h^{1/2}\cW+h\cW$. We have $G=G_\rII$ iff $\inf G=0$.
\item
$G$ has a minimum iff $z\in h\cW$. This minimum is at $-h^{-1}z$, and
then \[G_\rII(w)=\frac12\left(w+h^{-1}z|h(w+h^{-1}z)\right).\]
\een\eet
\subsection{Quantum dynamics}
\label{s3.2}
Let $h$, $z$ be as above.
Assume (\ref{wqw}), that is $z\in\cW+h\cW$. Define the family of
unitary operators on $\Gamma_\s(\cW)$
\[V(t):=\Gamma(\e^{\i
th})\exp\left((1-\e^{-\i th})h^{-1}z(a^*)
-(1-\e^{\i t\bar h})\bar h^{-1}\bar z(a)\right).\]
Define for $B\in\Gamma_\s(\cW)$
\beq\begin{array}{rl}
\beta^t\left(B\right):=V(t)BV(t)^*.\end{array}\end{equation}
It is easy to check that $\beta$ is a 1-parameter
group of $*$-automorphisms of $B(\Gamma_\s(\cW))$ continuous in the
strong operator topology.
In order to make the relationship with the classical dynamics
more clear, one can note that
\[\beta^t(w(a^*))=((\e^{\i th}-1)h^{-1}z|w)+(\e^{\i th}w)(a^*).\]
Let $\cW=\cW_1\oplus \cW_2$, as in Subsection \ref{s3.1}. Then
we have the identification
$\Gamma_\s(\cW)=\Gamma_\s(\cW_1)\otimes\Gamma_\s(\cW_2)$.
The dynamics $\beta$ factorizes
\beq\beta^t(B_1\otimes B_2)=\beta_1^t(B_1)\otimes\beta_2^t(B_2),\ \
B_1\in B(\Gamma_\s(\cW_1),\ B_2\in B(\Gamma_\s(\cW_2).\label{facto}\eeq
%\[\beta^t\left(\e^{\i(w(a^*)+\bar w(a))}\right):=\exp\left(\i2\Re(w|(\e^{\i
%th}-1)h^{-1}z)\right)\e^{\i(\e^{\i th}w(a^*)+\e^{-\i t\bar h}\bar w(a))}.\]
\subsection{Quantum Hamiltonian}
We say that a self-adjoint operator $H$ is a Hamiltonian of the
dynamics $\beta$ iff
\beq\beta^t(B)=\e^{\i tH}B\e^{-\i tH}.\label{hav}\eeq
Clearly, (\ref{hav}) defines $H$ only up to an additive constant.
For $z\in\cW+h^{1/2}\cW$, define
\[U_\rI(t):=
\e^{\i \Im(h^{-1}z|\e^{\i th}h^{-1}z)-\i t(z|h^{-1}z)}V(t).\]
For $z\in h^{1/2}\cW+h\cW$, define
\[U_\rII(t):=
\e^{\i \Im(h^{-1}z|\e^{\i th}h^{-1}z)}V(t).\]
We easily check that both $U_\rI(t)$ and $U_\rII(t)$ are 1-parameter
strongly continuous unitary groups. Therefore, there exist
self-adjoint operators $H_\rI$ and $H_\rII$ such that
\[U_\rI(t)=\e^{\i tH_\rI},\ \ \ \ U_\rII(t)=\e^{\i tH_\rII}.\]
Clearly, both $H_\rI$ and $H_\rII$
are Hamiltonians of $\beta$.
They are both well defined iff $z\in
h^{1/2}\cW$, and then
\[H_\rII=H_\rI+(z|h^{-1}z).\]
A Hamiltonian of $\beta$ will be called a van Hove Hamiltonian.
The operator $H_\rI$ will be called the van Hove Hamiltonian of the
first kind and $H_\rII$ will be called the van Hove Hamiltonian of the
second kind.
\bet Let $z\in
\cW+h\cW$. Then the
following statements are true:
\ben
\item There exist Hamiltonians of $\beta$.
\item Let $H$ be a
Hamiltonian of $\beta$.
$\Omega$ belongs to $\Dom |H|^{1/2}$ (the form domain of
$H$) iff
$ z\in\cW+ h^{1/2}\cW$.
Under this condition $H=H_\rI$ iff $(\Omega|H\Omega)=0$.
\item The operator $H$ is bounded from below iff
$ z\in h^{1/2}\cW+ h\cW$.
Under this condition $H=H_\rII$ iff $\inf H=0$.
\item The operator $H$ has a ground state ($\inf H$ is an eigenvalue
of $H$, where $\inf H$ denotes the infimum of the spectrum of $H$) iff
$ z\in h\cW$.
Then we can define the ``dressing operator''
\[U:=\exp\left(-h^{-1}z(a^*)+\bar h^{-1}\bar z(a)\right),\]
and
\beq H_\rII= U\d\Gamma(h)U^*.\label{ham1}\end{equation}
\een\eet
\noindent
{\bf Proof of (1).}
We split the dynamics $\beta$ as in (\ref{facto}) with the projections
$p_1$, $p_2$ given by (\ref{facto1}). Then we
can define the Hamiltonian $H_{1,\rI}$ for $\beta_1$ and the
Hamiltonian $H_{2,\rII}$ for $\beta_2$. We
set\[H:=H_{1,\rI}\otimes1+1\otimes H_{2,\rII}.\]
\qed
Easy calculations show that, at least formally,
\beq H_\rI=\d\Gamma(h)+z(a^*)+\bar z(a),\label{hov1}\end{equation}
\beq H_\rII=\d\Gamma(h)+z(a^*)+\bar z(a)+(z|h^{-1}z).\label{hov2}\end{equation}
Below we will make these formulas precise.
\ber
By the spectral theorem, one can find a measure space $(K,\d k)$ such that
$\cW$ is isomorphic to $L^2(K,\d k)$ and
$h$ is a multiplication operator by a measurable
function $K\ni k\mapsto h(k)$.
Then we can
introduce
$K\ni
k\mapsto a^*(k), a(k)$ and write
$\int z(k) a^*(k)\d k$, $\int \bar z(k)a(k)$, $\int h(k)a^*(k)a(k)\d
k$ instead of $z(a^*)$, $\bar z(a)$ and $\d\Gamma(h)$.
We used this notation in the introduction.
For example, Condition (\ref{con0}) of the introduction,
\[\begin{array}{l}
\int_{h(k)<1}|z(k)|^2\d k+\int_{h(k)\geq 1}\frac{|z(k)|^2}{h(k)^2}\d
k<\infty,\end{array}\]
corresponds to the condition
$ z\in\cW+h\cW$.
The notation involving the operator valued measures
$a^*(k)$ and $a(k)$ is very common and often convenient,
but it depends on a
non-canonical
choice of the measure space
$K$, and therefore we do not use it. \eer
\subsection{3 types of the ultraviolet problem}
\label{s3.3}
In this subsection we describe van Hove Hamiltonians without the
infrared problem.
\bet Assume that $h$ is a self-adjoint (possibly unbounded) operator
bounded from below by a positive constant. Then the dressing operator
$U$ and the van Hove Hamiltonian of the second kind $H_\rII$
are well
defined. $H_\rII$
possesses a unique ground state at $0$. Moreover, we
can distinguish 3 cases:
\ben
\item Let
$z\in \cW$.\\ Then
$z(a^*)+\bar z(a)$ is a $\d\Gamma(h)$-bounded operator
with the infinitesimal bound. $H_\rI$ and $H_\rII$ are self-adjoint on
$\Dom\,\d\Gamma(h)$ and can be defined by the formulas (\ref{hov1}),
(\ref{hov2}) and
by the
Kato-Rellich theorem.
\item Let
$z\in h^{1/2}\cW\backslash\cW$.\\
Then $z(a^*)+\bar z(a)$ is not an operator. Instead of
$z(a^*)+\bar z(a)$ we should consider the form with the domain
$\Gammal_\s(\Dom\, z)$ equal to $z(a^{*\f})+\bar z(a)$. This form is
$\d\Gamma(h)$-form bounded
with the infinitesimal bound. The operators $H_\rI$ and $H_\rII$ are
bounded from below and their form domains equal
$\Dom\,\d\Gamma(h)^{1/2}$. They can be defined by the formulas
(\ref{hov1}), (\ref{hov2}) and by the
KLMN theorem.
\item Let
$z\in h\cW\backslash h^{1/2}\cW$.\\
Then the form
$z(a^{*\f})+\bar z(a)$ is not $\d\Gamma(h)$-form bounded. $H_\rI$ is
not defined.
\een
\label{ultra}\eet
\subsection{3 types of the infrared problem}
\label{s3.4}
In this subsection we describe van Hove Hamiltonians without the
ultraviolet problem.
\bet Assume that $h$ is a bounded
positive operator. Then the formula (\ref{hov1}) defines
the operator $H_\rI$ as an
essentially self-adjoint operator on $\Dom\,\d\Gamma(h{+}1)$ by
Nelson's commutator theorem.
Moreover, we can distinguish the
following three cases:
\ben \item Let
$z\in h\cW$\\ Then
$z(a^*)+\bar z(a)$ is $\d\Gamma(h)$-bounded
with the infinitesimal bound.
The operators $H_\rI$ and $H_\rII$ are self-adjoint
on $\Dom\,\d\Gamma(h)$ and they can be defined by the
formulas (\ref{hov1}), (\ref{hov2})
and the Kato-Rellich theorem.
They have ground states and
the dressing operator $U$ is
well defined.
\item Let
$z\in h^{1/2}\cW\backslash h\cW$\\ Then
$z(a^*)+\bar z(a)$ is $\d\Gamma(h)$-bounded
with the infinitesimal bound. Again,
the operators $H_\rI$ and $H_\rII$ are self-adjoint
on $\Dom\,\d\Gamma(h)$ and they can be defined by the
formulas (\ref{hov1}), (\ref{hov2})
and the Kato-Rellich theorem.
They are bounded from below but have no
ground state and the dressing operator $U$ is not defined.
\item Let
$z\in\cW\backslash h^{1/2}\cW$.\\ Then
$z(a^*)+\bar z(a)$ is not $\d\Gamma(h)$-form bounded.
$H_\rI$ is not bounded from
below and the operator $H_\rII$ is not defined at all.
\een
\label{infi2} \eet
In the following subsections we will show various elements of the
above theorems.
\subsection{Relative form boundedness of field operators}
\label{s3.5}
%Recall that given an operator $h$ on $\cW$ we have a natural operator on $\cY$
%defined by $(h+\bar h)\Big|_\cY$. We will use the same letter $h$ to denote
%both operators.
\bep Let $h$ be a positive operator on $\cW$ and $z\in h^{1/2}\cW$.
Then
\ben\item
\[\|\bar z(a)\Psi\|^2\leq(z|h^{-1}z)(\Psi|\d\Gamma(h)\Psi).\]
\item $z(a^{*\f})+\bar z(a)$ is form bounded wrt $\d\Gamma(h)$ with the
infinitesimal bound. More precisely,
for any $\epsilon>0$, we have
\[|(\Psi|(z(a^{*\f})+\bar z(a))\Psi)|\leq
\epsilon(z|h^{-1}z)(\Psi|\d\Gamma(h)\Psi)+\epsilon^{-1}\|\Psi\|^2.
\]
\een
\label{opo}\eep
\proof
(1) If $z$ is an antilinear functional on $\cW$ with the domain $\Dom
z$, then
\[\Dom\, z\times \Dom\, z\ni(w_1,w_2)\mapsto (w_1|z)(z|w_2)\]
defines a sesquilinear form, that we will denote by $|z)(z|$. Note
that the following inequality is true:
\beq
|z)(z|\leq(z|h^{-1}z)h,\label{ineq}\end{equation}
Clearly,
\[\|\bar
z(a)\Psi\|^2=\left(\Psi|\d\Gamma\left(|z)(z|\right)\Psi\right).\]
(\ref{ineq}) implies
\[\d\Gamma\left(|z)(z|\right)\leq(z|h^{-1}z)\d\Gamma(h).\]
(2)
\[\begin{array}{rl}
\pm(\Psi|(z(a^{*\f})+\bar z(a))\Psi)&=\pm 2\Re\Big(\Psi|\bar z(a)\Psi)
\\[3mm]
&\leq2\|\Psi\|\|\bar z(a)\Psi\|\leq
\epsilon^{-1}\|\Psi\|^2+\epsilon\|\bar z(a)\Psi\|^2.\end{array}\]
\qed
\begin{corollary} If $z\in h^{1/2}\cW$, then both (\ref{hov1}) and
(\ref{hov2}) (with $z(a^*)$ replaced by $z(a^{*\f})$)
define by the KLMN theorem the self-adjoint
operators $H_\rI$ and
$H_\rII$ with the form domains $\Dom\, \d\Gamma(h)^{1/2}$.
\end{corollary}
\subsection{Relative boundedness of field operators}
\label{s3.6}
\bep Let $h$ be a positive operator on $\cW$ and $z\in h^{1/2}\cW\cap\cW$.
Then
\ben\item
\[\|
(z(a^*)+\bar z(a))\Psi\|^2\leq
4(z|h^{-1}z)(\Psi|\d\Gamma(h)\Psi)+2\|z\|^2\|\Psi\|^2.\]
\item
$z(a^*)+\bar z(a)$ is bounded wrt $\d\Gamma(h)$ with the
infinitesimal bound. More precisely,
for any $\epsilon>0$, we have
\[\|(z(a^*)+\bar z(a))\Psi\|^2\leq
2\epsilon(z|h^{-1}z)\|\d\Gamma(h)\Psi\|^2+
\left(2\epsilon^{-1}(z|h^{-1}z)+2\|z\|^2\right)\|\Psi\|^2
.\]
\een\eep
\proof
(2) follows immediately from (1). To see (1) we compute using
Proposition \ref{opo} (1):
\[\begin{array}{rl}
\|(z(a^*)+\bar z(a))\Psi\|^2&
\leq2\|z(a^*)\Psi\|^2+\|\bar z(a)\Psi\|^2\\[3mm]&
=4\|z(a^*)\Psi\|^2+2\|z\|^2\|\Psi\|^2\\[3mm]
&\leq4(z|h^{-1}z)(\Psi|\d\Gamma(h)\Psi)+2\|z\|^2\|\Psi\|^2.\end{array}\]
\qed
\begin{corollary} If $z\in \cW\cap h^{1/2}\cW$, then both (\ref{hov1}) and
(\ref{hov2}) define by the Kato-Rellich theorem the self-adjoint
operators $H_\rI$ and
$H_\rII$ with the domains $\Dom\, \d\Gamma(h)$.
\label{ada}\end{corollary}
\subsection{The infimum of van Hove Hamiltonians}
\label{s3.7}
\bep Assume $z\in \cW+ h^{1/2}\cW$. Then
the operator $H_\rI$
satisfies
\[\begin{array}{l}
\inf H_\rI=-(z|h^{-1}z),\\[3mm]
\sp H_\rI=-(z|h^{-1}z)+\sp\,\d\Gamma(h).\end{array}\]
\label{infi}\eep
\proof We drop $\rI$ from $H_\rI$. It is sufficient to assume that
\beq 0\ \hbox{
is not an isolated point in the spectrum of }\ h.\label{assa}\eeq
This in particular
implies that $\sp\,\d\Gamma(h)=[0,\infty[$. Thus we need to show that
\[\sp H=[-(z|h^{-1}z),\infty[.\]
\\
\noindent{\bf Step 1)}
Setting $\epsilon=\pm(z|h^{-1}z)^{-1}$ in
Proposition \ref{opo} (1) with we get
\beq \d\Gamma(h)+z(a^*)+\bar z(a)\geq -(z|h^{-1}z).\label{qoq}\end{equation}
\noindent {\bf Step 2)}
For any $n$ define $\cW^{n}:=1_{[\frac{1}{n},\infty[}(h)\cW$, $h^{n}:=
1_{[\frac{1}{n},\infty[}(h)h$, $z^{n}:=1_{[\frac{1}{n},\infty[}(h)z$,
and the operators on $\Gamma_\s(\cW^{n})$,
$H^{n}:=\d\Gamma(h^{n})+ z^{n}(a^*)+\bar{z^{n}}(a)$ and $U^{n}:=
\exp(-h^{-1}z^{n}(a^*)+\bar h^{-1}\bar z^{n}(a))$.
Then $U^{n*}H^{n}U^{n}=\d\Gamma(h^{n})-( z^{n} |h^{-1}z^{n}).$
Clearly, $\inf\d\Gamma(h^{n})=0$, hence
\beq\begin{array}{l}
\inf H^{n}=
-( z^{n} |h^{-1}z^{n}),\\[3mm]
\sp H^n=-(z^n|h^{-1}z^n)+\sp\d\Gamma(h^n)
.\end{array}\label{spec1}\end{equation}
\noindent {\bf Step 3)}
Likewise, define
$\cW_{n}:=1_{[0,\frac{1}{n}[}(h)\cW$, $h_{n}:=
1_{[0,\frac{1}{n}[}(h)h$, $z_{n}:=1_{[0,\frac{1}{n}[}(h)z$,
and the operator on $\Gamma_\s(\cW_{n})$,
$H_{n}:=\d\Gamma(h_{n})+ z_{n}(a^*)+\bar z_{n}(a)$.
We have $\Omega\in\Dom H_{n}$ and $(\Omega|H_{n}\Omega)=0$.
Hence,
\beq\inf H_{n}\leq0.\label{spp2}\end{equation}
Arguing as in Step 1) we get
\beq H_n\geq-(z_n|h^{-1}z_n).\label{qoqo}\end{equation}
\noindent {\bf Step 4)}
$\Gamma_\s(\cW)$ can be identified with $\Gamma_\s(\cW_{n})\otimes
\Gamma_\s(\cW^{n})$ and \beq
H=H_{n}\otimes1+1\otimes H^{n}.\label{spe1}\end{equation}
Therefore,
\beq\begin{array}{l}
\inf H=\inf H_{n}+\inf H^{n},\\[3mm]
\sp H=\sp H_n+\sp H^n.\end{array}\label{spp3}\end{equation}
\noindent {\bf Step 5)}
If $t>0$, then by (\ref{assa}), we will find a sequence
$t_n\in\sp\,\d\Gamma(h_n)$ such that $t_n\to t$. By Step 4), $\inf
H_n+t_n\in\sp\, H$. But by Step 3), $\inf H_n\to0$. By the closedness of
$\sp H$, $t\in\sp\, H$.
\qed
\subsection{Essential self-adjointness of van Hove Hamiltonians}
\label{s3.8}
\bep Suppose that $z\in\cW $.
Then $H_\rI$ is essentially self-adjoint on
$\Dom\,\d\Gamma(1+h)$.
\eep
\proof
First assume that $h$ is bounded.
We apply Nelson's commutator theorem with the comparison
operator $B:=1+\d\Gamma(1+h)$ \cite{RS2}. In fact,
\[\|(z(a^*)+\bar z(a))\Psi\|\leq c\|N\Psi\|\leq c\|B\Psi)\|,\]
\[\|\d\Gamma(h)\Psi\|\leq\| B\Psi\|.\]
Moreover,
\[[B,(z(a^*)+\bar z(a))]=(1+h)z(a^*)+(1+\bar h)\bar z(a).\]
Hence
\[|(\Psi|[B,(z(a^*)+\bar z(a))]\Psi)|
\leq(\Psi|B\Psi).\]
Hence $H$ is essentially self-adjoint on $\Dom B$.
Next consider an arbitrary $h$.
As described in Subsection \ref{s3.2}, we can split
$\cW=\cW_1\oplus\cW_2$, where $\cW_1=\Ran1_{[0,1]}(h)$.
We can define the operator $H_\rI$ on
$\Gammal_\s(\cW)=\Gammal_\s(\cW_1)\otimes \Gammal_\s(\cW_2)$
and it splits as
\[H_\rI=H_{\rI,1}\otimes 1+1\otimes H_{\rI,2},\]
with
\[H_{\rI,i}=\d\Gamma(h_i)+z_i(a^*)+\bar z_i(a).\]
We proved above that $H_{\rI,1}$ is essentially self-adjoint on
$\Dom\, \d\Gamma(h_1+1)$.
By corollary \ref{ada}, $H_{\rI,2}$ is self-adjoint on
$\Dom\, \d\Gamma(h_2)$. This implies that $H_\rI$ is essentially
self-adjoint on $\Dom\, \d\Gamma(h_1+1)\otimesal
\Dom\, \d\Gamma(h_1)$, which is dense in $\Dom\, \d\Gamma(h+1)$.
($\otimesal$ denotes the algebraic tensor product).
\qed
\subsection{Absence of a ground state}
\label{s3.9}
Let us recall the following well known result about coherent states:
\bet
Suppose that $\widetilde\cW$ is a dense subspace of $\cW$ and $f$ is an
antilinear functional on $\widetilde\cW$. Let
$\Psi\in\Gamma_\s(\cW)$, for any $w\in\widetilde\cW$,
$\Psi\in\Dom \bar h(a)$ and
\[\bar h(a)\Psi=(h|f)\Psi.\]
Then the following is true:
\ben\item If $f\in\cW$, then $\Psi$ is proportional to $\exp(f(a^*)-\bar
f(a))
\Omega$.
\item If $f\not\in\cW$, then $\Psi=0$.
\een\label{coho}\eet
\proof
By induction we show that for $w_1,\dots,w_n\in\widetilde\cW$,
$\bar w_{n-1}(a)\cdots \bar w_1(a)\Psi\in\Dom \bar w_n(a)$ and
\[\bar w_{n}(a)\cdots \bar w_1(a)\Psi=(w_1|f)\cdots(w_n|f)\Psi.\]
This implies
\beq(w_{1}(a^*)\cdots
w_n(a^*)\Omega|\Psi)=(w_1|f)\cdots(w_n|f)(\Omega|\Psi) .\label{asas}
\end{equation}
In particular,
\[(w|\Psi)=(w(a^*)\Omega|\Psi)=(w|f)(\Omega|\Psi),\ \ w\in\widetilde\cW.\]
Using the fact that $\widetilde\cW$ is dense in $\cW$ we see that $(\Omega|\Psi)f$
is a bounded functional on $\cW$, hence it belongs to $\cW$. Thus
either $f\in\cW$ or $(\Omega|\Psi)=0$. In the latter case, (\ref{asas})
implies that $\Psi=0$.
\qed
\bep
$H_\rI$ has a ground state iff $z\in h\cW$.
\eep
\proof
If $z\in h\cW$, then we can define the dressing transformation and
$U\Omega$ is a ground state of $H_\rI$.
Suppose that $z\not\in h\cW$.
We use the notation of the proof of Proposition \ref{infi}.
Let $\Psi$ be a ground state of $H$. Then it is also a ground state of
$H_{n}\otimes1$ and of $1\otimes H^{n}$.
Being a ground state of $1\otimes H^{n}$, it must be equal to
$\Psi_{n}\otimes U^{n}\Omega$.
Therefore, for $w\in\widetilde\cW:=\bigcup\limits_{n=1}^\infty \cW^{n}$
\[\bar w(a)\Psi=(w|h^{-1}z)\Psi.\]
But $\widetilde\cW$ is dense in $\cW$.
By Theorem \ref{coho}, this means that either
$h^{-1}z\in\cW$ or $\Psi=0$. \qed
\section{Scattering theory}
\label{s4}
\subsection{The usual formalism}
\label{s4.1}
The most common setup for scattering theory starts with a pair of
self-adjoint operators $H_0$ and $H$. The wave
operators $\Omega^\pm$ are defined
by the formulas
\beq \Omega^\pm:=\slim_{t\to\pm\infty}\e^{\i tH}\e^{-\i
tH_0}.\label{wave}\end{equation}
Note that $\Omega^\pm$ are automatically isometric and
\beq \Omega^\pm H_0=H\Omega^\pm.\label{twine}\end{equation}
The scattering operator is defined as
\beq S:=\Omega^{+*}\Omega^-.\label{sca}\end{equation}
It satisfies
\beq
S=\wlim_{t_+,t_-\to\infty}\e^{\i t_+H_0}
\e^{-\i (t_++t_-)H}\e^{\i t_-H_0}\label{sca1}\end{equation}
and commutes with $H_0$. If we have
\[\Ran\Omega^+=\Ran\Omega^-,\]
then $S$ is unitary.
\subsection{Wave operators defined by the Abelian limit---general
formalism}
\label{s4.2}
The
limits in (\ref{wave}) often do not exists. This happens, for instance,
if
$H_0$ has an eigenvector, which is not an eigenvector of $H$.
This is the case of many models of quantum field theory,
whose free vacuum (the unique eigenvector of $H_0$)
is often different from the interacting
vacuum (the unique eigenvector of $H$).
Nevertheless, sometimes even in this situation some kind of
a scattering theory can be developed. In this subsection we will
describe one of possible approaches to scattering theory,
which, as we will see, works in the case of van Hove Hamiltonians.
One can argue that this approach, or some its variation, is implicit in most
textbook presentations of QFT.
Again, we start with a pair of self-adjoint operators $H_0$ and
$H$. We suppose that
there exists the Abelian limit
\beq
\Omega_\ur^\pm:=\slim_{\epsilon\downarrow0}2\epsilon\int_0^\infty
\e^{\pm\i tH}\e^{\mp\i
tH_0}\e^{-2\epsilon t}\d t.\label{wave1}\end{equation}
Note that
there is no guarantee that $\Omega_\ur^\pm$ are isometric. One knows only that
$\Omega_\ur^\pm$ are contractions. One can easily
see that $\Omega_\ur^\pm$ have the
intertwining property:
\beq \Omega_\ur^\pm H_0=H \Omega_\ur^\pm.\label{twine1}\end{equation}
We will call $\Omega_\ur^\pm$ the ``unrenormalized wave operators''.
Of course, it may happen that $\Omega_\ur^\pm=0$.
Define the ``renormalization of wave function operator''
\[Z^\pm:=\Omega_\ur^{\pm*}\Omega_\ur^\pm.\]
It is easy to see hat $Z^\pm$ commutes with $H_0$. Assume that $\Ker
Z^\pm=\{0\}$. Then we define the ``renormalized wave operators''
\[\Omega_\rn^\pm :=\Omega_\ur^\pm (Z^\pm)^{-1/2}.\]
Note that $\Omega_\rn^\pm$ are isometric and have the intertwining
property:
\beq \Omega_\rn^\pm H_0=H \Omega_\rn^\pm.\label{twine1a}\end{equation}
The unrenormalized scattering operator is defined as
\beq
S_\ur:=\Omega_\ur^{+*}\Omega_\ur^-.\label{haha1}\end{equation}
Note that it can be also obtained as the following weak limit:
\[S_\ur=\wlim_{\epsilon_-,\epsilon_+\downarrow0}
4\epsilon_-\epsilon_+\int_0^\infty\d t_-\int_0^\infty\d t_+
\e^{\i t_+H_0}\e^{-\i(t_-+t_+)H}\e^{\i
t_-H_0}\e^{-2(\epsilon_-+2\epsilon_+)t }.\]
$S_\ur$ is a contraction that commutes with $H_0$.
We can define
the renormalized scattering operator as
\[S_\rn:=(Z^+)^{-1/2}S_\ur(Z^-)^{-1/2}=\Omega_\rn^{+*}\Omega_\rn^-.\]
$S_\rn$ also commutes with $H_0$ and if
\[\Ran\Omega_\rn^+=\Ran\Omega_\rn^-,\]
then it is unitary.
We will see in the following subsection that van Hove Hamiltonians
provide an example where the formalism based on the
Abelian limit is applicable.
\subsection{Wave operators for the van Hove Hamiltonian}
\label{s4.3}
Let $z\in h^{1/2}\cW+h\cW$.
Let $H=H_\rII$ be the van Hove Hamiltonian of the second kind and
\[H_0:=\d\Gamma(h).\]
It is easy to see that
in the case of the van Hove Hamiltonian, the limit in (\ref{wave})
does not exist unless $H=H_0 $. Hence the construction of wave
operators cannot be based on the approach described in Subsection \ref{s4.1}.
We will show, however, that the formalism of Subsection \ref{s4.2} works
for van Hove Hamiltonians if $h$ has an absolutely continuous
spectrum. It will turn out that the two renormalized wave
operator coincide with one another and are equal to the dressing
operator $U$. The operators $Z_\pm=:Z$ coincide and are just
constants. The renormalized scattering operator equals identity.
If $z\not\in h
\cW$, then $Z=0$. This is one of the manifestations of the infrared
problem.
All these statements are described in the following theorem.
\bet
Suppose that $h$ has absolutely continuous spectrum and
$z\in h^{1/2}\cW+h\cW$.
Then $\Omega_\ur^\pm$ exists and
\[\Omega_\ur^\pm=Z^{1/2}U,\ \ \ S_\ur=Z,\]
where
\[Z=\e^{-\|h^{-1}z\|^2}.\]
$Z\neq0$ iff $z\in h\cW$, and then
\[Z=(\Omega|U\Omega)^2.\]
We can then define the renormalized operators, which are equal
\[\Omega_\rn^\pm=U,\ \ \ S_\rn=1.\]
\eet
\proof
Let us assume that $z\in h\cW$. (The general case can be
obtained by the limiting argument). Set $g:=h^{-1}z$.
\beq\begin{array}{rl}
\e^{\i tH}\e^{-\i tH_0}&=U\e^{\i tH_0}U^*\e^{-\i tH_0}\\[3mm]
&=U
\exp(-\e^{\i th}g(a^*)+\e^{-\i t\bar h} \bar g(a))\\[3mm]
&=U\e^{-\frac12\|g\|^2}
\exp(-\e^{\i th}g(a^*))\exp(\e^{-\i t\bar h} g(a)).
\end{array}\label{rtr1}\end{equation}
Let $P_m$ be the projection onto the states with $\leq m$ particles.
Suppose that $P_m\Psi=\Psi$.
Now
\beq\begin{array}{l}
2\epsilon
\int_0^\infty
\e^{-2\epsilon t}\exp(-\e^{\i th}g(a^*))\exp(\e^{-\i t\bar h} \bar g(a))\Psi\d
t=\Psi\\[3mm]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+2\epsilon \int_0^\infty
\e^{-2\epsilon t}\bigl(\exp(-\e^{\i th}g(a^*))-1\bigr)\Psi\d
t\\[3mm]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+2\epsilon \int_0^\infty
\e^{-2\epsilon t}\exp(-\e^{\i th}g(a^*))\bigl(\exp(\e^{-\i t\bar h}
\bar g(a))-1\bigr)\Psi
\d t.\end{array}\label{dfd}\end{equation}
The norm of the third term can be estimated from above by
\[2\epsilon
\int_0^\infty
\e^{-2\epsilon t}\|\exp(-\e^{\i th}g(a^*))P_m\|
\|\bigl(\exp(\e^{-\i t\bar h}
\bar g(a))-1\bigr)\Psi\|
\d t.\]
Clearly, $\|\exp(-\e^{\i th}g(a^*))P_m\|$ is bounded uniformly in
time. Besides,
$\|\bigl(\exp(\e^{-\i t\bar h}
\bar g(a))-1\bigr)\Psi\|
\to0$, by the absolute continuity of $h$ and the
Riemann-Lebesgue lemma. Therefore, the third term of (\ref{dfd}) goes
to zero.
The second term equals
\beq\begin{array}{l}
\sum\limits_{n=1}^\infty2\epsilon\int\limits_0^\infty
\e^{-2\epsilon t}\frac{(-1)^n}{n!}
(\e^{\i th} g)^{\otimes n}(a^*)\Psi\d t\\[5mm]
=\sum\limits_{n=1}^\infty
\frac{(-1)^n}{n!}2\epsilon\bigl((2\epsilon-\i\d\Gamma(h))^{-1}
g^{\otimes n}\bigr)(a^*)\Psi.\end{array}\label{sgs}\end{equation}
Note that the $n$th term on the right goes to zero as
$\epsilon\searrow0$ and can be estimated by
\beq
\frac{\sqrt{(m+1)\cdots(m+n)}}{n!}\|g\|^n\|\Psi\|.
\label{sere}\end{equation}
The series with elements (\ref{sere}) is convergent. Hence by the dominated
convergence theorem, (\ref{sgs}) goes to zero
as
$\epsilon\searrow0$.
This shows that, for a finite particle $\Psi$, the left hand side of
(\ref{dfd}) goes to $\Psi$. By density, we can extend
this to all $\Psi\in\Gamma_\s(\cW)$. \qed
\comment{
\proof Let us assume that $z\in h\cW$. (The general case can be
obtained by the limiting argument).
\beq\begin{array}{rl}
\e^{\i tH}\e^{-\i tH_0}&=U\e^{\i tH_0}U^*\e^{-\i tH_0}\\[3mm]
&=UW(\i h^{-1}\e^{\i th}z+\coc)\\[3mm]
&=U\e^{\frac12\|h^{-1}z\|^2}
\exp(-\e^{\i th}h^{-1}z(a^*))\exp(\e^{-\i t\bar h}\bar h^{-1}\bar z(a)).
\end{array}\label{rtr1}\end{equation}
Clearly, $\wlim_{t\to\infty}\e^{\i th}h^{-1}z=0$. Hence the right hand
side of (\ref{rtr1}) goes to $U\e^{\frac12\|h^{-1}z\|^2}$.
Similarly,
\beq\begin{array}{rl}
\e^{-\i tH_0}\e^{\i 2tH}\e^{-\i tH_0}&=\e^{-\i tH_0}
U\e^{\i 2tH_0}U^*\e^{-\i tH_0}\\[3mm]
&=W(-\i h^{-1}\e^{-\i th}z+\coc)
W(\i h^{-1}\e^{\i th}z+\coc)\\[3mm]
&=\e^{\i\Im(h^{-1}z|\e^{\i2 th}h^{-1}z)}
W(\i h^{-1}(\e^{\i th}-\e^{-\i th})z+\coc)\\[3mm]
&=\e^{\i\Im(h^{-1}z|\e^{\i2 th}h^{-1}z)}
\e^{\frac12\|(\e^{\i th}-\e^{-\i th})z\|^2}\\[3mm]
&\times\exp\left(\i h^{-1}(\e^{\i th}-\e^{-\i th})z(a^*)\right)
\exp\left(-\i \bar h^{-1}(\e^{-\i t\bar
h}-\e^{\i t\bar h}) \bar z(a)\right)
.
\end{array}\label{rtr2}\end{equation}
The last line of (\ref{rtr2}) converges to 1.
\qed}
\subsection{Asymptotic fields---general formalism}
\label{s4.4}
There exists an alternative approach to scattering in quantum field
theory.
Instead of starting from wave operators, one looks at
the limits of certain observables in the interaction picture. There are
various forms of this approach, some of them go under the name of
the LSZ formalism, see e.g. \cite{Schwa}.
Let us present the abstract framework of one of the versions of
this approach developed in
\cite{HK} and used eg. in
\cite{DG2}.
Suppose that $H$ is a self-adjoint operator on the Fock space
$\Gamma_\s(\cW)$ and $h$ is a self-adjoint operator on $\cW$. Assume
that for some subspace $\cW_1\subset\cW$ there exists
\[\slim_{t\to\pm\infty}\e^{\i tH}W(\e^{-\i th}w)\e^{-\i
tH}=:W^\pm(w),\ \ \ w\in\cW_1.\]
Then \beq
\cW_1\ni w\mapsto W^\pm(w)\in U(\Gamma_\s(\cW))\label{ccr}\end{equation}
are two representations of Canonical Commutation Relations (CCR), that
means
\[W^\pm(w_1)W^\pm(w_2)=\e^{-\frac{\i}{2}\Im(w_1|w_2)}W^\pm(w_1+w_2).\]
Moreover, they satisfy
\[\e^{\i tH}W^\pm(w)\e^{-\i tH}=W^\pm(\e^{\i th}w).\]
Suppose that the representations (\ref{ccr}) are unitarily equivalent
to the Fock representation, which means that
there exist unitary operators $\Omega^\pm$ such that
\beq W^\pm(w)=\Omega^\pm W(w)\Omega^{\pm*}.\end{equation}
Then the operators $\Omega^\pm$ are
defined up to a phase factor. They are called wave operators.
The scattering operator is defined as
$S:=\Omega^{+*}\Omega^-$. Again, the scattering operator is defined up
to a phase factor.
Suppose that both the formalism of Subsection
\ref{s4.2} and of Subsection \ref{s4.4} apply. One can ask whether the
renormalized wave operators $\Omega_\rn^\pm$, defined as in Subsection
\ref{s4.2}, and the wave operator $\Omega^\pm$ defined in
this section coincide up to a phase factor. In general, there seems to
be no guarantee for this to hold. Nevertheless we will see that this
is true in the case of van Hove Hamiltonians.
\subsection{Asymptotic fields for van Hove Hamiltonians}
\label{s4.5}
The formalism of asymptotic fields works very well in the case of
van Hove Hamiltonians.
\bet Let $h$ have an absolutely
continuous spectrum, $0\leq \beta\leq1$ and
$ z\in h^{1-\beta}\cW+ h\cW$. Let $w\in\Dom\,h^{-\beta}$.
\ben\item
There
exists the norm limit
\[\lim_{|t|\to\infty}
\e^{\i tH}W(\e^{- ith} w)\e^{i tH}
=W(w)\e^{\i2\Re(w|h^{-1}z)}=:W^\as(w).\]
\item $\Dom\,h^{-\beta}\ni w\mapsto W^\as(w)$ is a regular
representation of CCR.
\item This representation is unitarily equivalent to the Fock
representation iff $z\in h\cW$, and
then
\[W^\as(w)=UW(w)U^*,\]
where $U$ is the dressing operator.
\een\eet
\proof
\[\e^{\i tH}W(\e^{- ith} w)\e^{i tH}
=W(w)\exp\left(\i2\Re(w|(1-\e^{-\i th})h^{-1}z)\right).\]
Now $h^{-1}z\in h^{-\beta}\cW+\cW$ and $w\in\Dom\,h^{-\beta}$. Hence
$\lim_{|t|\to\infty}
(w|\e^{-\i th}h^{-1}z)=0$ by the Riemann-Lebesgue lemma. This proves
(1).
\qed
\section{Examples}
\subsection{Harmonic oscillators}
In this section we describe van Hove Hamiltonians in a somewhat more
concrete setting, typical for physical applications. We will
restrict ourselves to the classical case, since it is parallel to the
quantum case.
We will describe a system of harmonic oscillators with a linear
perturbing potential.
Up to now, we assumed that our system is described by phase space
$\cW$. There was no need to introduce the configuration space.
For a system of harmonic oscillators it is however natural to start
from a configuration space, which will be described by a real Hilbert
space $\cX$
with the scalar product denoted by the dot. The preliminary phase
space is
$\cX\oplus\cX$. It has the structure of a symplectic space with the
symplectic form
\beq(x_1,\xi_1)\omega(x_2,\xi_2)=x_1\cdot\xi_2-x_2\cdot\xi_1.\label{sympo}\end{equation}
Note, however, that we will have to take a slightly different phase space.
Let $r$ denote a positive
operator on $\cX$ and $q$ is a linear functional on $\cX$ (possibly
unbounded and not everywhere defined). A system of harmonic
oscillators with a linear perturbing potential is described by
the (classical) Hamiltonian
\[G(x,\xi)=\frac{1}{2}|r x|^2+\frac{1}{2}|\xi|^2+q\cdot x,\]
defined for $x\ni\Dom\, r\cap\Dom\, q$, $\xi\in\cX$. It is easy to see that
$\cX\oplus\cX$ is not an appropriate space for the Hamiltonian $G$.
It is natural to replace it by the space
$\cW:=r^{-1/2}\cX\oplus r^{1/2}\cX$. We keep the symplectic form (\ref{sympo})
We equip $\cW$ with the complex
structure
\[\i(x,\xi):=(-r^{-1} \xi,r x).\]
We can view $\cW$ as a complex Hilbert space:
\[\begin{array}{r}(w_1|w_2)=
r^{1/2}x_1\cdot r^{1/2}x_2+r^{-1/2}\xi_1\cdot r^{-1/2}\xi_2+\i
x_1\cdot\xi_2-\i x_2\cdot\xi_1,\\[3mm]
w_1=(x_1,\xi_1),\ \ w_2=(x_2,\xi_2).\end{array}\]
Note that the symplectic form (\ref{sympo}) is the imaginary part of
the scalar product.
Introduce $z:=(r^{-1/2}q,0)$ and a positive self-adjoint operator $h$
on $\cW$ defined by $h:=r\oplus r$. Then we can rewrite $G$ as
\[G(w)=
\frac12\bigl((w|hw)
+\left(z|w\right)
+\left(w|z\right)\bigr).\]
Note that the infrared and ultraviolet conditions expressed in
terms of $q$ instead of $z$ have their power shifted by $1/2$. More
precisely,
$z\in h^\alpha\cW$ iff $q\in r^{1/2+\alpha} \cX$.
\subsection{Scalar massless field theory}
Suppose that $\cX=L^2(\rr^d)$, $r=|\i\nabla|$.
Then the Hamiltonian $\frac{1}{2}|r x|^2+\frac{1}{2}|\xi|^2$
describes the so-called scalar massless
field theory. After taking the Fourier transformation, the operator $r$
becomes the multiplication by $|\xi|$, where $\xi$ is the momentum variable.
Suppose that we add a linear perturbation given by
$q\in\cS(\rr^d)$. After taking the Fourier transformation we get $\hat
q\in\cS(\rr^d)$ and we see that the ultraviolet problem is absent. The
infrared problem will depend on whether $\hat q(0)$ equals zero or
not. $\hat q(0)$ equals the integral of $q$ over the whole
configuration space. Since in some physical examples $q$ can be
interpreted as the density of the charge, we will call $\hat q(0)$ the
total charge. Note that if $\hat q(0)=0$, then $|\hat
q(\xi)|=O(|\xi|)$ around zero.
Concerning the type of the infrared behavior, we easily get the
following table (the number in the round brackets corresponds to the
part of Theorem \ref{infi2}):
\medskip
\begin{tabular}{c|c|c}
\underline{Dimension of configuration space}& \underline{Nonzero total
charge}
& \underline{Zero total charge}\\[3mm]
$d=1$&Hamiltonian undefined&(2)\\[3mm]
$d=2$&(3)&(1)\\[3mm]
$d=3$&(2)&(1)\\[3mm]
$d\geq4$&(1)&(1)
\end{tabular}
\ber
As we see from the table, in dimension 3, in the nonzero charge case
we get the infrared behavior of type (2). Thus the Hamiltonian is
bounded from below, but the ground state is absent. This is the type
of the infrared problem widely discussed in the literature \cite{Ki}.
Some authors say, however, that the type (2) behavior is an artifact
of the model and disappears if one takes a more physical Hamiltonian.
In fact, in \cite{BFS2} it is proven that
the (ultraviolet cut-off)
Hamiltonian of QED with
some types of a potential (e.g. the Coulomb potential)
possesses a ground state.
This is related to the fact that
in above considerations we considered a scalar field, whereas
photons in QED have spin one and are coupled to the charge by the
minimal coupling prescription.
\eer
\section{Time dependent van Hove Hamiltonians}
In this section we describe a certain class of strongly continuous
dynamics on the Fock space. One can say that these are the dynamics
generated by a time dependent family of van Hove
Hamiltonians.
Let $\rr\ni t \mapsto g^t\in\cW$ be a continuous vector valued function and $
\rr\ni t\mapsto u^t\in U(\cW)$ be a strongly continuous function with
values in unitary operators. We assume that $g^0=0$ and $u^0=1$.
Set
\[V(t):=\Gamma(u^t)\exp\bigl(
\i g^t(a^*)+\i \bar g^t(a)\bigr)
.\]
For $A\in B(\Gamma_\s(\cW))$ we set
\[\beta^t(A):=V(t)AV(t)^*.\]
Note that
\[\beta^t(w(a^*))=u^t w(a^*)+\i (g^t|w).\]
\bet \ben \item
$V(t)$ is a strongly continuous family of unitary operators
on $\Gamma_\s(\cW)$ such that $V(0)=1$.
\item
$\beta^t$ is a pointwise strongly continuous family of
$*$-automorphisms of $B(\Gamma_\s(\cW))$ such that $\beta^0$ is the
identity.
\item $V(t)$ is the distinguished implementation of $\beta^t$ in the
following sense: if $\tilde V(t)$ is a family of unitary operators
such that $\beta^t(A)=\tilde V(t)A\tilde V(t)^*$ and $(\Omega|\tilde
V(t)\Omega)>0$, then $\tilde V(t)=V(t)$.
\een\eet
One can ask what is the time-dependent generator of $V(t)$. To answer
this question we proceed formally, without worrying about the exact
meaning of various objects involved in our formulas.
Suppose that the dot denotes the temporal derivative. It is easy to
check the following identities:
\[\begin{array}{rl}
\frac{\d}{\d t}\e^{\i g^t(a^*)+\i \bar g^t(a)}&
=\bigl(\frac{\i}{2}\Im(\dot g^t|g^t)
+\i\dot g^t(a^*)+\i\dot{\bar g}^t(a)\bigr)\e^{\i g^t(a^*)+\i \bar
g^t(a)},\\[3mm]
\frac{\d}{\d t}\Gamma(u^t)&=\d\Gamma(\dot u^tu^{t*})\Gamma(u^t).
\end{array}\]
Therefore,
\[\frac{\d}{\d t}V(t)=\i\Big(\frac{1}{2}\Im(\dot g^t|g^t)
+u^{t}\dot g^t(a^*)+\bar u^{t}\dot{\bar g^t}(a)
-\i\d\Gamma(\dot u^tu^{t*})
\Big)V(t)\]
Now suppose that
$t\mapsto z^t$ is a family of vectors and $t\mapsto h^t$ is a
family of self-adjoint operators. Suppose that
$u^t$ is the solution of
\[ \frac{\d}{\d t}u^t=\i h^tu^t,\ \ \ u^0=1;\]
and
\[\begin{array}{l}
g^t:=\int\limits_0^tu^{s*}z^s\d s,\\[3mm]
\sigma^t:=\frac12\Im\int\limits_0^t(z^t|u^tu^{s*}z^s)\d s.\end{array}\]
Then
\[\frac{\d}{\d t} V(t)=\i H(t)V(t),\]
where
\[H(t):=\d\Gamma(h^t)+z^t(a^*)+\bar z^t(a)+\sigma^t.\]
Thus, at least
on a formal level, $V(t)$ is generated by van Hove Hamiltonians.
\begin{thebibliography}{aaaaa}
\bibitem[Ar]{Ar} Arai, A.: A note on scattering theory in
non-relativistic quantum electrodynamics, J. Phys. A: Math. Gen. 16
(1983) 49-70
\bibitem[AHH]{AHH} Arai, A., Hirokawa, M., Hiroshima, F.: On the
absence of eigenvectors of Hamiltonians in a class of massless quantum
field models without infrared cutoff, J. Funct. Anal. 168 (1999) 470-497
\bibitem[BFS1]{BFS1} Bach, V., Fr\"{o}hlich, J., Sigal, I.:
Quantum electrodynamics of confined non-relativistic particles,
Adv. Math. { 137} (1998) 299
\bibitem[BFS2]{BFS2} Bach, V., Fr\"{o}hlich, J., Sigal, I.:
Spectral analysis for systems of atoms and molecules coupled to the
quantized radiation field, Commun. Math. Phys. { 207} (1999)
249
%\bibitem[BSZ]{BSZ} Baez, J.C., Segal, I.E., Zhou, Z.: {\em
%Introduction to Algebraic and Constructive Quantum Field Theory},
%Princeton NJ, Princeton University Press (1991).
\bibitem[Be]{Be} Berezin, F.: {\em The Method of Second Quantization},
second edition,
Nauka, 1986
\bibitem[BN]{BN} Bloch, F., Nordsieck, A.: Phys. Rev. 52 (1937) 54
\bibitem[BR]{BR} Brattelli, O, Robinson D. W.: {\it Operator Algebras and
Quantum Statistical Mechanics, vol. I and II}, Springer, Berlin (1981).
\bibitem[DG1]{DG1} Derezi\'nski, J., G\'erard, C.: Asymptotic
completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians,
Rev. Math. Phys. { 11} (1999) 383
\bibitem[DG2]{DG2}Derezi\'{n}ski, J., G\'erard,
C.: Spectral and scattering theory of spatially cut-off
$P(\varphi)_{2}$ Hamiltonians, Comm. Math. Phys. 213 (2000) 39-125
\bibitem[DJ]{DJ}
Derezi\'nski, J., Jak${\check {\rm s}}$i\'c, V.: Spectral theory of
Pauli-Fierz operators, Journ. Func. Analysis (2001) 241-327
\bibitem[EP]{EP} Edwards S., Peierls P. E.: Field equations in
functional form, Proc. Roy. Soc. Acad. 224 (1954) 24-33
\bibitem[Frie]{Frie} Friedrichs, K. O. {\em Mathematical aspects of
quantum theory of fields}, New York 1953
\bibitem[Fr]{Fr} Fr\"ohlich, J.: On the infrared problem in a model of
scalar electrons an massless, scalar bosons,
Ann. Inst. H. Poincar\'{e} 19 (1973) 1-103
\bibitem[GS]{GS} Greenberg O. W., Schweber S.S.: Clothed particle
operators in simple models of quantum field theory, Nuovo Cimento 8
(1958) 378-406
%\bibitem[He]{He} Heitler, W.: {\em The Quantum Theory of Radiation,}
%Oxford, Oxford University Press (1954).
\bibitem[HK]{HK} H\o egh-Krohn R.:
Asymptotic limits in some models of quantum field theory, I J. Math.
Phys. J. Math. Phys. 9 (1968) 2075-2079
\bibitem[KM1]{KM1} Kato, Y, Mugibayashi, N.: Regular perturbation and
asymptootic limits of operators in quantum field theory,
Prog. Theor. Phys. 30 (1963) 103-133
\bibitem[KM2]{KM2} Kato, Y, Mugibayashi, N.: Regular perturbation and
asymptootic limits of operators in fixed source theory,
Prog. Theor. Phys. 31 (1964) 300-310
\bibitem[Ki]{Ki} Kibble, T. W. B.: J. Math. Phys. 9 (1968) 15,
Phys. Rev. 173 (1968) 1527; Phys. Rev. 174 (1968) 1882; Phys. Rev. 175
(1968) 1624
\bibitem[Kato]{Kato} Kato, T.: {\em Perturbation Theory for Linear Operators},
second edition,
Springer-Verlag, Berlin (1976).
\bibitem[Mi]{Mi} Miyatake, O.: On the non-existence of solution of
field equations in quantum mechanics, J. Inst. Polytech. Osaka City
Univ. Ser. A Math. 2 (1952) 89-99
%\bibitem[RS1]{RS1} Reed, M., Simon, B.:
% {\em Methods of Modern Mathematical Physics, I. Functional
%Analysis}, London, Academic Press (1980).
\bibitem[RS2]{RS2} Reed, M., Simon, B.: {\em Methods of Modern
Mathematical Physics, II. Fourier Analysis, Self-Adjointness},
London, Academic Press (1975).
\bibitem[RS3]{RS3} Reed, M., Simon, B.: {\em Methods
of Modern Mathematical Physics, III. Scattering Theory},
London, Academic Press (1978).
%\bibitem[RS4]{RS4} Reed, M., Simon, B.: {\em Methods
%of Modern Mathematical Physics, IV. Analysis of
%Operators}, London, Academic Press (1978).
\bibitem[Schwa]{Schwa} Schwarz, A. S.: {\em Mathematical foundations
of quantum field theory}, Atomizdat 1975, Moscow (Russian)
\bibitem[Sch]{Sch} Schweber, S. S.: {\em Introduction to nonrelativistic
quantum field theory}, Harper and Row 1962
\bibitem[To]{To} Tomonaga, S.: On the effect of the field reactions on
the interaction of mesotrons and nuclear particles
I. Progr. Theor. Phys. 1 (1946) 83-91
\bibitem[vH]{vH} van Hove, L.: Les difficult\'es de divergences pour
un modele particulier de champ quantifi\'e, Physica 18 (1952) 145-152
\bibitem[Ya]{Ya} Yafaev, D.: {\em Mathematical Scattering Theory}, AMS
\end{thebibliography}
\end{document}
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