Spectral Analysis for Systems of Atoms and
Molecules coupled to the Quantized Radiation
Field
Volker Bach
FB Mathematik MA 7-2; TU Berlin; Str. d. 17 Juni 136;
D-10623 Berlin; Germany (bach@math.tu-berlin.de)
Jurg Frohlich
Inst. f. Theoretische Physik; ETH Honggerberg;
CH-8093 Zurich; Switzerland (juerg@itp.phys.ethz.ch)
Israel Michael Sigal y
Dept. of Math.; Univ. of Toronto;
Toronto, M5S 3G3; Canada (sigal@math.toronto.edu)
November 20, 1998
Abstract
We consider systems of static nuclei and electrons {atoms and
molecules{ coupled to the quantized radiation eld. The interactions
between electrons and the soft modes of the quantized electromagnetic
eld are described by minimal coupling, p~ ! p~ eA~ (~x), where A~ (~x) is
the electromagnetic vector potential with an ultraviolet cuto . If the
interactions between the electrons and the quantized radiation eld are
turned o , the atom or molecule is assumed to have at least one bound
state. We prove that, for suciently small values of the feinstructure
constant , the interacting system has a ground state corresponding
to the bottom of its energy spectrum and that the excited states of the
atom or molecule above the ground state turn into metastable states
Heisenberg Fellow of the DFG, supported by SFB 288 of the DFG, the TMR-Network
on \PDE and QM".
y Supported by NSERC Grant NA 7901
0
01/11/1998
EXT-2000-055
whose life-times we estimate. Furthermore the energy spectrum is
absolutely continuous, except, perhaps, in a small interval above the
ground state energy and around the threshold energies of the atom or
molecule.
Keywords: Renormalization Group, Spectrum, Resonances, Fock
space, QED.
BFS-3, November 20, 1998
I
1
Introduction and Survey of Results
I.1 The Quantum Theory of Photons and Nonrelativistic, Bound Electrons
In this paper we continue our mathematical analysis of the standard model
of nonrelativistic, quantum-mechanical matter interacting with the quantized
radiation eld initiated in [4, 5, 6]. The purpose of the present paper is to
re ne the techniques developed in [4, 5, 6] in such a way that they can be
applied to the study of physically relevant models.
The physical system we are analyzing consists of a nite number of nuclei,
treated as static sources, and a nite number of nonrelativistic electrons, e.g.,
atoms, ions, or molecules in a Born-Oppenheimer approximation, interacting
with the soft modes of the quantized electromagnetic eld which is cut-o
in the ultraviolet. The Hilbert space of pure state vectors of the system is
given by
H := Hel F
(I.1)
;
where Hel is the Hilbert space of some nite number, , of electrons, and
F is the photon Fock space. Thus, in the Schrodinger (con guration-space)
representation, Hel is given by the subspace of totally antisymmetric wave
functions in 2 [(R 3 Z2)N ], where R 3 is the con guration space of a single
N
L
electron, and Z2 describes its spin, i.e.,
Hel := AN
2
L
(R Z2)N
3
;
(I.2)
with AN being the orthogonal projection onto the subspace of totally antisymmetric wave functions, as required by the Pauli principle.
The one-photon Hilbert space is given by 2 [R 3 Z2], where R 3 is the
photon momentum space and Z2 describes the two independent transversal
polarizations of a photon. (Here and above, the integration measure on R 3
is Lebesgue measure.) The photon Fock space is then de ned by
1
M
3
n
2
(I.3)
F := Sn (R Z2)
L
n=0
L
;
where Sn is the orthogonal projection onto the subspace of totally symmetric -photon wave functions, in accordance with the fact that photons are
bosons. It is convenient to represent the Hilbert space H as the space of antisymmetric, square-integrable wave functions on the -electron con guration
n
N
BFS-3, November 20, 1998
2
space with values in the photon Fock space F , i.e.,
H = AN L2 (R 3 Z2)N ; F :
(I.4)
The dynamics of the system is generated by the Hamiltonian
H 0 :=
N
X
~ ~xj
~j ir
j =1
2
21=2 3=2 A~ ( ~xj ) + Vc(x) + Hf ;
(I.5)
where we use units in which ~ = 1 and the electron mass equals 1=2. In
(I.5), ~j = (jx; jy ; jz ) denotes the three Pauli matrices associated with the
j th electron, xj is its position (in suitable units of length described below),
is the feinstructure constant, and A~ (~x) denotes the quantized vector potential
of the transverse modes of the electromagnetic eld in the Coulomb gauge,
i.e.,
X Z d3 k (j~kj=K ) ~ i~k~x ~
q
~"(k)e a(k) + ~"(~k) ei~k~xa (~k) ;
A~ (~x) :=
=1;2
2!(~k)
(I.6)
where is an entire function of rapid decrease on the real line, e.g., (r) :=
exp( r4 ), cutting o the vector potential in the ultraviolet domain, !(~k) :=
j~kj is the frequency of a photon with wave vector ~k, and ~"(~k), = 1; 2, are
photon polarization vectors satisfying
~"(~k) ~"(~k) = ; ~k ~"(~k) = 0 ; for ; = 1; 2.
(I.7)
Moreover, a(~k), a (~k) are standard creation- and annihilation operators
(see, e.g., [25]) on F obeying the canonical commutation relations
[a# (~k1 ); a# (~k2 )] = 0; [a (~k1); a(~k2)] = (~k1 ~k2 ) ;
(I.8)
where a# = a or a . These objects are densely de ned, operator-valued
tempered distributions on Fock space F . Fock space F contains a vector ,
the vacuum vector, uniquely determined, up to a phase, by the properties
that k k = 1 and a (~k) = 0, for all and ~k. A dense set of vectors in F
is obtained by applying polynomials in a ( = 1; 2), smeared out with test
functions, to the vacuum vector .
The term Vc on the right side of Eqn. (I.5) de ning the Hamiltonian is the
Coulomb potential describing the electrostatic interactions between electrons
and nuclei. In our units, it is given by
N X
M
X
Zm + X
1 ;
V (x) :=
(I.9)
c
j =1 m=1 j~xj
R~ m j
1i<j N j~xi
~xj j
BFS-3, November 20, 1998
3
where x (~x ; : : : ; ~xN ) 2 R N , and R~ ; : : : ; R~ M 2 R are the positions of M
static nuclei with atomic numbers Z ; : : : ; ZM .
Finally, the Hamiltonian of the free quantized electromagnetic eld, Hf ,
is given by
3
1
3
1
1
Hf :=
X Z d k a(~k) !(~k) a(~k) :
(I.10)
3
=1;2
The r.s. of Eqn. (I.10) de nes a densely de ned, positive, self-adjoint operator
on F with absolutely continuous spectrum on the positive half-axis and a
simple eigenvalue at 0 corresponding to the eigenvector .
Next, we comment on the units chosen in (I.5). Length (and hence the
positions, ~xj , of the electrons and, R~ m , of the nuclei) is measured in units
of half a Bohr radius, r = m~el2 e2 , expressed in Gaussian units, where
mel is the mass of an electron and e its electric charge. Photon wave
lengths are measured in units of times half a Bohr radius, r , i.e., the
unit for photon wave vectors is 22 r
. The energy unit is chosen to
e . The ultraviolet cuto of the radiation
be 4 Rydberg, with 4 Ry = rBohr
eld imposed by the function (jkj=K ) used in the de nition of the vector
potential A~ (~x) turns o interactions between electrons and photons with
energy large compared to K 4 Ry. The physical value of the feinstructure
constant is 1=137. In this paper, plays the r^ole of a small, dimensionless
number. Our results hold for suciently small values of . We shall not
verify that the radii of convergence in of our analytical constructions cover
the physical point =
1=137 (such a veri cation would presumably
require numerical work on a computer).
Our main concern in this paper is to analyze properties of the energy
spectrum of the Hamiltonian H 0 and to study resonances for the dynamics
generated by H 0 and estimate their life times. We shall show that, for
suciently small values of > 0, H 0 has a normalizable ground state corresponding to the minimum of its spectrum and that \most" of its spectrum
is absolutely continuous. Furthermore, we shall show that the excited bound
states of the atom or ion (for purely technical reason, we exclude molecules
here) with electrons decoupled from the radiation eld, i.e., the bound states
of H , turn into metastable states of nite life time when that coupling is
turned on. We show that Fermi's golden rule [24] yields an accurate estimate
of the life times of metastable states, and we provide a rigorous justi cation
of Bethe's formula [10, 9] for the Lamb shift.
In order to describe our results and methods more precisely, we start
with a discussion of the spectral properties of the unperturbed Hamiltonian
1
2
Bohr
2
1
2
(phys)
0
Bohr
1
2
Bohr
BFS-3, November 20, 1998
4
(electrons decoupled from the radiation eld),
H0
H
0
=0
= Hel + Hf ;
(I.11)
where Hel is the usual atomic (or molecular) Hamiltonian de ned by
Hel
:=
N
X
j =1
~x + Vc(x) :
j
(I.12)
We recall some key properties of Hel , viewed as an operator on Hel . For
details and proofs, see [24, 13] and references given there. We rst note
that
potential Vc is a perturbation of the kinetic energy operator x :=
PN the
j =1 ~x with zero relative bound. Thus Hel is a semibounded, self-adjoint
operator on the domain D(Hel) = D(x) = Hel \ H 2[(R 3 Z2)N ], where
H 2 is the usual Sobolev space. The essential spectrum of Hel is given by
ess (Hel ) = [; 1), where is the in mum of the spectrum of Hel , with N
replaced by N 1, as follows from the HVZ theorem (see, e.g., [13]). Thus,
is the ionization threshold.
In what follows, we shall make the following assumption about the atom,
ion, or molecule under consideration, represented by the parameters N ,
~ 1; : : : ; R
~ M , and Z1 ; : : : ; ZM . For the proof of existence of a ground state of
R
H in Section II, we assume that Hel has at least one eigenvalue E0 below
the ionization threshold , i.e., we require that
j
0
E0
:= inf (Hel ) < := inf ess(Hel ) :
(I.13)
The contents of Section III on the life times of metastable states is nontrivial only if, besides E0 , Hel has at least one further eigenvalue E1, with
E0 < E1 < , i.e., in Section III we additionally require that the spectrum
(Hel )
= fE0 ; E1; : : : g [ [; 1)
(I.14)
of Hel consists of eigenvalues E0 < E1 < E2 < , of nite multiplicity
below , possibly with an accumulation point at and (absolutely) continuous spectrum in [; 1); see Fig. 1. In Section III we shall also require an
assumption saying, roughly speaking, that there are no accidental selection
rules in the system described by Hel, which could prevent excited eigenstates
of Hel from decaying radiatively (in 2nd order in the relevant coupling constant).
For positive ions and (neutral) atoms or molecules our assumption (I.14)
is justi ed, as it is known that if N PMm=1 Zm then Hel has in nitely
many eigenvalues of nite multiplicity below [31]. In the case of negatively
BFS-3, November 20, 1998
5
gr. states
(abs.) cont. spectrum
exc. states
E0
E1
E2
Figure 1: The Spectrum of Hel
E0
E1
E2
Figure 2: The Spectrum of H0 = Hel 1 + 1 Hf
ionized atoms or molecules, i.e., if N > PMm=1 Zm, the question, whether Hel
has isolated eigenvalues below the ionization treshold, is more subtle. Indeed,
if N PMm=1(2Zm + 1) then Hel has no eigenvalues at all [23] (see [27, 28]
for earlier results).
Assuming that the atomic numbers of the nuclei are such that E0 < ,
i.e., that Hel has isolated eigenvalues below the ionization threshold, it tends
to be an open question whether the ground state corresponding to the energy
E0 is unique or not, except when N = 1 or = 2, in which case a standard
Perron-Frobenius argument proves uniqueness. (Non-uniqueness for N 3
may arise as a consequence of the Pauli principle.)
The spectrum of the photon Hamiltonian Hf consists of a simple eigenvalue at 0, corresponding to the vacuum vector 2 F , and absolutely
continuous spectrum (of in nite multiplicity) covering the half-axis [0; 1).
Consequently, by separation of variables, the unperturbed Hamiltonian H0 =
Hel + Hf on F has spectrum
(H0 ) = (Hel ) + (Hf ) :
(I.15)
The point spectrum of H0 is the same as the point spectrum of Hel , i.e., it
consists of the eigenvalues fEj gj=0;1;2;::: (corresponding to the eigenvectors
'j;`
, where f'j;`g`=1;2;:::;n is an orthonormal basis of eigenvectors of Hel
corresponding to the eigenvalue Ej of multiplicity nj ). The continuous spectrum of H0 covers the half-axis [E0; 1) and consists of a union of branches
[Ej ; 1) starting at the eigenvalues Ej and the branch [; 1); as indicated
in Fig. 2.
We note that the ground state energy E0 = inf (Hel ) of the atom or
molecule in the absence of the quantized radiation eld coincides with the
ground state energy E0 E0 ( = 0) = inf (H0) of the system of an atom
j
BFS-3, November 20, 1998
6
or molecule in the presence of photons, but decoupled from them. But, while
E0 is an isolated eigenvalue of Hel , it lies at the tip of a branch of continuous
spectrum of the Hamiltonian H0. Similarly, the energies E1 ; E2; : : : are isolated eigenvalues of Hel; but they are eigenvalues of H0 imbedded in continuous
spectrum of H0, and each Ej is the threshold of a branch of continuous
spectrum of H0 .
These spectral properties of H0 make it a dicult problem to analyze,
mathematically rigorously, the fate of the eigenvalues Ej of H0 , and the
nature of the energy spectrum of the interacting system described by the
Hamiltonian H introduced in (I.5), for > 0. Although the perturbation,
W := H H0, is a small perturbation of H0, general analytical methods
to deal with this type of problem in perturbation theory do not appear to
be available. In [5, 6], we have started to develop such methods, tailor-made
to analyze a class of Hamiltonians describing interactions between nonrelativistic quantum-mechanical matter and the radiation eld. In this paper we
extend those methods to the Hamiltonian H (= H0 + W ) of Eqn. (I.5),
which describes much of the physics of light atoms or molecules interacting
with the quantized electromagnetic eld (within the Born-Oppenheimer approximation). For background material concerning the physics described by
H , see e.g., [11, 12] and references given there.
0
0
0
0
0
0
I.2
Survey of Main Results
In the next section, I.3, we consider the structure and properties of the
perturbation W in the Hamiltonian H = H0 + W of (I.5). The strength
of the perturbation W relative to H0 is measured by the parameter
0
0
0
0
g := ( K )3=2 ;
(I.16)
where K is the ultraviolet cuto in the electromagnetic vector potential A~ , as
described in Eqn. (I.6). The parameter K is a \dimensionless energy scale"
given by the photon energy above which interactions between electrons and
the radiation eld are cut o , divided by 4 Ry. In Section I.3 we prove that if
g is suciently small then the interaction Wg is bounded by H0 , in the sense
of Kato [22, 25], with relative bound strictly smaller than 1. This proves
that, for small , H is bounded from below and self-adjoint on the domain
of H0 ; see Corollary I.2. (Under somewhat weaker assumptions one can prove
that H de nes a semibounded quadratic form on an appropriate core. It
is not easy, however, to characterize the domain of the corresponding selfadjoint operator.) Since g := ( K )3=2 is the relevant coupling parameter, we
0
0
BFS-3, November 20, 1998
henceforth write
7
Hg := H 0 =g2=3 =K ;
Wg := W 0 =g2=3 =K :
(I.17)
We should emphasize that our bounds on g = ( K ) = become poor, as
the number of electrons, N , becomes large. This does not mean that the
Hamiltonian Hg is ill-de ned or unbounded from below, for large values of
, as long as the ultraviolet cuto K is kept nite. In fact, using pathspace methods one can rigorously construct the semi-group exp( tHg ), for
t 0, and prove that it is self-adjoint, strongly continuous in t > 0 with
exp( tHg ) ! 1, as t ! 0, for arbitrary values of and N , as long as
K < 1; (see [17, 30] for various ingredients of the proof). However, for the
purposes of the analysis presented in Section III, Kato- (or quadratic-form-)
boundedness of Wg in terms of H is an important property.
In Section II we address the question whether Hg has a ground state corresponding to an eigenvalue E (g) := inf (Hg ). Our main result is Theorem
II.1, which answers this question in the armative, as long as the e ective
coupling constant g is suciently small.
The method used in Section II is "non-perturbative" but non-constructive.
For this reason, they do not enable us to estimate the multiplicity of the
ground state energy E (g). However, if the number N of electrons is = 1 or
= 2, and if the Zeeman terms in the Hamiltonian Hg are set to 0, then we
can construct an L -space representation of the photon Fock space F (\electric eld" representation) with the property that exp( tHg ) is positivitypreserving on Hel F , where Hel is taken in the usual Schrodinger con guration-space representation, and, for N = 2, only spin-singlet states are
considered. Modulo ergodicity of exp( tHg ) (see [24]), which we have not
checked, a Perron-Frobenius argument then establishes uniqueness of the
groundstate.
It is worthwhile to point out some (fairly standard, but) fundamental
consequences of the existence of a ground state of Hg for scattering theory:
Using straightforward variants of methods developed in [18, 19, 20, 2, 3], one
can construct Moller (wave) operators, , such that the range of ( )
describes asymptotic states of the system consisting of an atom (or molecule)
in a ground state accompanied by an outgoing cloud of freely moving photons.
The obvious conjecture is that Ranf g = Ranf g =: Hasy , where Hasy
is isomorphic to Hg F , and where Hg is the space of ground states
of Hg . This conjecture, called \asymptotic completeness", would imply the
unitarity of the scattering matrix for the scattering of photons o an atom
or molecule below the ionization threshold. We are miles away from proving
this conjecture! But, in a drastically simpler model of massive photons and
con ned electrons, it has recently been proven in [16, 14].
3 2
0
0
0
2
+
(0)
+
(0)
BFS-3, November 20, 1998
8
In Section III, we prove that, outside small neighbourhoods of E0 and
and below , the spectrum of Hg is purely absolutely continuous (Outside
small neighbourhoods of the thresholds of Hel and above , the spectrum of
Hg can be shown to be purely absolutely continuous by using the methods
in [5, 7]. In particular, Hg does not have any eigenvalues in the vicinity of
the imbedded eigenvalues E1 < E2 < < En < , of H0 , where is a
small positive constant depending on g and n. We will, however, make the
idea rather precise that imbedded eigenvalues of H0 give rise to metastable
states of Hg , and we shall estimate the life time, / g2, of these metastable
states up to an error term O(g2+"), for some " > 0.
Our notion of resonance is based on dilatation analyticity. In order to
state our ideas simply, we assume that there is only one atomic nucleus
(M = 1 in Eqn. (I.9)) of atomic number Z = Z1 2N located at R~ 1 = 0;
(the general case of an arbitrary, nite number M of nuclei can be studied,
too, by borrowing ideas developed in [21]). To describe dilatation analyticity,
we start by scaling the positions, ~xj , of the electrons and the momenta, ~k, of
the photons by
~
xj
7!
~
k
e ~
xj ;
7!
e
(I.18)
~
k :
When is real, the transformations (I.18) determine a unitary transformation
U on the Hilbert space H de ned in Eqns. (I.1){(I.4). It is easy to see that
the subspace, D H, of vectors, , with the property that () := U is
analytic in , for jImj < =2, is dense in H. Furthermore, one easily checks
that
U Hf U
=
e
(I.19)
Hf :
These facts, combined with well-known results [1, 8] on dilatation analyticity
for Schrodinger operators, show that, for arbitrary ; ' 2 D, the function
F 0;' (; z )
:=
D
z
H0 ( )
1
E
'( ) ;
(I.20)
where H0 () = U H0U 1 , is independent of , for jImj < =2, and, for
Im =: # xed, F 0;' (; z) is analytic in z in the complement of the shaded
region depicted in Fig. 3
In Section III, we construct the function
F
g
;' (; z ) :=
D
z
Hg ( )
1
E
'( ) ;
(I.21)
where Hg () = U Hg U 1 , with Hg as in (I.5), and we show that, for our
choice of an ultraviolet cuto (see (I.6) and below), and for arbitrary
BFS-3, November 20, 1998
E0
9
E1
E2
Im()
Figure 3: A projection of the Riemann surface of z 7! F 0;' (; z) onto the
energy plane
E0
E0 (g )
E1
Ej
Ej;` (g )
E1 (g )
Figure 4: A projection of the Riemann surface of z 7! F g;' (; z) onto the
energy plane
and ' in D, F g;'(; z) is independent of , for jImj < =4 small enough, and,
for Im =: # xed, F g;' (; z) is analytic in z in the complement of the shaded
region depicted in Fig. 4. It thus provides an analytic continuation of the
matrix element F g;' (0; z) of the resolvent of Hg z in z from the upper half
plane into the lower half plane outside the spectrum of Hg (). This implies
the absolute continuity of the spectrum of Hg for those energies which are
contained in the resolvent set of Hg () (see Corollary III.2).
In Section III, Eqns. (III.7){(III.10), we introduce a notion of resonance
energy Ej (g) corresponding to the energy Ej of the j th excited state of the
atom or molecule. We show that the Lamb shift, RefEj (g) Ej g, is given
by Bethe's formula and ImfEj (g)g is given by Fermi's Golden Rule, to order
g 2 , with error terms that we prove to be O (g 2+" ), for any 0 < " < 1=3. If
there are no miraculous selection rules forbidding the decay of the j th excited
state of the atom to rst order in g then
ImfEj (g)g =
jg
2
+ O(g2+") ;
(I.22)
with j strictly positive.
It then follows by standard reasoning that, for g > 0 suciently small,
the spectrum of Hg is purely absolutely continuous in a neighbourhood of
every eigenvalue Ej , j 1, of Hel for which j > 0.
What we are really looking after, from a physics point of view, is a precise
understanding of the decay of the excited states of theatom under the time
evolution exp[ it Hg ], as t becomes large. Let := dist Ej ; (Hel) nfEj g >
BFS-3, November 20, 1998
10
0, set Ij (=2) := (Ej =2; Ej + =2), and
notice that Ij (=2) is an open
interval containing Ej and such that dist Ij ; (Hel ) n fEj g = =2. Let Fj
denote a smooth characteristic function of Ij (=2) (see Subsection III.1). We
shall identify a "j th excited state" of the atom with a vector of the form
j
:=
1=2 (H
Fj
g
) Pel;j exp
h
g
2H
i
f
(I.23)
(see (III.14)), where Pel;j is the projection onto the eigenspace
of Hel corh
i
2
responding to the eigenvalue Ej , and the operator exp g Hf essentially
eliminates high-energy photons of energy larger than g2 in the state , which
is assumed to be dilatation analytic. An example for a state is given by
=
'j;`
(I.24)
;
where 'j;` is an eigenvector of Hel with corresponding to the eigenvalue Ej .
We then show that, for any 0 < " < 1=3, there exists a constant C 0
and, for any N 2 N , a constant CN 0 such that
j
exp[
B2
C
itHg ] j
ln(1=g) exp
t
g2
Cg 2+" )
j
+
CN t
N
g4
;
(I.25)
where B := 1 + supjj0 k()k. This estimate implies that, given " > 0,
there is a nite constant C" > 0 such that, for t > C" j 1 g 2 ln ln(1=g),
D
j
e itHg
E
j
":
(I.26)
We remark that it is known from other methods (see e.g. [15]) that, given
0, there is a constant D" > 0 such that, for 0 t < D"g 2,
">
D
itHg
j e
E
j
":
(I.27)
Estimate (I:26) shows that
the state j decays, with a life-time bounded
2
above by O g ln ln(1=g) . It is a typical example for the kind of estimates
we are able to prove with the help of the methods developed in Section III.
I.3 Relative Bounds, Self-Adjointness, and Dilatation
Analyticity
We return to Eqn. (I.5), which we write as
Hg
=
H0
+
Wg ;
(I.28)
BFS-3, November 20, 1998
11
where H0 is de ned in (I.11), and we obtain
Wg =
N
X
(
j =1
41=2
+ 2
3=2
~ ~xj ) + 2 3A~ 2( ~xj )
A~ ( ~xj ) ( ir
1=2 5=2
~j r~ A~ ( ~xj )
)
=g2=3 =K
(I.29)
from expanding the square in (I.5). Our rst goal in this subsection is to
prove in Corollary I.2 that Wg is de ned on D(H0) and obeys the bound
Wg jH0 + iC (N )j
1
C 0(N ) ( K )3=2 ;
(I.30)
for some constants C (N ); C 0(N ) 0. This establishes the semiboundedness
and self-adointness of Hg on D(H0) for g := ( K )3=2 < C 0(N ) 1 . Our second
goal is to establish the dilatation analyticity of Wg () := U Wg U 1 (see
Corollary I.3), where U is the dilatation operator de ned in (I.18), i.e., we
prove that 7! Wg () is an analytic function on D(0; 0) := fz 2 C jjzj < 0g
with values in B(D(H0 ); H), the bounded operators from D(H0) to H, for
some 0 > 0. We establish this property by observing that the coupling
functions in Wg () are analytic in , pointwise in the other parameters, and
by verifying a bound similar to (I.30), namely,
Wg () jH0 + iC (N; )j
1
C 0(N; ) g ;
(I.31)
for some constants C (N; ); C 0(N; ) 0. In fact, Eqn. (I.30) is just the special case = 0 in Eqn. (I.31). Finally, we establish the dilatation analyticity
of Hg () := U Hg U 1 in Corollary I.4, assuming that Hel = x + Vc(x) is
the Hamiltonian of an atom, i.e., M = 1. This simplifying assumption could
be avoided by using exterior dilatations [21, 29], at the expense of having to
deal with more involved estimates; we do not carry out this analysis here.
We characterize the coupling functions in Wg () in terms of the following
functions,
G~ ~x(~k; ; ) :=
and
B~ ~x(~k; ; ) :=
p
2 eq=2 (e j~kj=K ) exp[ i ~k ~x] ~" (~k)
K 3 !(~k)
p
(I.32)
2 eq3=2 (e j~kj=K ) exp[ i ~k ~x] ~k ~" (~k) ;
i K 3 !(~k)
(I.33)
BFS-3, November 20, 1998
12
where we introduce a dilatation parameter 2 D(0; 0) C , for some 0 > 0
suciently small. Note that, pointwise for every ~x; ~k 2 R 3 and 2 Z2, the
maps 7! G~ ~x(~k; ; ) and 7! B~ ~x(~k; ; ) are analytic in D(0; 0). Further~ G~ ~x (~k; ; 0). By means of G
~ ~x (~k; ; )
more, we notice that B~ ~x(~k; ; 0) = r
3
and B~ ~x(~k; ; ) we de ne the following functions on R Z2 with values in
the operators on Hel ,
N
X
~
~
~
~
~
~
2G~xj (k; ; ) ~pj + ~j B~xj (k; ; ) ;
w1;0 (k; ; ) := w0;1 (k; ; ) :=
j =1
(I.34)
w2;0 (~k1 ; 1 ; ~k2; 2 ; ) := w0;2 (~k1 ; 1 ; ~k2; 2 ; )
(I.35)
N
X ~ ~
:=
G~xj (k1 ; 1 ; ) G~ ~xj (~k2 ; 2 ; ) ;
j =1
N
X
~
~
~ ~xj (~k2 ; 2 ; )
G~ ~xj (~k1 ; 1 ; ) G
w1;1 (k1 ; 1 ; k2; 2 ; ) :=
j =1
(I.36)
+ G~ ~xj (~k1; 1; ) G~ ~xj (~k2 ; 2; ) ;
and these, in turn, serve as coupling functions for the operators de ned by
Wm;n () :=
X
1 ;:::;m+n =1;2
Z
d3 k1 d3 km+n wm;n (~k1 ; 1 ; : : : ; ~km+n ; m+n ; )
a1 (~k1 ) am (~km ) am+1 (~km+1 ) am+n (~km+n ) : (I.37)
Then we observe that, after normal ordering, the (dilated) interaction Wg ()
reads
Wg () =
X
m+n2
g m+n Wmn () + g 2 Cno ;
(I.38)
R
where Cno is de ned by Cno := 2N jG~ ~x(~k; 1; 0)j2 d3 k, which is independent
of x, and g := ( K )3=2 . Henceforth and consistent with our previous de nitions, we omit in our notation in the undilated case, = 0, writing
~ ~x(~k; ) := G~ ~x (~k; ; 0) ; B~ ~x (~k; ) := B~ ~x (~k; ; 0) ;
G
(I.39)
wm;n (~k1 ; 1 ; : : : ; ~km+n ; m+n ) := wm;n (~k1 ; 1 ; : : : ; ~km+n ; m+n ; 0) ; (I.40)
Wm;n := Wm;n (0) :
(I.41)
BFS-3, November 20, 1998
13
Next, we de ne J : R ! R to be the smallest function such that
w ; (~k; ; ) ( x + 1) = ; w ; (~k; ; ) ( x + 1) = J (~k)
(I.42)
3
+
1 2
10
1 2
01
holds, for all jj and (~k; ) 2 R Z , and such that
3
0
(I.43)
w ; (~k ; ; ~k ; ; ) ;
w ; (~k ; ; ~k ; ; ) J (~k ) J (~k )
w ; (~k ; ; ~k ; ; ) ;
20
1
1
2
2
11
2
1
1
02
2
1
1
2
2
2
1
2
holds, for all jj and (~k ; ); (~k ; ) 2 R Z . Note that, due to
ir~ ~x ( ~x + 1) = , we have that
0
1 2
j
J (~k) C ( ) N K
1
=
3 2
0
j~kj
1
=
1 2
2
3
2
2
(e j~kj=K ) + j~kj (e j~kj=K ) ;
(I.44)
for some constant C ( ) 0. The rapid decay of implies that
0
:=
Z
J (~k) !(~k) d k
2
=
!1 2
3
< 1;
(I.45)
for any > 2. In particular, is uniform in K 1, for any 1 > 2.
This uniformity in K 1 is actually the basic requirement that determines
p = 3=2 in the coupling parameter g = = K p.
The main relative bound that we use is described in the following lemma.
Lemma I.1. For all m; n 2 N with 1 m + n 2 and all 2 D(0; ),
the operators Wm;n() are de ned on D(H ) and obey the bound
3 2
0
0
0
Wm;n() jH + iC (N; )j
0
0
1
4(1 + + )
2
0
2
m+n)=2
(
1
;
(I.46)
for some constant C (N; ) > 0.
Proof: We rst note that the canonical commutation relations (I.8) allow
us to convert estimates on W1;0() into those for W0;1(). Indeed, for any
2 D(H0),
kW ; () k = kW ; () k +
10
2
01
2
X
=1;2
Z
d k kw ; (~k; ; ) k
3
2
10
kW ; () k + 2 ( x + 1) =
01
2
0
1 2
2
:
(I.47)
BFS-3, November 20, 1998
Furthermore,
kW0;1() k
14
X Z
=1;2
2
d3 k J (~k) ( x + 1)1=2 a (k)
1=2
1 ( x + 1)1=2 Hf
;
(I.48)
and hence
kW1;0() k ; kW0;1() k 2 (0 + 1) ( x + Hf + 1) : (I.49)
Similarly, we convert estimates on W2;0 and W1;1 into those for W0;2. For
W2;0 , for instance, the canonical commutation relations (I.8) imply that
W2;0 () W2;0 () = W0;2Z() W0;2 ()
(I.50)
+ 2 d1d2 w2;0(1; 2; )w2;0(1; 2; )
Z
+4 d1d2d3 w2;0(1; 3; ) w2;0(2; 3; ) a(1)a(2) ;
where we denoted := (~k; ), d := P=1;2 d3k, and a#( ) := a# (~k). This
yields
kW2;0() k 3kW0;2() k + 420 k k2 ;
(I.51)
R
R
and, as in (I.48), we obtain that
kW0;2() k 2 1 kHf k 2 1 ( x + Hf + 1) :
(I.52)
The bound (I.46) follows from (I.49), (I.51), (I.52), and the additional use of
the fact that x is relatively Hel -bounded with relative bound arbitrarily
close to 1.
Now self-adjointness of Hg on D(H0) and dilatation analyticity of Wg ()
are just two immediate implications of Lemma I.1.
3=2 < (1 + 2 + 2 ) 1=2 =10 then Hg is
Corollary I.2. If 0 < g = ( K )
0
1
self-adjoint and semibounded on D(H0 ).
Corollary I.3. The map Wg : D (0; 0 ) ! B (D (H0 ); H), 7! Wg ( ) is analytic.
Finally, we establish the dilatation analyticity of Hg () := U Hg U 1 , assuming that the potential Vc(x) is dilatation analytic, i.e., D(0; 0) 3 !
Vc (e x) 2 B(D(H0 ); H) is an analytic function. This property holds in case
BFS-3, November 20, 1998
15
that Hel = x + Vc(x) is the Hamiltonian of an atom, i.e., M = 1, for arbitrary . Indeed, in the atomic case we may choose without loss of generality
the position of the nucleus to be the origin of the one-electron con guration
space, and then we obtain
Hel () := U Hel U 1 = e 2 x ) + e Vc(x) :
(I.53)
H0 () := U H0 U 1 = Hel () + e Hf
(I.54)
Therefore,
is an analytic family of type A and, by Corollary I.3, so is Hg () = H0() +
Wg (). We summarize this discussion and a simple consequence of (I.54) in
the following corollary.
Corollary I.4. The family fHg ()j 2 D(0; 0)g is dilatation analytic, i.e.,
the map Hg : D(0; 0 ) ! B(D(H0 ); H), 7! Hg () is analytic. Moreover,
there exists a constant b 0 such that
Hel () (Hel i) 1 b jj ;
(I.55)
where Hel () := Hel() Hel , for all 2 D(0; 0 ).
II Soft Photon Bound
and Existence of a Ground State
In this section we derive a new soft photon bound; see Inequalities (II.5){(II.6)
and Theorem II.3 below. It is taylored for the minimal coupling model, and
we use this bound to prove the existence of a ground state.
Theorem II.1. There exists a constant C (N; E0 ) 0 such that, for all
0 < g = ( K )3=2 C (N; E0), the Hamiltonian Hg has a ground state,
i.e., E0 ( ) := inf (Hg ) is an eigenvalue.
T
Proof: We introduce the notation a# (F~ ) := a# (F1 ); a# (F2 ); a# (F3) ,
R
R
P
P
a (F ) := =1;2 d3 k F (~k; )a (~k), and a(F ) := =1;2 d3 k F (~k; ) a (~k).
Here, F is a function on R 3 Z2 with values in the operators on Hel such
2
R
that P=1;2 d3k F (~k; )( x + 1) 1=2 < 1. We further denote (F ) :=
BFS-3, November 20, 1998
16
~ ~x . In this notation, the
a (F ) + a(F ), (F~ ) := a (F~ ) + a(F~ ), and ~pj := ir
interaction Wg reads
j
Wg =
N
X
2g(G~ ~x ) p~j + g (G~ ~x ) + ~j g(B~ ~x ) ;
2
j
j =1
2
j
(II.1)
j
where G~ and B~ are de ned in Eqns. (I.32), (I.33), and (I.39).
Next, we introduce an infrared regularization by switching o the interaction for photons of small momenta. Speci cally, we pick a \photon mass",
m > 0, that is, we replace G~ ~x(~k; ), B~ ~x(~k; ), and Wg in (I.32), (I.33), and
(II.1) by
Wg m :=
(
)
PN
j =1
G~ ~xm (~k; ) := [!(k) m] G~ ~x(~k; ) ;
(II.2)
m
B~ (~k; ) := [!(k) m] B~ ~x(~k; ) ;
(II.3)
~x
2g (G~ ~xm ) p~j + g (G~ ~xm ) + g ~j (B~ ~xm ) ;(II.4)
(
)
(
)
(
)
(
2
j
) 2
(
j
)
j
and we denote Hgm := H + Wg m and E m (g) := inf (Hgm ). We remark
that Hgm ! Hg = Hg in norm resolvent sense, as m ! 0. This easily
follows from an estimate similar to Lemma I.1 (see [5]). In Theorem II.2
below we show that, for g = ( K ) = suciently small, Hgm has a ground
state, m , i.e., there exists a normalized solution of Hgm m = E m (g) m,
for all m > 0. Since k mk = 1, the family f mgm> contains a weakly
convergent subsequence, f m n gn2N , where limn!1 m(n) = 0. We put :=
w limn!1 m n . Then one easily shows [5], that 2 D(Hg ) and that
Hg = E ( ) . To conclude, it remains to show that 6= 0.
To show that 6= 0, we employ a soft photon bound, as in [5]. There is an
important di erence, though. The soft photon boundPin [5]Restimated the photon number expectation h mjNf m i, where Nf := ; a (~k)a(~k)d k, in
terms of sup~x k! G~ ~xk . It was derived from a virial type argument, using
the commutator of a (~k) and Hg . This bound does not directly apply to the
present problem because k! G~ ~xk = 1, for all ~x. Modifying the argument
slightly by using the commutator of a (~k) iFx(~k; ) and Hg , for a suitably
chosen Fx, we avoid the appearance of sup~x k! G~ ~xk on the right side of
the estimate, which we trade for a factor of k jxj m k.
More precisely, in Theorem II.3 below we show that
(
(
)
)
(
0
(
0
)
)
(
)
(
)
(0)
3 2
(
)
(
0
)
0
0
( )
( )
0
0
0
0
0
0
3
=1 2
1
2
1
2
1
2
2
h mj Nf mi C (N ) g (1 + jxj) m ;
1
2
(II.5)
BFS-3, November 20, 1998
17
for some constant C1(N ) 0. In [5] we showed that m is exponentially
localized in the electron variables. More generally, there exists an " > 0 such
that
e"jxj (Hg(m) ) C2 < 1 ;
(II.6)
where (Hg(m) ) is the spectral projection onto := ( 1 ; ( + E0 )=2),
provided that g2(1+ E0 ) is suciently small (with C2 and " independent
of g). Since m 2 Ranf (Hg(m) )g, this implies the boundedness of k jxj mk.
Thus there exists a constant, C3 C3(N; E0 ), such that
h mj Nf mi C3 g2 ;
(II.7)
for all m > 0. Next, we introduce the projection Pel onto all bound states of
Hel below 21 (+E0) < . Note that Pel has nite rank and that ( E0 )Pel?
2(H0 E0 ). The latter implies that
h mj Pel? mi 2( E0 ) 1 h mj (E0(m) (g) E0 Wg(m) ) mi
C3 g 2 :
(II.8)
From (II.7){(II.8) and Pel P 1 Pel? Nf we draw the important consequence that
h mj Pel P m i 1
2C3 g2 ;
(II.9)
where P := j ih j denotes the rank-one projection onto the photon vacuum
vector . Thus, if g = ( K )3=2 is suciently small then h mj Pel P mi
1=2, for all m > 0. Since Pel P has nite rank, it follows that 0 6= 0.
In the following theorem, we review the proof in [5] of the existence of a
ground state of Hg(m) , for m > 0.
Theorem II.2. There exists a constant C C (N; E0 ) > 0 such that,
for all 0 < g = ( K )3=2 C , the Hamiltonian Hg(m) has a ground state, i.e.,
E0(m) (g) := inf (Hg(m) ) is an eigenvalue, for any m > 0.
Proof: We only sketch the argument, see [5, Section II.2] for details.
Alternatively, one may proceed as in [16]. The assertion is proven if we can
nd some m~ > 0 such that the
sum of the negative eigenvalues
of Hg(m)
n
o
E0(m) (g) m~ is nite, i.e., Tr [Hg(m) E0(m) (g) m~ ] > 1, where the
negative part of a real number is de ned as [] := minf; 0g. To this
BFS-3, November 20, 1998
18
end, we rst employ a discretization. Given " > 0 and a locally integrable
function F , we de ne its "-average by
hF (~k)i" := " 3
Z
~n(~k)+Q"
F (~k0; )d3k0 ;
(II.10)
where Q" = [ "=2; "=2)3 and ~n(~k) 2 ("Z)3 is the integer part of ~k, i.e.,
~k ~n(~k) 2 Q". We de ne the corresponding interaction, Wg(m;"), by replacing
~G~x(m) (~k; ) and B~ ~x(m) (~k; ) in (II.2){(II.4) by G~ ~x(m) (~k; ) and B~ ~x(m) (~k; ) ,
"
"
respectively. Then, by the bound (I.46) it follows that
(Wg(m;") Wg(m) )
(II.11)
o("0) (Hg(m) E0(m) (g) + 1)1=2 1 + jxj (Hg(m) E0(m) (g) + 1)1=2 ;
where o("0) denotes a function which possibly depends on g, N , , m and
tends to zero as " ! 0. Here, our original manuscript contained a small
mistake in that the factor 1 + jxj was missing, as was kindly pointed out to
us by F. Hiroshima. Next, we de ne Hf(m;") by replacing !(~k) in Hf(m) by
h!(~k)i". Since j!(~k) h!(~k)i"j = O(") and !(k) m, we obtain that
1
o("0)
H (m)
f
H (m;")
f
1 + o("0)
Hf(m) :
(II.12)
Denoting Hg(m;") := Hel + Hf(m;") + Wg(m;") , we hence obtain that
Hg(m)
1 o("0) Hg(m;")
(II.13)
o("0) (Hg(m) E0(m) (g) + 1)1=2 (1 + jxj) (Hg(m) E0(m) (g) + 1)1=2 :
Next, we introduce the interval e := ( 1 ; E0(m) (g) + m~ ) and observe that
[Hg(m) E0(m) (g) m~ ] = e (Hg(m) ) Hg(m) E0(m) (g) m~ e (Hg(m) ) :
(II.14)
Furthermore, we note that, thanks to (II.6) and e = e , for m~ > 0
suciently small, we have
e (Hg(m) ) (Hg(m) E0(m) (g) + 1)1=2
(II.15)
1 + jxj (Hg(m) E0(m) (g) + 1)1=2 e (Hg(m) ) < 1 :
BFS-3, November 20, 1998
Thus,
n
19
o
Tr [Hg(m) E0(m) (g) m~ ]
n
o
Tr e (Hg(m) ) Hg(m;") E0(m;")(g) m~ o("0) e (Hg(m))
n
h
i
o
Tr e (Hg(m) ) Hg(m;") E0(m;") (g) m~ o("0) e (Hg(m) )
nh
i o
Tr Hg(m;") E0(m;") (g) m=
~ 2 ;
(II.16)
for " > 0 suciently small, and the niteness of the right side in (II.16) (and
hence the claim) follows if we can show that, for any " > 0, the discretized
Hamiltonian Hg(m;") has only nitely many eigenvalues below E0(m;") (g)+ m=
~ 2.
The key point of the discretization by means of the "-average is the tensor
? ], where Hdisc is spanned
product representation F
= F [Hdisc] F [Hdisc
by Q"+~n, ~n 2 ("Z)3 (see [5] for details). Note that, with respect to this
representation, we have Hf(m;")
= Wg(m;") 1,
= Hf(m;") 1 + 1 Hf(m;") , Wg(m;")
and hence
(II.17)
Hg(m;")
= Hg(m;") 1 + 1 Hf(m;")
E0(m;") (g) + m P ? + Hg(m;") P ;
? ] by P . Hence, for
denoting the projection onto the vacuum in F [Hdisc
m~ m,
Hg(m;") E0(m;") (g) m=
~ 2 Hg(m;") E0(m;") (g) m=
~ 2 P : (II.18)
Next, for suciently small " > 0, an estimate similar to Lemma I.1 together
with an interpolation argument implies that, on Hel F [Hdisc], we have
Wg(m;") C10 g Hel + Hf(m;") + C (N ) ;
(II.19)
for some constants C10 ; C (N ) 0. Thus, we obtain that
Hg(m;") E0(m;") (g)
(II.20)
(1 C10 g)Hel (1 + C10 g)E0 + (1 C10 g)Hf(m;") 2C10 C (N )
n
o
21 ( E0 ) 2C10 g(1 + 2jE0j) o("0) Pel?
n
o
+ (1 C10 g)Hf(m;") 2C10 g(1 + 2jE0j) o("0) Pel ;
where Pel is the nite dimensional projection onto the bound states of Hel of
energy < 21 ( + E0 ). Now, if g is suciently small such that
m^ := 21 ( E0 ) 2C10 g(1 + 2jE0j) > 0
(II.21)
BFS-3, November 20, 1998
20
then, for any 0 < m~ < minfm; m^ g, we obtain from inserting (II.20) into
(II.18) that
Hg(m) E0(m) (g) m=
~ 2
n
(1 o("0) 2C10 g)Hf(m;")
(II.22)
P ;
o
2C10 g(1 + 2jE0j) o("0) Pel
for " > 0 suciently small. The right side, however, has clearly only nitely
many negative eigenvalues, for any " > 0, which proves that, for any " > 0,
the discretized Hamiltonian Hg(m;") has only nitely many eigenvalues below
E0(m;") (g) + m=
~ 2.
TheoremPII.3. RLet m be a ground state of Hg(m) , for m > 0, and denote
by Nf := =1;2 a(~k)a (~k)d3 k the photon number operator. Assume that
(K + 1) 1. Then there exist constants C (N ); C1 (N ) 0 such that
(Z
1
2
~
~
~
h m j N f m i
sup Tr33 jr~xG~x(k; )j 1 + ~ 2 d3k
~x;
!(k)
3 )
Z
Z
2
+ jG~ ~x(~k; )j2d3k + jB~ ~x(~k; )j2 + j~xG~ ~x(~k; )j2 d~k 2 (1 + jxj) m
! (k )
2
(II.23)
C1(N ) g2 (1 + jxj) m :
C (N ) g2
Before giving the proof of Theorem II.3, we remark that using the de nitions (I.32), (I.33), and (I.39) of G~ and B~ , one easily checks the integrals
on the right side of Eqn. (II.23) to be bounded by a constant, uniformly in
K 1.
Proof of Theorem II.3: Throughout the proof we omit the superscript
\(m)". To prove the asserted bound, we rst observe the following commutation relations,
a (~k) Hg =
Hg + !(~k) a(~k)
+
N
X
(
j =1
(II.24)
)
~j gB~ ~x (~k; )
2gG~ ~x (~k; ) ~pj g(G~ ~x ) ;
j
j
j
F~x (~k; ) Hg = Hg F~x (~k; )
(II.25)
~ ~x F~x (~k; ) p~j g(G~ ~x ) ;
+ ~x F~x (~k; ) + 2i r
j
j
j
j
j
j
j
21
BFS-3, November 20, 1998
on C0(R 3N ; F ) \D(H0), for any F 2 C 2(R 3 ; L2(R 3 Z2)). We therefore have
N
a (~k) i F~x (~k; ) Hg E0 ( )
(II.26)
!
X
j
j =1
=
E0 (
Hg
+
) + !(~k)
a (~k)
E0 (
i Hg
)
N(
X
j =1
N
X
j =1
F~xj (~k; )
!
~j g B~ ~xj (~k; ) + ~xj F~xj (~k; )
N
+ R ! (k )
X
2
j =1
r~ ~xj F~xj (~k; )
!
X
(
j
~j g B~ ~xj (~k; ) + ~xj F~xj (~k; )
j =1
~ ~xj )
g (G
~pj
~ ~xj (~k; )
gG
)
2 p~j
;
We apply (II.26) to (jxj=R) m, where 2 C01(R ; [0; 1]) and 1 on [ 1; 1].
Letting R ! 1 and using (Hg E0 ( )) m = 0, we derive that
N
F~x (~k; ) m
(II.27)
a (~k) m = iR ! (k) Hg E0 ( )
~ ~xj )
g (G
)
gG~ ~x (~k; ) r~ ~x F~x (~k; ) m ;
j
j
j
where R(!) := Hg E0 ( ) + ! 1 . We choose F~x(~k; ) := g ~x G~ ~x(~k; ).
Then (II.27) reads
N
~
~xj G~ ~x (~k; ) m
(II.28)
a (k) m = igR ! (k) Hg E0 ( )
+ R !(k)
N
j =1
(
X
j =1
!
X
j
~j g B~ ~xj (~k; ) + ig~xj ~xj G~ ~xj (~k; )
+ 2g (p~j
)
~ ~xj G
~ ~xj (~k; ) ~xj m :
g (G~ ~xj )) r
We observe that R !(k) Hg E0( ) 1 and that Nj=1 ~xj G~ ~x
1=2
jxj Nj=1 jG~ ~x j2 , denoting jxj2 := Nj=1(~xj )2 . Hence, for 2 f1; 2g,
P
j
P
P
j
Z
d3 k R ! (k) Hg
E0 (
)
N
X
j =1
N sup
~x
~ ~xj (~k; ) m
~xj G
(
Z
jG~ ~x(~k; )j2d3 k
(II.29)
=2
)1
jxj m :
BFS-3, November 20, 1998
22
Additionally using k R !(k) k !(~k) 1, we similarly obtain
Z
d k R ! (k)
3
N sup
N
X
~ ~xj (~k; ) m
~j B~ ~xj (~k; ) + i~xj ~xj G
j =1
(Z
jB~ ~x(~k; )j + j~xG~ ~x(~k; )j
~x
2
2
(II.30)
)
d3 k 1=2
(1 + jxj) m :
! (~k)2
Furthermore, we note that
R ! (k) (Hg
R ! (k)
N
X
j =1
2
g (G~ ~xj )
p~j
E0 ( ) + 1)
1 + !(~k) 1 ;
(II.31)
C (N ) ;
(II.32)
for some constant C (N ) 0 which only depends on N . These bounds yield
the following estimate,
Z
d k R ! (k )
3
N
X
j =1
N sup
~x
(Z
(p~j
)
~ ~xj G
~ ~xj (~k; ) ~xj m
g (G~ ~xj )) r
~ ~xG~ ~x (~k; )j2
Tr33 jr
1 + !(~k)
2
d3 k
(II.33)
)1=2
jxj m :
Thus we arrive at the assertion.
III Resonances and Time-Decay Estimates
In the present section, we study spectral properties of the Hamiltonian Hg .
We also study the propagator exp[ itHg ] applied to states whose spectral
support is localized about the excited atomic energy level Ej , j 1. As we
describe in the introduction, our main tool for this analysis is the complex
dilatation, Hg () = U Hg U 1 , of the Hamiltonian Hg , where U is the dilatation de ned in (I.18). We prove below that, for = i#, 0 < #, and g #,
a complex neighbourhood of an interval Ij about Ej does not contain any
spectrum of Hg (). By the dilatation analyticity of Hg () in , this implies
that the spectrum of Hg in Ij is purely absolutely continuous,nand it allows
o
for an estimate of the time decay rate of certain states in Ran I (Hg ) .
j
BFS-3, November 20, 1998
23
To state this estimate more precisely, we recall from (I.32){(I.38) that
Hg () = H0 () + Wg () ;
(III.1)
H0 () = Hel () + e Hf ;
(III.2)
2
X
Wg () =
m+n=1
g m+n Wm;n () + g 2 Cno ;
(III.3)
R
where g := ( K )3=2 , and Cno := 2N jG~ ~x(~k; 1)j2 d3k is an energy shift resulting from normal-ordering Wg . We absorb this constant by rede ning
Wg () ! Wg () + g 2 Cno ;
Hel () ! Hel () g 2 Cno ;
(III.4)
and since it only shifts all energies by Cno, we henceforth ignore the constant
Cno by setting it equal to zero. Thus we obtain
Hg () := U Hg U 1 = H0 () + Wg () = H0 () +
X
m+n2
g m+n Wm;n () :
(III.5)
Recall that we assumed in Eqn. (I.14) that the j th atomic energy level is
of nite degeneracy nj and isolated from the rest of the spectrum of Hel by
a positive distance
n
o
:= dist Ej ; (Hel ) n fEj g > 0 :
(III.6)
As in [5, Eqns. (IV.84), (IV.85)], we de ne two nj nj matrices by
:=
j
Z d ( )
j
X Z
?
U Pel;j w0;1 (~k; )Pel;j
(III.7)
=1;2
1
? w (~k; )P U 1 d3 k ;
~
Hel Ej + ! (k) i0 Pel;j
1;0
el;j
Z od ()
:=
X Z
d3 k
U Pel;j w0;1 (~k; ) Pel;j w1;0 (~k; )Pel;j U 1 ~ ; (III.8)
! (k)
=1;2
Pnj
`=1 j'j;`ih'j;`j
where Pel;j :=
is the projection onto the eigenspace of Hel
corresponding to the eigenvalue Ej . Note that the matrices Zjod () and Zjd()
are similar to Zjod(0) and Zjd (0), respectively, for all 2 D(0; 0). We remark
that Zjd(0) and ImfZjod(0)g (Zjod (0) Zjod(0))=2i are self-adjoint. We
require ImfZjod()g to be invertible and to obey
0 < c
j
n
o
:= min (ImfZjod (0)g) c0 ;
(III.9)
BFS-3, November 20, 1998
24
for some constants 0 < c c0. Given a small constant " > 0, and a large
constant C > 0, we de ne the following union of \comet-shaped" sets Rj
Rj ("; C ) C in the lower halfplane,
Rj :=
Ej
+ g2(Zjd())
ig 2 j
+e
+ D(0; Cg2+") ;
R+
(III.10)
Furthermore, for any 0 < < , we de ne the interval
Ij () := ( Ej ; Ej + ) ;
and we remark that distfIj (=2) ; (Hel ) n fEj g g =2.
(III.11)
Now we are ready to formulate the rst main spectral result of this section
Theorem III.1. Let 0 < " < 1=3. For = i# and # > 0 and g > 0
suciently small, there exist constants c; C > 0 such that
Aj := Ij (=2) + i[
where
that
Hg ( )
cg 2 " ;
1) n Rj ("; C )
Hg ( ) ;
(III.12)
is the resolvent set of Hg (), and for any z 2 Aj , we have
Hg ( )
z
1
n
C 0 dist z ;
Rj
o 1
(III.13)
;
for some C 0 > 0.
Theorem III.1 has two important consequences. The rst is immediate
from the analytic continuation in , and we state it as a corollary.
Corollary III.2. For g > 0 suciently small, the spectrum of Hg in Ij (=2)
is absolutely continuous, Ij (=2) ac (Hg ).
III.1 Time-Decay Estimates
To formulate the second main result of this section,
of
another consequence
1
Theorem III.1, we pick a smooth function F 2 C0 [0; 1=2); [0; 1] such that
F 1 on [0; 1=4], and we de ne Fj 2 C01 Ij (=2) ; [0; 1] , with Fj 1 on
Ij (=4), by Fj () := F ( 1j Ej j). Furthermore, for 2 H, we set
j
j ( ) :=
Pel;j
exp
h
g
2H i :
(III.14)
f
Theorem III.3. Let 2 H be normalized, dilatation analytic in D(0; 0).
Denote () := U , and assume that B := 1 + supjj k ()k < 1.
0
Moreover, assume that g > 0 is suciently small, t > 1, and 0 < " < 1=3.
BFS-3, November 20, 1998
25
Then there exist a constant C 0 and, for any N 2 N , a constant CN 0
such that
exp[ itHg ] Fj (Hg )
j
j
B C ln(1=g) exp t g
2
2
j
Cg
2+"
) + CN t
N
g
4
: (III.15)
Proof of Theorem III.3, given Theorem III.1: We rst use the fact that
Fj () =
1
Z =2
F 0 (!= ) Ij (!) () d!
=4
(III.16)
to rewrite the matrix element on the left side of (III.15) as
j
exp[ itHg ] Fj (Hg )
=
=
1
Z =2
=4
Z =2
=4
j
F 0 (!= )
F 0 (!= ) Z
e
Ij ( ! )
exp[ itHg ] I
j
! (Hg )
j( )
it
D
Hg ()
j ( )
Im
j
(III.17)
d!
1
j ( )
E
d d! ;
where we use Stone's formula (see, e.g., [26]) together with Theorem III.1,
which implies that the limit lim"& (Hg () i") = (Hg () ) exists
and is bounded. Indeed, Theorem III.1 even implies that is 7! (Hg ()
+ is) is bounded analytic, provided s 2 R is not too large. We exploit
this fact by deforming the integration contour Ij (!) C into the lower
half-plane. To this end, we de ne a number
1
0
1
1
S := g 2
(III.18)
CS g 2+" ;
j
where CS > 0 is later chosen suciently large, and we assume that 2
Ij (=2) and s 2 [ S; g ]. Then, Theorem III.1 yields that
2
Hg () is
1
C0
n
dist + is ; Rj
o 1
;
(III.19)
for g = ( K ) = suciently small and some C 0 0, where Rj is de ned in
(III.10) above. Since Zj () is similar to a self-adjoint nj nj -matrix, we may
write
3 2
d
Rj :=
[
2
e (Zjd ( ))
R j ( e) ;
(III.20)
BFS-3, November 20, 1998
26
with
Rj (e) := Ej + g e ig
2
2
j
+ e R + D(0; Cg ") :
(III.21)
2+
+
Hence
Hg () is
1
2
o 1
n
2
min dist + is ; Rj (e)
e (Zjd ( ))
n
2 e2max
dist Ej eg + iCS g " ; e
Z
2
2+
i#
( jd ( ))
R+
o
Cg
(III.22)
2+"
1
;
where Cg " takes the disc D(0; Cg ") in the de nition of Rj (e) into account. Thus, choosing CS in (III.18) larger than 4C , we obtain that
2+
2+
1
Hg () is
( Ej eg ) + (CS g "=4)
2# e2max
Z
2 2
2+
2
( jd ( ))
1=2
:
(III.23)
Therefore, z 7! (Hg () z) is analytic in the rectangular domain Ij (=2)+
i[ S; g ], and by Cauchy's integral formula, Eqn. (III.17) can be written as
1
2
j
exp[ itHg ] Fj (Hg )
where
A :=
Z =2
=4
D
Z =2
=4
D
j
= A
A
+
Ak ;
(III.24)
F 0 (!=) S exp[ it(E ! is)]
j
4i
Hg () Ej ! + is
j ( )
j ( )
Z
(III.25)
0
1
E
Hg () Ej ! + is
j ( )
F 0 (!=) Z exp[ it( iS )]
4i Ij !
E
Hg () + iS
j ( )
j ( )
D
Ak :=
1
j ( )
E
ds d! ;
(III.26)
( )
1
D
Hg () + iS
j ( )
1
j ( )
E
d d! :
We rst estimate Ak. Since Pel;j and e Hf =g2 are dilatation analytic and
k()k B , we have for all jj that k j ()kk j ()k C B , for some
constant C depending on . Inserting this and (III.23) into the de nition of
0
0
2
BFS-3, November 20, 1998
27
Ak , we obtain that
0
=
)
jAkj = = F 2(!=
I ! exp[ it( iS )]
+ iS
j ( ) Hg ( )
j ( )
Z
Z
2
4
j( )
D
Hg () + iS
j ( )
D
C B e
2
tS
2
1
max
Ij (=2)
h
E
j ( )
1
d d!
Z
e (Zjd ( ))
E
(Ej + g e ) + (CS g =4)
2
2+"
2
2
1=2
d
i
C 0 B ln(1=g) exp t(g j CS g ") ;
(III.27)
for some constants C; C 0 0 which depend on , , and nj . Note that
2
2
2+
0
Hg () + iS is bounded invertible because the spectrum of Hg () = Hg ()
lies in the upper half plane.
Second, we estimate A . We omit the similar estimate on A . We use
the fact that, for any N 2 N and t > 0,
+
h
i
exp it(Ej + ! is) = ( it)
Thus, an integration by parts yields
N
dN exp it(E + ! is) : (III.28)
j
d!N
i
h
i
e it E ! is
(III.29)
A = 2i
tN
=
dN F 0(!=)
Hg () Ej ! + is
j ( )
j ( )
d!N
Hg () Ej ! + is
ds d!
j ( )
j ( )
i N S = e it E ! is N N ! N k F N k (!=)
= 2i
tN
(N k)!
=
k
k
Ej ! + is
j ( ) Hg ( )
j ( )
Z S Z =2
N
+
( j+
4
D
0
)
1
D
Z
Z
0
D
( j+
)
4
X
=0
+
Hg () Ej ! + is
j ( )
D
E
1
2
(
+1)
E
1
k
E
1
E
j ( )
ds d! :
Since all derivatives of F are bounded and S g j , there exists a constant
CN 0 such that
D
E
jA j CNNg sup j () Hg () Ej ! + is k j ()
2
2
+
t
1
D
Hg () Ej ! + is
j ( )
k
1
j ( )
E
0 k N ; ! ; 0 s S : (III.30)
4
2
BFS-3, November 20, 1998
28
Thus, Theorem III.3 follows if we can nd a constant C > 0 such that
D
j
()
Rg ( )k+1 j ( )
E
D
j
()
Rg ()k+1 j ()
E
C B2 g 2 ;
(III.31)
for all k 2 f0; 1; : : : ; N g, ! 2 [=4; =2], and s 2 [0; S ], where we denote
Rg ( )
:=
H g ( )
Ej
!
+ is
1
1
:
(III.32)
;
(III.33)
To this end we introduce an unperturbed resolvent,
Q0 ( )
:=
H0 ( )
Ej
ig 2
!
and we observe that, for g=jj suciently small,
j j 1;
Wg ( ) Q0 ( )
Cgjj 1 ;
Q 0 ( )
Rg ( ) ;
Wg ( ) Rg ( ) ;
(III.34)
(III.35)
C
and some constant C > 0, which is uniform in ! 2 [=4; =2] and s 2 [0; S ].
Using the second resolvent equation, we obtain that
Rg ( )
=
i
h
Q0 ( ) Wg ( ) + is + ig 2 Q0 ( )
(III.36)
+ Q0 () Wg () + is + ig2 Rg () Wg () + is + ig2 Q0 () :
Q 0 ( )
h
i
h
i
We expand Rg ()k+1 by means of (III.36),
Rg ( )k+1
=
Q0 ( )k+1
kX
+1
=1
Q0 ( ) Wg ( ) Q0 ( )k+2
+ Rem ; (III.37)
and (III.34){(III.35) show that there is a constant C 0, depending on k
and , such that
k Rem k
C g2 :
(III.38)
Similarly, we nd that
Rg ()k+1
=
Q0 ( )k+1
kX
+1
=1
Q0 ( ) Wg () Q0 ( )k+2
k Rem0 k
C g2
+ Rem0 ; (III.39)
(III.40)
BFS-3, November 20, 1998
29
(note that Q0() 6= Q0 ()). Inserting the two identities (III.37) and (III.39)
into (III.31) and using (III.38) and (III.40), we observe that it suces to
prove that there is a constant C 0 such that
D
Im
j ()
Q0 ( )k+1 j ( )
E
k+2 ()
j () Q0 () Wg () Q0 ()
j
D
E
C B2 g 2 ;
(III.41)
C B2 g 2 ;
(III.42)
for all k 2 f0; 1; : : : ; N g, 2 f1; 2; : : : ; k + 1g, and ! 2 [=4; =2]. We
remark that (III.41) and (III.42) hold trivially for j = 'j , where 'j is
an normalized eigenvector of Hel corresponding to the eigenvalue Ej .
To prove (III.42), we observe that, for any two vectors 2 D(Hfm=2),
n=2
' 2 D (Hf ), we have
D
E
Pel;j ( ) Wg ( ) Pel;j ( ) '
2
X
m+n=1
g m+n
C
C0
(III.43)
Pel;j ( ) wm;n ( (m) ; ~(n) ; ) Pel;j ( )
ka( (m) ) k ka(~(n))'kd (m) d~(n)
2
X
Z
m+n=1
Z
g m+n
2
X
(
m+n)=2 D
!(
! (k )
1g
m+n=1
J (k )2 d3 k
)m+n
Hfm
E D
' Hfn '
E
n=2 ' ;
m=2
Hf
Hf
P2
P2
by
Schwarz'
inequality.
Here
we
abbreviate
the
summation
=1
m =1
1
R
d3 k1 d3 km by d (m) , and (m) := (k1 ; 1 ; : : : ; km ; m ). Thus we have
k+2 ()
(III.44)
j
j () Q0 () Wg () Q0 ()
D
E
k+2 ()
=
Q0 ( )
j () Pel;j () Wg () Pel;j () Q0 ()
j
n=
2
m=
2
k
+2
m
+n
g
Hf
Q0 ( )
C mmax
j () ;
j () Hf Q0 ()
+n=1;2
E
D
Next, we observe that
Q0 ( )
j ()
=
e
H
f
! + ig 2
exp
h
i
g 2 e Hf Pel;j () () ;
(III.45)
BFS-3, November 20, 1998
30
and hence, for suitable constants C; C 0; C 00 0,
m=2 Q () ()
Hf
0
j
C B
sup
C0 B
1
r0
#
r m=2 ei# r
sup
r0
r m=2
! + ig 2
exp
h
exp
g e# r
2
i
h
(III.46)
g 2 ei# r
i
C 00 B # 1 g 2 e#
m=2
:
Inserting this and a similar estimate for k Hfn=2 Q0()k+2 j ()k into (III.44),
we obtain that
o
n
D
E
2 m+n
W () Q ()k+2 ()
2 2
(
Cg
)
(
)
Q
(
)
j
j
B
max
j
0
g
0
j
m+n=1;2
= C jj 2 B2 g2 ;
(III.47)
for some constant C 0 and g > 0 suciently small. This proves (III.42).
Finally, we establish (III.41) by using the fact that we may analytically
continue in , since the spectral parameter Ej + ! + ig2 in Q0 () is in the
upper half-plane. Thus
E
E
D
D
2 k 1
k+1 () =
j ()
j
j () (e Hf ! ig )
j () Q0 ()
D
E
= j (0) (Hf ! ig2) k 1 j (0)
(III.48)
E
D
= Pel;j e 2Hf =g2 (Hf ! ig2) k 1 Pel;j :
Therefore,
o
n
E
D
2r=g2
2 k 1
k
+1
Im (r ! ig )
:
Im j () Q0 ()
sup e
j ( )
r0
(III.49)
Now, we use that ! =4. If r !=2 then ! r =8 and thus
8g2 :
r ! ig 2 )j
(III.50)
j arg(r ! ig2)j jjIm(
Re(r ! ig2)j
Hence, for r !=2,
n
o
k+2
exp[ 2g 2r] Im (r ! ig2) k 1 (k + 1) 8= g2 : (III.51)
We point out that only for the derivation of (III.51) we need to estimate the
imaginary part of a matrix element rather than its magnitude. It remains to
consider the case r !=2 =8. We estimate as follows,
o
n
exp[ 2g 2r] Im (r ! ig2) k 1 g 2k 2 exp[ g 2=8] C g2 ;
(III.52)
BFS-3, November 20, 1998
31
for some constant C 0. Inserting (III.51), (III.52) into (III.49), we obtain
(III.41) which, together with (III.42), nishes the proof.
III.2 Resolvent Norm Estimates and the Proof of Theorem III.1
The purpose of this subsection is to prove Theorem III.1, i.e., that Hg () z
is invertible for z 2 Aj , where Aj is de ned in (III.12), and its inverse obeys
the norm estimate
Hg ( )
z
n
C dist
1
z;
Rj
o
;
(III.53)
for some C > 0 and with Rj as de ned in (III.10). Before we turn to the
actual proof, we outline our strategy. We distinguish the spectral parameters
z close to the eigenvalue Ej of Hel ( ) from those which are away from Ej by
a certain minimal distance, =2, where
0
0
:=
2"
g2
(III.54)
;
and 0 < " < 1=3 is arbitrary but xed. Note that, given any ; #; c > 0, we
have
0
sin(#=2) and
0
cg 2
"
;
(III.55)
provided g 0 is suciently small.
For z 2 Aj n D(Ej ; =2), a Neumann series expansion shows the invertibility of Hg () z and proves (III.53) for that region. This is proven in
Lemma III.12 below.
For z 2 D(Ej ; =2), the situation is more delicate, and we contruct the
inverse of Hg () z by means of the Feshbach map discussed in detail in
[5, 6]. For this construction, we specify a partition of unity given by the
(non-orthogonal) projections
0
P ( )
:=
Pel;j ( )
Hf <0
and
P ( )
:= 1
P ( ) ;
(III.56)
where Pel;j () := U Pel;j U and Pel;j is the (orthogonal) projection onto the
eigenspace of Hel corresponding to the eigenvalue Ej . In Lemma III.9 below
we prove that
1
Hg ( )P ()
z
is invertible on RanfP ()g ;
(III.57)
BFS-3, November 20, 1998
32
where we denote AP := PAP . This property and some further relative
bounds of more technical nature stated in Lemma III.10, choosing :=
2"(1 ") 2 (0; 1), insure the existence of the Feshbach operator de ned
by
1
FP := FP Hg () z
( )
( )
:= Hg ()P
P ()Wg P () Hg ()P
z P ( )
( )
( )
1
1
z
(III.58)
P ()Wg P () :
Moreover, the following identity holds,
Hg () z
1
F
P () Hg ()P
= P ()
1
P ( )
( )
z
P ()Wg P () Hg ()P
P ( )
z
+ P () Hg ()P
( )
( )
1
Wg P ()
z
P ( ) :
1
(III.59)
Thus using bounds collected in Lemma III.10 below, we obtain that, for
z 2 D(Ej ; =2),
0
Hg () z
= 1 + O(g # ) FP + O(1) :
1
0
1
1
1
( )
(III.60)
Next, a careful analysis of the Feshbach operator in Lemma III.11 yields that
FP
( )
(Ej + g Zj () + g Zj () + e Hf )P () O(g ") ; (III.61)
2
d
2
od
2+
for any 0 < " < 1=3, which immediately implies (III.53).
III.2.1 Estimates on the Dilated Atomic Hamiltonian
In this subsection, we start to implement the strategy of the proof of Theorem
III.1 outlined above. To begin with, we recall some de nitions and notation.
The pure point spectrum of Hel is given by the set fE ; E ; : : : ; Ej ; : : : g
contained in ( 1; ), and its essential spectrum
is contained in [; 1). We
assume that 0 and, we denote Ri := dist Ei ; (Hel) nfEig > 0. Fixing
j 1 { and we shall keep j 1 xed henceforth { we set := Rj . So
denoting Hel() := U HelU , we can construct the projection Pel;i() onto
the eigenspace of Hel() corresponding to the eigenvalue Ei by using the
Dunford integral,
0
1
1
Z
i
Pel;i() = 2 jz
j
Ei =Ri =2
dz ;
Hel() z
(III.62)
BFS-3, November 20, 1998
33
Next, we de ne a nite-rank projection Pdisc() by
X
Pdisc ( ) :=
Pel;i ( ) ;
(III.63)
i:Ei
where is some xed, strictly positive number. Note that
k Pdisc() Pdisc(0) k C b jj ;
thanks to the relative bound (I.55),
Hel ()
Hel + i
1
(III.64)
b j j ;
(III.65)
where Hel () := Hel () Hel.
Our rst result is the following bound.
Lemma III.4.
Let
z 2 C with Refz g < . Then, for j j 1 + (
Refzg) 1 suciently small, Hel () z is invertible on RanfP disc()g and
(III.66)
(Hel () z) 1 P disc() 2 Refzg :
Proof: We rst observe that Q := P disc(0)Hel (0) z is globally invertible
on Hel , and since 0 we have
o
n
kQ 1k max j zj 1 ; j zj 1 Refzg 1 :
(III.67)
Similarly, we obtain
r+i
r+ +i
1
k (Hel(0) + i) Q k = max sup r + z ; sup z
r0
E0 r
1
C1 1 + Refzg ;
(III.68)
for some constant C1 0. Inserting this and (III.64){(III.65), we obtain
P disc (0)Hel (0) P disc ( )Hel ( ) Q 1
(III.69)
jj 1+
C2
Refzg
1
(III.70)
;
for some constant C2 0. Thus a Neumann series expansion yields
(III.71)
(Hel() z) 1 P disc() = (P disc()Hel () z) 1 P disc()
Q 1
1
X
n=0
1
(P disc(0)Hel(0)
Refzg
1
X
n=0
n
P disc ( )Hel ( )) Q 1 P disc ( )
jj+
C2
jj
C2
Refzg
n
2
Refzg
;
BFS-3, November 20, 1998
for jj 1 + (
Refzg)
34
1
(2C2) 1.
Next, we extend Lemma III.4 to a global bound for the resolvent of Hel.
Lemma III.5. Let > 0. For = i# and # > 0 suciently small, there
exists a constant C 0 such that Hel () Ej + e is invertible on Hel and
(Hel()
Ej
+e
) 1
C (#) 1 :
(III.72)
Proof: We observe that Hel () commutes with Pdisc() = Pi:E
and that
i
(Hel ()
Ej
+e
) 1 Pdisc ( )
=
X
i:Ei
(Ei
Ej
+e
Pel;i ( )
) 1 Pel;i ( ) :
(III.73)
Thus, for some constant C 0 0,
(Hel ()
Ej
+e
max
i:E
) 1 Pdisc ( )
C0
#
i
Pel;i ( )
(III.74)
n
#
Ei
2 (Hel ) Ei
o
:
1
Using the integral representation Pel;i() = (2i) 1 jz E j=R =2 z Hel () dz
together with the relative bound (III.65), we obtain that Pel;i() 1 +
O(jj). Conversely, on RanfP disc()g we apply Lemma III.4 and obtain
R
i
(Hel()
Ej
+e
) 1 P disc ( )
i
Ej + Refe g 1
2 cos(#) 1 :
(III.75)
Lemma III.6. Let = i#, 0 < # < 0 and 0 < 0 (=3) sin #. There
exists a constant C 0 such that, for all z 2 D(Ej ; 0 =2) and all r 0,
Hel ( )
Ej
+ e (0 + r)
P el;j ( )
Hel ( )
z
+e
r
C:
(III.76)
BFS-3, November 20, 1998
35
Using 1 = Pdisc() + P disc() as in the proof of Lemma III.5, we
obtain that
Proof:
X P el;j ( )
X P disc ( )
X P disc ( )
Ei
X
+
i:Ei
Ej
Ei
6
";i=j
+
t + e (0 + r )
C sup
t + + e r
(
t
(III.77)
+ e (0 + r)
z + e r
2R n(
; ) ;
Pel;j ( )
2 D(0; 0=2)
)
;
where we denote
X
:=
Hel ( )
Ej
+ e (0 + r)
Hel ( )
z
+e
r
1
(III.78)
:
We observe that minimization with respect to r 0 yields
t+
+ e r = 1
t + e (0 + r )
1
1
e
t+e
0
(III.79)
(0 + r)
30 t + ( + r) cos #2 + ( + r)2 sin2 #
0
0
2
30 (1 cos #)t2 + ( + r)3 1=2 1 ;
0
2
2
1=2
additionally taking into account that 0 (=3) sin #. Hence
X P el;j ( )
X P disc ( )
+ 2C :
(III.80)
Next, we write
X P disc ( )
=
P disc ( )
+ (e
0
+z
Ej ) Hel ( )
and we obtain from Lemma III.4 that
30
X P disc ( ) P disc ( ) 1 +
2
and hence we arrive at the claim.
+e
1
Refzg
z
r
1
P disc ( ) ;
(III.81)
C;
(III.82)
BFS-3, November 20, 1998
36
III.2.2 Relative Bounds on the Interaction
In this subsection we use the estimates on the dilated electron Hamiltonian
derived in the previous subsection to obtain suitable relative bounds on the
interaction. To this end, we recall Eqns. (I.37){(I.45), and we introduce the
operator
B () := H () Ej + e = Hel() Ej + e (Hf + ) : (III.83)
We shall generally assume that 0 < # and that 0 < (=3) sin #. As
before, we denote Hg ()P := P ()Hg ()P () and pH ()P := H ()P ().
Finally, for a closed operator A we denote jAj := AA. We start with a
0
0
0
0
( )
0
( )
preparatory lemma.
Lemma III.7. For = i# and # 2 (0; ) suciently small there exists a
constant C 0 such that
0
C
!
# 1+ ;
C# 1 + 1 ;
jB ()j (Hf + !)
1
jB ()j (Hel(0) + i)
1
(III.84)
(III.85)
for all > 0 and ! 0.
Proof: By the functional calculus and Lemma III.5 we have
jB ()j (Hf + !) = sup
Hel() Ej + e (r + )
(r + !)
r
r + ! C max1 ; ! ; (III.86)
C sup
#(r + )
#
r>
1
1
0
0
which implies (III.84). To establish (III.85) we start with a similar observation, namely, that
jB ()j (Hel(0) + i) = sup Y
1
r 0
where
Y := Hel () Ej + e (r + )
1
Hel ;
(III.87)
(Hel (0) + i) :
(III.88)
We observe the following identity,
Y = 1
Y (Hel(0) + i) Hel ()
(III.89)
+ i + Ej e (r + ) Hel () Ej + e (r + ) :
1
1
BFS-3, November 20, 1998
37
Solving for Y and applying Lemma III.5, we obtain
kX ()k 1 1bjj 1 + C ji + Ej e j 1 + #1 C 1 + #1 :
(III.90)
Now, we come to the main relative bound used in this section.
Lemma III.8. For ; 1 ; 2 2 fi#g, 0 < < 1, and # 2 (0; 0) suciently
small there exists a constant C 0 such that
jB ()j
1=2
1
jB ()j
Wm;n () jB2 ()j 1=2
P () Wm;n() jB ()j 1=2
P () Wm;n() P ()
1=2
1
Wm;n () P ()
2
C
C
C
C
1+
1=2
1+
1=2
1+
1=2
1+
1=2
(III.91)
;
( + 0 )1=2 n0
=2 ;
(III.92)
( + 0 )1=2 m0
=2 ;
(III.93)
( + 0 ) (0m+n)
=2 ; (III.94)
for all > 0.
Proof: We rst observe that a (~k)B () = B ( + ! (k))a(~k) and thus
a (~k) jB ()j2 = a (~k) B() B ()
= B( + !(k)) B ( + !(k)) a(~k)
(III.95)
= jB ( + !(k))j2 a (~k) :
Thus, functional calculus implies the Pull-Through Formulae
= jB ( + !(k))j 1=2 a(~k) ;
= a (~k) jB ( + !(k))j 1=2 :
Using (III.96){(III.97), we observe that, for any 2 H,
jB ()j
1
=
1=2
2 Z
X
=1
= sup
=1;2
a (~k) jB ()j 1=2
jB ()j 1=2 a(~k)
(III.96)
(III.97)
W0;1 () jB2 ()j 1=2
(III.98)
d3 k jB1 ()j 1=2 w0;1 (~k; ; ) jB2 ( + ! )j 1=2 a (~k)
(Z
jB ()j
1
1=2
(
2 Z
X
=1
Hf
w0;1 (~k; ; ) jB2 ( + ! )j
+!
1=2
a (~k)
2
3
!d k
1=2
)1=2
Hf
;
+!
1=2
2
d3 k
!
)1=2
BFS-3, November 20, 1998
38
where here and henceforth we denote ! := !(k) and !0 := !(k0). Note that
there is an additional constraint !(k) 0 in the integrals on the right side of
(III.98) if we require that 2 Ran Hf <0 . The last factor in (III.98) equals
(
2
X
Z
1=2 ~
1=2 ?
P ? Hf
a (k )a (~k ) Hf
P
=1
! d3 k
1=2
)
k
k
(III.99)
:
Since furthermore Hf + ! commutes with and B1 () and w0;1(~k; ; ), we
may use (I.42) and Lemma III.7 to estimate
B1 ()j 1=2 w0;1 (~k; ; ) jB2 ( + ! )j 1=2 Hf
j
j
B1 ()j 1=2 Hf
+!
1=2
Hel (0) + ij1=2 jB2 ( + ! )j 1=2 Hf
1=2
1
1+
1+ +!
J (k )
#
1=2
C
1
1 + (1 + !)1=2 J (k) ;
#
C
! 1=2
for some constant
obtain that
C
1=2
+!
1=2
0. Inserting (III.100) and (III.99) into (III.98), we
2
B1 ()j 1=2 W0;1 ( ) jB2 ()j 1=2
(III.101)
j
C
#2
1+
1
(III.100)
w0;1 (~k; ; ) jHel (0) + ij 1=2
j
+!
Z
1 + ! (k )
C (0 + 1 )
1 ;
1
+
#2
1 J (k)2 d3 k
for some constant C 0. Similarly, by additionally requiring that
Ran Hf <0 , we obtain that
j
2
2
B1 ()j 1=2 W0;1 ( ) jB2 ()j 1=2 Hf <0
C
#2
1+
C 0
#2
C (
1
Z
!(k)0
1+ 1
+
#2
1
Z
1 + ! (k )
! (k )
1 J (k)2 d3 k
+ ! (k )
) 0 1 +
1 :
1
J (k )2 d3 k
(III.102)
The estimate for W1;0() is similar. Next, we derive (III.91) in the case of
BFS-3, November 20, 1998
W0;2 ( ).
j
Picking
B1 ()j
2 H
39
, we observe that
W0;2 ( ) jB2 ()j 1=2
(Z
1=2
jB1 ()j
w0;2 (~k; ~k 0 ; ; 0 ; )
(III.103)
1=2
sup
;0 =1;2
0
jB2 ( + ! + ! )j
(
Z
2
X
;0 =1
Hf
1=2
+ ! + !0
1
Hf + ! + ! 0
a (~k 0 )a (~k )
0
2
2
)1=2
d3 k d3 k 0
! !0
)1=2
! ! 0 d3 k
:
Again, we have the additional constraint !(k) 0 in the integrals on the
right side of (III.103) if we require that 2 Ran Hf <0 . The last factor is
bounded by k k. Thus Eqn. (I.43) and Lemma III.7 imply that, for some
constants C 0,
B1 ()j
Z
j
2
W0;2 ( ) jB2 ()j 1=2
1=2
1=2
Hf + ! + ! 0
jB1 ()j
1=2
(III.104)
2
1=2 2 J (k )2 J (k 0 )2 d3 k d3 k 0
B2 ( + ! + ! 0 )j 1=2 Hf + ! + ! 0
! !0
Z
! + !0
! + !0
J (k )2 J (k 0 )2 d3 k d3 k 0
C
j
#2
C (0
1+
1+
+ ! + !0
+ 1)2 1 + 1 ;
#2
! !0
and
j
B1 ()j 1=2 W0;2 ( ) jB2 ()j
C (
1=2
Hf <0
+
1
#2
2
) 20 1 +
(III.105)
1
:
Estimates similar to (III.103){(III.104) establish (III.91) in the remaining cases, i.e., for W1;1() and W2;0 (). Finally, we observe that P () = Hf <0 P ()
and
B () P ( )
= (Hf + ) P () =
0 + ;
which together with (III.102) and (III.105) yields (III.92){(III.94).
(III.106)
BFS-3, November 20, 1998
40
III.2.3 Domain of the Feshbach Map
In the following subsection we apply the relative bounds from Lemma III.8
to prove that, for z suciently close to Ej , the Feshbach map with projection
P ( ) is applicable to Hg ( ) z .
Lemma III.9. Let 0 < (=3) sin #, and assume that # 2 (0; 0 ) is sufciently small. Then, for g0 1=2 > 0 suciently small and for all z 2
D (Ej ; 0 =2), the operator Hg ( )P () z is invertible on RanfP ( )g, and
Hg ( )P ()
z
1
P ( )
C
# 0
(III.107)
;
for some constant C 0.
Proof: We construct
mann series,
Hg ( )P ()
z
1
Hg ( )P ()
P ( )
=
z
1
1
P ( )
!"
P ( )
X
n=0
H0 ( )
z
by a norm-convergent NeuWg ( )
! #n
P ( )
H0 ( )
z
:
(III.108)
We estimate the norm of the term in nth order by means of Lemma III.8,
!"
P ( )
H 0 ( )
B (0 )
1=2
jB(0)j
C
# 0
Wg ( )
z
C g 0
H0 ( )
P ( )
H0 ( )
1=2
z
jB (0)j
1=2 n
! #n
P ( )
(III.109)
z
B(0 )
1=2
1=2 n+1
jB(0)j
1=2
j
j
Wg ( ) B (0 ) 1=2
n
:
This proves the convergence of the Neumann series (III.108) in norm.
Lemma III.9 is the main ingredient used to prove the existence of the
Feshbach operator de ned in (III.58){(III.60).
Lemma III.10. Let 0 < (=3) sin #, and assume that # 2 (0; 0) and g > 0
are suciently small. Then, for all z 2 D(Ej ; 0 =2), the Feshbach operator de ned in (III.58) exists and obeys Eqn. (III.59). Moreover, for some
BFS-3, November 20, 1998
41
constant C 0, we have
P () Wg () P () Hg ()P ()
P () Hg ()P ()
z
1
z
P ( )
1
;
P () Wg () P ()
>
;
Cg
:
# 10=2
Cg 10=2
:
#
P () Wg () P ()
9
>
=
(III.110)
(III.111)
Proof: The proof of (III.110) is similar to the one for Lemma III.9. Then
Lemma III.9 and (III.108) imply the existence of the Feshbach operator dened in (III.58) and that it obeys Eqn. (III.59) (see, e.g., [5, 6]).
We nally establish two lemmata which, inserted into (III.61) and (III.53),
prove Theorem III.1.
Lemma III.11. Let 0 < " < 1=3, := g ", 0 < < 1, and assume that
# 2 (0; ) and g > 0 are suciently small and such that < (=3) sin #.
Then, for all z 2 D(Ej ; =2),
2
0
2
0
0
0
FP (Ej + g Zj () + g Zj () + e Hf )P () (III.112)
"
"
C g "+g
+g
+g " ;
for some constant C 0.
2
( )
2+
d
2+2 (1
2
od
)
1+(1+ )(1
)
4
2
Proof: Recall from (III.58) and (III.7){(III.8) that
FP := FP
( )
Hg ()
( )
:= Hg ()P
z P ( )
( )
and
Z j ( )
od
X Z
:=
=1;2
Hel
Zj ()
d
:=
z
X Z
=1;2
P ()Wg P () Hg ()P ()
z
1
(III.113)
P ()Wg P ()
?
U Pel;j w0;1 (~k; )Pel;j
Ej + ! (~k)
i0
1
(III.114)
? w (~k; )P U 1 d3 k ;
Pel;j
1;0
el;j
d3 k
;
U Pel;j w0;1 (~k; ) Pel;j w1;0 (~k; )Pel;j U 1
! (~k)
(III.115)
BFS-3, November 20, 1998
42
where Pel;j = Pel;j ( = 0) = Pn` j j'j;`ih'j;`j is the orthogonal projection onto
the eigenspace of Hel corresponding to the eigenvalue Ej . As in [5], we write
the di erence to be estimated as a sum of six error terms,
=1
FP
(Ej + g
( )
2
Zjd ( ) + g 2 Zjod ( ) + e
Hf )P ( )
5
X
=
=0
Rem ; (III.116)
where (compare to [5, (IV.58), (IV.60), (IV.68), (IV.77), (IV.101), (IV.86),
and (IV.87)])
Rem :=
P ( )Wg ( )P ( )
0
"
P ( )Hg ( )P ( )
z
P ( )H0 ( )P ( )
P ( )Wg ( )P ( ) P ( )Hg P ( )
Rem :=
1
g 2 P ( ) W0;1 ( ) + W1;0 ( )
Rem :=
g P ( ) W0;1 ( ) + W1;0 ( )
2
g
2
2
2 Z
X
P ( )w0;1 (~k; )
=1
"
z
P ( )
H0 z
z
H0
+e
P ( )Wg ( )P ( ) ;
!
(III.118)
W0;1 ( ) + W1;0 ( ) P ( ) ;
!
(k )
W0;1 ( ) + W1;0 ( ) P ( )
#
P ( ; ! (k ))
i# !
1#
P ( )Wg ( )P ( )
1
P ( )
H0 z
(III.117)
1
z
w1;0 (~k; )P ( ) d3k ;
(III.119)
where P (; !) := Pel;j () Hf !<0 ,
Rem := P () Wg () P () ;
+
(III.120)
3
Rem :=
g
4
2 Z
X
=1
2
4
2
e
i#
Hf
dk Pel;j w0;1 (~k; )
+
B (! (k )) + Ej
Rem :=
5
g2
h
e
Z X
=1;2
i#
Ej
z
(III.121)
z P el;j Hf <0
Hel
Ej
+e
i# !
3
5 w1;0 (~k; ) Pel;j Hf <0 ;
(k )
Pel;j w0;1 (~k; ) Pel;j w1;0 (~k; )Pel;j
(Hf + !(k)) + Ej
z
i
1
Hf +!(k)0 dk
(III.122)
Zj ( ) Hf <0 :
d
BFS-3, November 20, 1998
43
We rst rewrite Rem using the second resolvent equation which yields
0
Rem =
P ( )Wg ( )P ( ) P ( )Hg ( )P ( )
0
P ( )H0 ( )P ( )
z
1
z
1
P ( )Wg ( )P ( )
P ( )Wg ( )P ( ) :
(III.123)
Then an application of Lemma III.8 gives
Cg 3
k Rem k
1=2
0
#2 0
= O
= O
(III.124)
g 2+" :
Second, a similar estimate yields
Cg 3
k Rem k
1=2
1
#2 0
(III.125)
g 2+" :
The derivation of these two estimates, (III.124) and (III.125), is similar to
[5, (IV.58){(IV.62)].
Third, we observe that
Rem =
2
Z
2
X
;0 =1
dk dk 0
(
(III.126)
P ( ) w1;0 (k; ; ) a (k )a (k 0 )
+ P () w ; (k; ; )
"
01
"
#
P (; ! (k ))
H0 + e i# ! (k )
+ P (") w ; (k; ; ) a(k0)
P (; ! (k ) + ! (k 0 ))
H + e i# (! (k ) + ! (k 0 ))
"
+ P () w (k; ; ) a (k) P ()
01
0
1;0
#
P (! (k 0 ))
H0 + e i# ! (k 0 )
H0
#
z
#
z
z
w1;0 (k 0 ; 0 ; ) P ( )
w0;1 (k 0 ; 0 ; ) a(k )a(k 0 ) P ( )
w1;0 (k 0 ; 0 ; ) a(k ) P ( )
z
w0;1 (k 0 ; 0 ; ) a(k 0 ) P ( )
)
f ; ( ) + W
f ; ( ) +
(compare
to [5, (IV.66)]), which is of the form P () W
f ; () P (). A somewhat lengthy estimate analogous to [5, Lemma IV.9]
W
yields, after using (III.92){(III.93),
20
02
11
k Rem k = O
2
g 2+2
(1
")
(III.127)
:
Fourth, we apply (III.94) and directly obtain
k Rem k = O
3
(1+ )=2
g 0
= O
g 1+(1+
)(1
")
:
(III.128)
BFS-3, November 20, 1998
44
In order to estimate Rem4, we observe that when restricted to RanP el;j ,
the resolvents of B (!(k)) + Ej z and Hel Ej + e i#!(k) are bounded by a constant C 0. Since, furthermore, z Ej 2 D(Ej ; 0 =2) and
k Hf Hf <0 k = 0 , the fraction in the integrand on the right side of (III.121)
is bounded in norm by 2C0, and we thus obtain
(III.129)
k Rem4 k O g2 0 O g4 2" :
Finally, a similar argument, which is along the lines of [5, Lemma IV.12],
yields
(III.130)
k Rem5 k O g2 0 O g2+2 (1 ") :
Adding up all error terms, taking into account that 0 < < 1, we arrive at
(III.112).
and assume that # 2 (0; 0 )
and g > 0 are suciently small and such that 0 < (=3) sin # and 0
2# 1 minfImAj g. Then, for all z 2 Aj n D(Ej ; 0 =2),
Lemma III.12. Let 0 < " < 1, 0 := g2
Hg ()
z
1
2"
h
C dist z ; Rj
i 1
(III.131)
;
for some constant C 0.
Proof: We rst consider all eigenvalues Ei of Hel() below . Since
the eigenvalue closest to z is Ej , there exists a constant c > 0 such that
h
min dist z ; Ei + e #R +
E
i
i
h
i
c # dist z ; Ej + e # R + ;
(III.132)
for any z 2 Aj n D(Ej ; 0=2). Thus, on RanfPdisc()g (see Eqns. (III.62){
(III.62)) we have
h
i 1
Pdisc ()
#
dist z ; Ei + e R +
C Emax
H0 () z
i
h
i
C dist z ; Ej + e # R + 1 :
(III.133)
c#
Moreover, since 0 2# 1 minfImAj g, for g > 0 suciently small, we
have #=2 arg(z Ej ) + #=2, for any z 2 Aj n D(Ej ; 0 =2). Therefore,
we also have the estimate
(
)
r + 0
B (0 ) Pdisc()
C sup max
C# 2;
i#
E
H 0 ( ) z
jEi z + e rj
r>0 i
(III.134)
BFS-3, November 20, 1998
45
for some C 0.
Second, on RanfP disc()g we employ Lemma III.4 to prove a similar
bound,
h
i 1
P disc ( )
C0
C
B (0 )P disc ( )
dist
z ; Ej + e # R +
;
;
H0 ( ) z
H0 ( ) z
#
#
(III.135)
for some constants C; Ch0 0. Summarizing
(III.133),
(III.134),
(III.135),
i
h
i
#
and observing that dist z ; Ej + e R + c dist z ; Rj , for some c > 0,
we obtain
B (0 )
C# 1
1
h
i
and
C # 1 ; (III.136)
H0 ( ) z
H
(
)
z
0
dist z ; Rj
for some constant C 0 and all z 2 Aj n D(Ej ; 0 =2). So, nally, we obtain
the invertibility of Hg () z and (III.131) from an expansion in a Neumann
series as in (III.108){(III.109).
Acknowledgements: We thank T. Chen and A. So er for numerous very
helpful discussions on the material presented in this paper and H. Zenk for
careful proofreading. We are also grateful to D. Buchholz, F. Hiroshima,
F. Klopp, Y. Last, and H. Spohn for valuable comments.
References
[1] J. Aguilar and J. M. Combes. A class of analytic pertubations for onebody Schrodinger Hamiltonians. Commun. Math. Phys. , 22:269{279,
1971.
[2] S. Albeverio. Scattering theory in a model of quantum elds. I.
J. Math. Phys., 14(2):1800{1816, 1972.
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