All
$\mathrm{E}\mathrm{x}_{1^{)}}1^{\cdot}\mathrm{e}l\mathrm{s}_{\backslash }\mathrm{s}\urcorner \mathrm{i}_{0}\mathrm{n}$of
$\mathfrak{t}_{1}\mathrm{h}\mathrm{e}\mathrm{G}1^{\cdot}\mathrm{o}\mathrm{l}111\mathrm{d}$St,ate
$\mathrm{E}\mathrm{n}\epsilon^{\tau}1^{\backslash }\mathrm{g}\mathrm{y}$
of
the
Spin-Boson
Model
Masao Hirokawa
(
廣川真男
)1
$Departrner\iota t$
of
Mathematics.
Tokyo
$Gak_{\mathrm{t}}\iota ge$
?
University.
Koganei
184.
Japan
Abstract
$\mathrm{A}\mathrm{I}1$
expression of
the
ground
state
energy
$E_{SB}$
of
the
spin-boson
Hamiltonian
$H_{5B}$
.
is considered. The expression in the
cases
of both massive and massless
bosons
is
given by a nonperturbative method.
The
$\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}- \mathrm{b}_{0}\mathrm{s}\mathrm{o}\mathrm{I}\mathrm{l}$nlodel.
which describes
a two-level
$\mathrm{S}_{\backslash }\mathrm{y}\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{n}1$coupled
to
a
quantized
Bose
field.,
$11’\$
been
investigated as a simplified model for atomic
$\mathrm{s}\mathrm{y}_{\mathrm{S}\mathrm{t}\mathrm{e}}\mathrm{n}\mathrm{l}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}_{\mathrm{I}}$with a
quantized
$\mathrm{r}\mathrm{a}(\{\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}$or
phonon
field
([LCDFCW,
Am,
Arl.
$\mathrm{D},$ $\mathrm{F}\mathrm{a}\mathrm{h}^{-}\mathrm{V}$,
G\’er,
$\mathrm{H}ii\mathrm{S}\mathrm{p}1$.
$\mathrm{H}\ddot{\mathrm{u}}\mathrm{S}\mathrm{p}2$,
Spl,
$\mathrm{A}\mathrm{r}\mathrm{H}\mathrm{i}\mathrm{l}$.
$\mathrm{J}\mathrm{P}3$]
and
references therein).
Several
properties of the ground states of the
model
are of interest. Especially. we are interested in
expressions
of the
ground state
energy
of
the
model,
because for each Hamiltonian
we can
actually
observe
its
energies
only at
every
state,
neither the Hamiltonian
nor the state according
to
the
standard quantum theory. For
the spin-bosoIl Hamiltonian
$H_{SB}.$
recently.
attention
has
been
paid
to.
the
ground
states
$\mathfrak{B}$
the
eigenvectors
of
$H_{SB}$
with eigenvalue equal to the infimum of its spectrum
to
$\mathrm{d}\mathrm{e}\mathrm{v}\mathrm{e}1_{0}\mathrm{p}$nonperturbative
method
(
$[\mathrm{S}\mathrm{p}3$.
$(\mathrm{i}\mathrm{i})$on
p.5]
.
$[\mathrm{A}\mathrm{r}\mathrm{H}\mathrm{i}\mathrm{l}$.
$\mathrm{A}\mathrm{r}\mathrm{H}\mathrm{i}2]$)
and analyze spectral properties
and the process of radiative decay
$([\mathrm{H}\ddot{\mathrm{u}}\mathrm{s}_{\mathrm{p}1}. \mathrm{H}\ddot{\mathrm{u}}\mathrm{S}\mathrm{p}2])$.
Talking
of the
ground states of this
type
model.
we
here note that
in [Tl,
T2.
T3, T4]
Tomonaga argued
the
ground state
of the
model which has relation
to
the
spin-boson
model in
order to
get rid of
physical
difficulties
caused by
applving the
perturbation theory
to the model.
Moreover,
recently
Bach,
$\mathrm{F}\mathrm{r}\ddot{\mathrm{o}}\mathrm{h}\mathrm{l}\mathrm{i}_{\mathrm{C}}\mathrm{h}$and
Sigal argued the ground
state,
spectrum and
resonance
for a model
of nonrelativistic quantum electrodynamics
$([\mathrm{B}\mathrm{a}\mathrm{F}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{l}, \mathrm{B}\mathrm{a}\mathrm{F}\mathrm{r}\mathrm{S}\mathrm{i}\mathrm{g}2, \mathrm{B}\mathrm{a}\mathrm{F}\mathrm{r}\mathrm{S}\mathrm{i}\mathrm{g}3, \mathrm{B}\mathrm{a}\mathrm{F}\mathrm{r}\mathrm{S}\mathrm{i}\mathrm{g}4])$.
Especially
they
$\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{i}_{\mathrm{S}}$}
$1\mathrm{e}\mathrm{d}$the
method of renormalization
group
to
investigate
resonances
in quantum electrodynamics, which is of great value for many who deal with problems
on
the
resonances in the case of
$\mathrm{m}\mathrm{a}s$sless bosons. For the
generalized
spin-boson
model.
Arai
and
the author showed
that,
under
certain
conditions,
there
exists a ground
state
of
the
generalized model in
$[\mathrm{A}\mathrm{r}\mathrm{H}\mathrm{i}2]$by
a nonperturbative method., and we
gave
a
formula
for
the asymptotic behavior of the
ground
state
energy
of the
generalized
model
in
the
strong
coupling region
(
$[\mathrm{A}\mathrm{r}\mathrm{H}\mathrm{i}\underline{9}$,
Proposition 1.4]). In this paper
we
focus
our attention on
the
expression of the
ground
state
energy
of the
(standard)
spin-boson
Hamiltonian
$H_{SB}$
in the
cases
of both
$\mathrm{n}\mathrm{l}\mathrm{a}$ssive
$\mathrm{a}\mathrm{I}\iota \mathrm{d}$
massless
bosons. Especially, it is important that
we
$\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{f}$}
$\gamma$the
expression in
the
case of
$\mathrm{m}\alpha$
sless
bosons:
because
we
cannot apply the
regular
perturbation
theory to
$H_{SB}$
in the
case.
Thus
we
try
nonperturbative approach to
our
problem in this
paper.
For physical
reality,
we consider the situation where
bosons
rnove
in the
3-dimensional
EuclideaIl space
$\mathrm{R}^{3}$.
We take a Hilbert space of bosons
to
be
$\mathcal{F}_{b}=\mathcal{F}(L^{2}(\mathrm{R}^{3}))=\bigoplus_{n=0}^{\infty}[\emptyset_{S}^{n}L2(\mathrm{R}3)]_{\}$
(1)
1Research
is supported
by the
Grarit-In-Aid
No.09740092 for Encouragement of Young
Scientists
from
the
$\mathrm{s}\mathrm{y}_{\mathrm{I}11}1\mathrm{I}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{F}_{0\mathrm{C}:}\mathrm{k}$space over
$L^{2}(\mathrm{R}^{1})(^{\prime\backslash }4_{-}^{1^{l}}..\mathcal{K}_{\mathrm{t}\mathrm{l}\mathrm{e}\mathrm{I}}1\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{S}\theta 1$the
$n$
-folcl
$,\backslash ’ \mathrm{v}\mathrm{I}1\iota 111\mathrm{e}\mathrm{t}_{\Gamma}\mathrm{i}_{\mathrm{C}}$.
tensor
$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{l}\iota \mathrm{l}\mathrm{C}^{\cdot}\mathrm{t}$of a Hilbert space
$\mathcal{K}$.
$5_{-s}\text{ノ^{}1}\kappa 0\equiv \mathrm{C}$).
Let
$\Omega_{0}$be tlle
$\mathrm{F}0\mathrm{C}^{\cdot}\mathrm{k}_{\mathrm{V}\mathrm{a}}\mathrm{t}\iota 1\mathrm{u}\mathrm{I}11\mathrm{i}\mathrm{I}1\mathcal{F}_{b}$.
In this paper. we
set
both
of
$\Gamma$,
and
$\mathrm{c}$
.
one,
i.e.,
$\gamma_{l}=c=1$
.
where
$\Gamma\iota$
is
the
$\mathrm{P}1_{\dot{\mathrm{c}}\mathrm{u}1}\mathrm{c}\mathrm{k}\mathrm{c}\cdot \mathrm{o}\mathrm{I}\mathrm{l}\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{I}\mathrm{l}\mathrm{t}$divided
by
$2\pi$
.
$\mathrm{a}\mathrm{I}\mathrm{l}\mathrm{C}1$(
the
$\mathrm{v}\mathrm{e}\mathrm{l}0\mathrm{C}^{\cdot}\mathrm{i}\mathrm{t}\mathrm{y}$of the
light. A
$\mathrm{f}\iota\iota \mathrm{n}\mathrm{c}:\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{I}1’\ ^{\dagger}\Gamma$is
given
by
$\omega_{f}(\mathrm{A}\cdot):=\sqrt{|k|^{2}+\gamma n^{2}}$
,
$m\geq 0$
.
$k\in \mathrm{R}^{\mathrm{I}}$
,
which is the
energy
of the relativistic bosons with mass
$rrl\mathrm{a}\mathrm{i}_{\mathrm{l}\mathrm{t}}1\mathrm{I}\iota \mathrm{l}\mathrm{O}\mathrm{I}\iota \mathrm{l}\mathrm{e}\iota \mathrm{l}\mathrm{t}\mathrm{t}\mathrm{l}\mathrm{n}\mathrm{l}k$.
We cfeIlote by
$d\Gamma(_{\ ^{1}r}.)$
the second quantization of the multiplicatioIl operator
$\omega_{r}$on
$L^{2}(\mathrm{R}^{3})$
and
set
$H_{b}=d \Gamma(’.\vee^{1}r)=\int \mathrm{R}^{3}.\omega_{r}dk(k)a(+k)a(k)$
,
where
$a(k)$
and
$a^{+}(k)$
are
the operator-valued distribution
kernels
of
the
smeared annihilation
and
creatioIl operators
respectively:
$a(f)= \int_{\mathrm{R}^{3}}dka(k)f(k)$
,
(2)
$a^{+}(f)= \int_{\mathrm{R}^{3}}dka^{+}(k)f(k)$
(3)
for
every
$f\in L^{2}(\mathrm{R}^{3})$
on
$\mathcal{F}_{b}$.
$Re^{J}$
mark
1. In
$[\mathrm{A}\mathrm{r}\mathrm{H}\mathrm{i}\mathrm{l}, \mathrm{A}\mathrm{r}\mathrm{H}\mathrm{i}2]$.
we used the
definition,
$a(f)= \int_{\mathrm{R}^{3}}dka(k)f(k)^{*}$
.
$f\in L^{2}(\mathrm{R}^{3})$
,
as the annihilation
operator
$a(f)$
according
to
the
custom
for
mathematics,
where
$f(k)^{*}$
denotes the complex conjugate of
$f(k)(k\in \mathrm{R}^{3})$
.
but
we
here
employ
(2)
as the definition
of
$\alpha(f)$
according
to
the
way of
physics.
The
Segal
field
operator
$\phi_{S}(f)(f\in L^{2}(\mathrm{R}^{3}))$
is given
by
$\phi_{s}(f):=\frac{1}{\sqrt{2}}(a(+f)+a(f))$
.
(4)
Let
$\lambda$be
a
real-valued continuous
function
on
$\mathrm{R}^{3}$satisfying
the
following
conditions:
(A)
$\lambda(k)=\lambda(-k)(k\in \mathrm{R}^{3})$
,
and
$\lambda,$$\lambda/\omega_{r}\in L^{2}(\mathrm{R}^{3})$
.
$Re$
mark
2. Since
$\lambda,$$\lambda/\omega_{\mathrm{r}}\in L^{2}(\mathrm{R}^{3})\iota$
.
we have
$\lambda/\sqrt{\omega_{f}}\in L^{2}(\mathrm{R}^{3})$
.
The Hamiltonian of the spin-boson
model
is
defined
by
$H_{SB}:= \frac{\mu}{2}\sigma_{3}\sqrt{\mathrm{d}}I+I\otimes Hb+\sqrt{2}\alpha\sigma 1\otimes\emptyset_{s}(\lambda)$
acting in the Hilbert space
$\mathcal{F}:=\mathrm{c}^{2}\otimes \mathcal{F}_{b}=\mathcal{F}_{b}\oplus \mathcal{F}b$
.
(5)
where
$\sigma_{1},$$\sigma_{3}$are the standard Pauli
matrices,
III
(5),
we
$\mathrm{i}\mathrm{t}[\mathrm{g}\mathrm{I}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{e}(1\mathrm{c}^{2\prime}4_{-^{\mathrm{y}}b}^{\cdot};\mathcal{F}$with
$\mathcal{F}_{b}arrow_{\perp}^{1}\mathcal{F}_{b}$.
So,
$H_{SB}11\dot{\epsilon}\iota \mathrm{b}$
the following
$\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}8\mathrm{e}\mathrm{I}\mathrm{l}\mathrm{t}\dot{\mathrm{c}}\mathrm{t}\mathrm{t}\mathrm{i}_{0}\mathrm{I}1$on
$\mathcal{F}_{b}\vdash_{\mathrm{L}}^{\mathrm{b}}\mathcal{F}\iota$and
we
$\mathrm{e}\mathrm{l}\mathrm{I}\mathrm{l}1^{\mathrm{J}1_{0_{\backslash }}}\mathrm{v}$it,
$\mathrm{i}\mathrm{I}1$this
paper:
$H_{SB}=(_{\sqrt{2}\alpha}^{H_{b}+\frac{l^{l}}{(2}}\Phi_{s}’\lambda)$
$\sqrt{2}\alpha\psi_{S}^{}\lambda)H_{b}-\frac{/l(}{2})$
.
For
a
$1\mathrm{i}\mathrm{I}\mathrm{l}\mathrm{e}_{\dot{\subset}}\iota \mathrm{r}$operator
$T$
on
a
Hilbert space,
we
denote its domain by
$D(T)$
.
It
is
well-known
that
$H_{SB}$
is self-adjoint with
$D(H_{SB})=D(I\Theta H_{b})$
and
$- \frac{|fl|}{\underline{?}}-\alpha^{2}||\frac{\lambda}{\sqrt{p_{\mathfrak{l}}}}||^{2}L\underline{\circ}\leq H_{SB}$
,
where
$||\cdot||_{L\underline’}$
denotes
the
norm of
$L^{2}(\mathrm{R}^{3})$
.
For a self-adjoint
operator
$T$
bounded from
below.
we denote by
$E(T)$
the
infimum of the
spectrum
$\sigma(T)$
of
$T$
:
$E(T)= \inf\sigma(T)$
.
In this paper, an eigenvector of
$T$
with
eigenvalue
$E(T)$
is called a ground
state
of
$T$
(if
it
exists).
We say that
$T$
has
a
(resp. unique)
ground
state
if
$\dim \mathrm{k}\mathrm{e}\mathrm{r}(T-E(T))\geq 1$
(resp.
dim ker
$(T-E(T))=1)$
.
We call
$E(T)$
the ground state
energy
of
$T$
if
$T$
is
a
Hamiltonian.
For
$H_{SB}$
we
set
$E_{SB}(\mu, \alpha):=E(H_{S}B)$
.
By the variational principle
(
$[\mathrm{A}\mathrm{r}1$, Theorem 2.4] and
$[\mathrm{D}$,
p.161]).
we
have
$E_{SB}( \mu, \alpha)\leq-\frac{|\mu|}{2}.e^{-2}\alpha 2||\lambda/\omega_{\gamma}||^{2}L^{2}-\alpha^{2}||\frac{\lambda}{\sqrt{v_{r}}}.||_{L}22$
(6)
Under
certain assumptions.
we
know that
$H_{SB}$
has
a ground
state
([H\"uSpl,
$\mathrm{A}\mathrm{r}\mathrm{H}\mathrm{i}\mathrm{l}$,
Sp3]
and
see Remark
4(1)
in this
paper).
DEFINITION
1.
We say a vector
$\Psi\in \mathcal{F}\equiv \mathcal{F}_{b}\oplus \mathcal{F}_{b}$
overlaps with a ground state
$\Omega_{SB}(\mu, \alpha)$
if
and only if there
exists the
groun.d
state
$\Omega_{SB(\mu},$
$\alpha$)
of
$H_{SB}$
such that
$\langle\Psi . \Omega_{SB}(\mu, \alpha)\rangle r\neq 0$
.
where
$\langle$,
$\rangle_{\mathcal{F}}$is the
standard
inner
product
of
$\mathcal{F}\equiv \mathcal{F}_{b}\mathrm{t}^{\mathrm{T}_{\backslash }}\supset \mathcal{F}_{b}$.
From
now
on.
according
to
the custom
for
the physicists, all the inner products of the
Hilbert
spues appearing in
this
paper
have
the linearity
on
the
right
hand side.
If
a ground state
$\Omega_{SB}(\mu.\alpha)$
of
$H_{SB}$
exists,
for
$\Omega_{SB}(\mu.\alpha)$
we
set
$\Omega_{SB}(.\mu.\alpha)=\in \mathcal{F}\equiv \mathcal{F}_{b^{\frac}}."|\mathcal{F}_{b}$
.
It
is
well kIlOWIl
that.
if
$f\in L^{2}(\mathrm{R}^{3})$
,
we
can define a
self-adjoint operator
$P(f)$
by
tlllls,
if
$\lambda/\ ^{1}\mathrm{r}\in L^{2}‘(\mathrm{R}^{s})$
,
we have
two
unitary operators
$[_{\Sigma}^{r_{\pm}}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{i}_{\mathrm{I}}1\mathrm{P}\mathrm{t}1$by
$\iota_{\pm \mathrm{p}}^{r}’:=\mathrm{e}\mathrm{x}[\pm i\alpha P(\lambda/\omega_{f})]$
.
$(\overline{/})$We
$\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{i}_{\mathrm{I}}1\mathrm{e}$two
$\mathrm{u}\mathrm{I}\iota \mathrm{i}\mathrm{t}_{\mathrm{V}\mathrm{e}}\mathrm{C}^{\cdot}\mathrm{t}\mathrm{o}\Gamma \mathrm{S}\Omega_{\pm}\in \mathcal{F}\equiv \mathcal{F}_{b\ddot{\dot{\oplus}}}\mathcal{F}_{b}$by
$\Omega_{\pm}:=\frac{1}{2}$
.
We
have
the followinff
$\mathrm{D}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{S}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$on an
upoer
bound of the
ground
state
energy:
$H_{sBS}/:=- \frac{\mu}{2}\sigma_{1}\sigma\cup^{i}I+I\otimes Hb+\sqrt{2}\alpha\sigma_{3}\otimes\emptyset’(\lambda)$
respectively. We
note here that
$H_{SB}$
and
$H_{SB}’$
are
unitary
equivalent
(see
Lemma
$2(\mathrm{i})$
in
this
paper).
So,
by
[
$\mathrm{D}$,
Theorem
10],
for sufficiently small
$|\mu|$
,
$E_{SB}( \mu, \alpha)=-\alpha^{2}\int \mathrm{R}3dk\frac{\lambda(k)^{2}}{\vee\prime b^{1}r(k)}-\frac{|\mu|}{2}\exp[-9.\alpha^{2}\int_{\mathrm{R}^{3}}dk\frac{\lambda(k)^{2}}{\omega_{r}(k)^{2}}]+0(\mu^{2})$
.
For
arbitrary
fixed
$m>0$
and
$\mu\neq 0$
(resp.
$\alpha\neq 0$
),
sufficiently
small
$|\alpha|$
(resp.
$|\mu|$
)
satisfies
the inequality in
(ii).
Thus Theorem
$3(\mathrm{i})_{\pm^{\mathrm{W}\mathrm{i}}}\mathrm{t}\mathrm{h}(\mathrm{i}\mathrm{i})$may be
legarded
as a result which
im-proves the
oIle
obtained
by
regular
perturbation theory. Note that
(10)
is a nonperturbative
$\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{i}_{\mathrm{I}\mathrm{n}}\mathrm{a}\mathrm{t}\mathrm{e}$
in
$\alpha$,
since
the left
hand side
of
(10)
is
non-polynomial
in
$\alpha$.
(3)
To
$\mathrm{a}\iota\iota \mathrm{t}1_{1\mathrm{O}}\mathrm{r}\mathrm{S}$best kIlowledge, Theorem
$3(\mathrm{i})_{\pm}$
with
(iii)
is the first which establishes a
concrete
expression
of the
ground state energy
of the spin-boson
model
$H_{SB}$
in the case of
COROLLARY.
$A_{\mathrm{S}}.’
s\mathrm{c}\iota rn,e(A)$
.
Fix
$\alpha\in \mathrm{R}$
.
.
(i)
Suppose that.
for
$l^{l}\cdot l\iota’>0$
.
$\Omega_{SB(l}\iota,$
$\alpha$)
and
$\Omega_{SB}(ll’, (\mathrm{r})e\prime x$
ist.
$.and\Omega_{+}$
overlaps
with
both
of
them. Then
$EsB(/l’ !\alpha)\leq ESB(/l, \alpha)$
if
$\ell\iota<\mu’$
.
(ii)
$Suppo\mathit{8}e$
that.
$for/\iota,$
$\mu’<0,$
$\Omega sB(\mu, \alpha)$
and
$\Omega_{SB(\mu’,\alpha}$
)
exist,
and
$\Omega$-overlaps with both
of
them. Then
$E_{SB}(\mu’, \alpha)\geq E_{SB(\mu,\alpha\cdot)}$
if
$\mu<\mu’$
.
The
basic idea
to
prove
our main
theorem
is
as follows: If
there
exist
a ground
state
$\Omega_{SB}(\mu, \alpha)$
of
$H_{SB}$
and a
vector
$\Psi\in \mathcal{F}\equiv \mathcal{F}_{b}\oplus \mathcal{F}_{b}$
such that
$\Psi$
overlaps
with
$\Omega_{SB}(\mu, \alpha)$
,
then
by Bloch
$\mathrm{s}$formula
(
$[\mathrm{B}\mathrm{l}\mathrm{o},$(12)
$]$and
see
Lemma
2.4 in
this
paper),
we
have
$E_{SB}( \mu, \alpha)=-\lim_{\betaarrow\infty}\frac{1}{\beta}\ln\langle\Psi, e^{-}\Psi\beta HsB\rangle_{\tau}$
.
So,
our
problenl
is
reduced to
that of
how to
find such
$\Psi$
that
we now
try to
calculate
$\langle\Psi .
e^{-\beta B}\Psi H_{S}\rangle_{f}$
in the concrete.
In
the
following
section,
we shall use several
unitary
trans-formations and the Du
Hammel
formula
so
that
we can
apply
the Feynman-Kac-Nelson
formu.la
for the free
field,
and we
shall find
that
either
$\Omega_{+}$
or
$\Omega_{-}$
is one of the
answers
for
the
problem above.
Here,
it is important that we employ the Feynman-Kac-Nelson formula
for the
free field
because
we can calculate
$\mathrm{a}\mathrm{c}\mathrm{t}\backslash$ually
and
concretely
the
ground
state
energy
of
the
spin-boson
model.
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A.Amann,
Ground
states
of
a
spin-boson model,
Ann.
Phys. 208 (1991),
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non-relativistic limit
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Pauli-Fierz and a spin-boson
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(1990),
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A.Arai,
Perturbation of embedded eigenvalues: a general class of exactly
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Hokkaido Math. Jour.
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A.Arai and
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[BaFrsigl]
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...
$[\mathrm{B}\mathrm{a}\mathrm{F}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{g}2]\backslash ^{r},.\mathrm{B}_{\dot{C}\iota}\mathrm{c}.1_{1}$
.
$\mathrm{J}.\mathrm{F}\mathrm{r}\ddot{0}111\mathrm{i}\mathrm{C}^{\cdot}1_{1}$and
$\mathfrak{l}.\mathrm{b}^{\mathrm{r}}\mathrm{I}.\mathrm{S}\mathrm{i}\mathrm{g}\mathrm{a}1$,
$\mathrm{Q}_{\mathfrak{U}\mathrm{a}\mathrm{I}\mathrm{l}\mathrm{t}\mathrm{u}}.111\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\iota\cdot 0\mathrm{t}\mathrm{l}\mathrm{y}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{I}\mathrm{l}\mathrm{l}\mathrm{i}(\mathrm{S}$of
(
$\mathrm{O}\mathrm{I}1\mathrm{f}\mathrm{i}_{\mathrm{I}\mathrm{l}\mathrm{e}\mathrm{c}1}$
nonrel-ativistic
particles
(to
appear in
$Adv$
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in
Math.),
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1996;
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