Monotonicity of the polaron energy II:General theory of operator monotonicity

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by  Tadahiro Miyao

Monotonicity of the polaron energy II:
General theory of operator monotonicity
Tadahiro Miyao
Department of Mathematics, Hokkaido University,
Sapporo 060-0810, Japan
3 [email protected]
1
0
2
n Abstract
a
We construct a general theory of operator monotonicity and apply it to the
J
Fr¨ohlichpolaronhamiltonian. This generaltheoryprovidesa consistentviewpoint
7
of the Fr¨ohlich model.

]
h
p 1 Introduction
-
h
t This paper is a sequel to [31]. To explain our motivation of this work, let us recall
a
m the result in [31] first. In the previous work [31], we studied the Fro¨hlich hamiltonian
[ defined by

1 1 1
v H = ∆ √αλ dk [eik·xa(k)+e−ik·xa(k)∗]+N . (1.1)
0 Λ −2 x− 0Z|k|≤Λ k f
| |
0
1 Here Λ is the ultraviolet cutoff. (The complete definiton of H will be recalled in the
Λ
1
subsequent section.) One of the main results in [31] is stated as follows.
.
1
0
Theorem 1.1 [31] Let E =infspec(H ). Then E is strictly decreasing in Λ.
3 Λ Λ Λ
1
: In the proof of Theorem 1.1, a new operator monotonicity played important roles. To
v
illustrate the meaning of our operator monotonicity, we introduce some terms. Let p
i
X
be a convex cone in the Hilbert space h. p is called to be self-dual if it satisfies
r
a
p= x h x,y 0 y p . (1.2)
{ ∈ |h i ≥ ∀ ∈ }
Let A,B be linear operators in h. For simplicity, suppose both are bounded. If A and
B satisfy (A B)p p, we denote this as A☎B w.r.t. p. It is easily checked that
− ⊆
the binary relation “☎” is a partial order. Thus we can regard the relation A ☎ B
w.r.t. p as an operator inequality. The operator monotonicity, our central subject in
this paper, is expressed as follows. Let A be a family of operators in h. We say
s s≥0
{ }
A is monotonically decreasing if
s
s > s′ = As′ ☎As w.r.t. p. (1.3)
⇒
To prove Theorem 1.1, we effectively applied this notion in [31], namely, we selected a
properself-dual cone so that a family of hamiltonians H becomes monotonically
Λ Λ≥0
{ }
1
decreasing under the choice. (It will be seen that we can extend the notion of operator
monotonicitytounboundedoperators,seeSection2.) Combiningthismonotonicityand
ageneraltheoremestablishedin[31],oneobtainedtheassertioninTheorem1.1. Inthis
paper, we search for further possibilities of the operator monotonicity. We construct a
theory of the singular perturbation by the operator monotonicity, and apply it to the
Fro¨hlich polaron hamiltonian. The results in the part I[31] are included in this theory.
Our theory in this paper provides a consistent viewpoint of the polaron as well.
Next we briefly background mathematical difficulties studying the Fro¨hlich polaron
model and then state the results concerned these difficulties. When we tackle the prob-
lem of investigating the mathematical aspects of the polaron model, we unavoidably
encouter the question how we define the hamiltonian. In various physical literatures,
one presupposes that the hamiltonian without the ultraviolet cutoff H is given.
Λ=∞
However it is not obvious whether the hamiltonian H is definable or not. To clar-
Λ=∞
ify the point, put Λ = in the electron-phonon interaction term in (1.1). Then since
∞
1/k is not in L2(R3), the interaction term can not be defined as a linear operator
| |
in the Hilbert space. (Remark the symbols dkf(k)a(k) and dkf(k)a(k)∗ are
R3 R3
well-defined linear operators only if f L2(RR3). ) If one looksRat only interaction
∈
term, the interaction term does not have mathematical meaning at Λ = , however if
∞
our vison broaden and we study the whole hamiltonian including not only the electron-
phonon interaction term but also the electron and phonon kinetic energy terms, the
limit lim H exists in a suitable sense. This is the standard way how to define
Λ→∞ Λ
the Fro¨hlich hamiltonian [8, 15, 16, 17, 21, 35, 36, 39]. In this paper, we prove the
existence of the limiting hamiltonian by applying the general theory of the operator
monotonicity. As far as we know, our proof is novel. In addition, this provides an
important example of our operator monotonicity.
Next problem is to study the ground state property of the hamiltonian without the
ultraviolet cutoff. (Here keep in mind that we are discussing the ground state of the
hamiltonian at a fixed total momentum now.) Remark that existence of the ground
statehasbeenalreadyestablished[15,17,34,41]. Inthispaperwearguetheuniqueness
of the ground state. This is known to be rather difficult because the hamiltonian is
defined through the limiting procedures. In [29], the author proved the uniqueness of
the ground state. In the present work, we will give a different proof by the general
theory of the operator monotonicity.
Once again, let us reconfirm the aim of this work. Through analysis of the Fro¨hlich
hamiltonian, we build a general theory of the operator monotonicity. We expect the
further validity of our operator inequalities is revealed through our attempt.
The organization of the paper is as follows. In Sects. 2 and 3, we establish a
general theory of our operator monotonicity. Terminologies from the self-dual cone
analysis [29, 30, 31, 32, 33] are introduced in this section. In Sect. 4, we summarize
basic facts of the second quantization. Sect. 5 is devoted to display applications of the
general theory in Sects. 2 and 3 to the Fro¨hlich polaron model. In Sects. 6-10, we give
detailed proofs of the results stated in Sect 5.
Acknowledgments
I would like to thank H. Spohn for useful discussions. Financial support by
KAKENHI (20554421) is gratefully acknowledged.
2
2 A general theory of operator monotonicity
In this section, we will provide an abstract theory of operator monotonicity. As we
will see in the later sections, this abstract theory has an important application to the
condensed matter physics.
2.1 Definitions
To express our ideas, results and proofs, we need to introduce some technical terms
from an earlier work [31]. The terms defined here serve as a language in this paper.
Let h be a complex Hilbert space and p be a convex cone in h. Then p is called to
be self-dual if
p = x h x,y 0 y p . (2.1)
{ ∈ |h i ≥ ∀ ∈ }
Henceforth p always denotes the self-dual cone in h. The following properties of p are
well-known [4, 22]:
Proposition 2.1 One has the following.
(i) p ( p) = 0 .
∩ − { }
(ii) There exists a unique involution j in h such that jx = x for all x p.
∈
(iii) Each element x h with jx = x has a unique decomposition x = x x , where
+ −
∈ −
x ,x p and x ,x = 0.
+ − + −
∈ h i
(iv) h is linearly spanned by p.
Remark 2.2 We clarify the precise meaning of (iv) in Proposition 2.1. Each x h
∈
can be expressed as x = x + i x, where x = 1(1l + j)x and x = 1(1l j)x.
ℜ ℑ ℜ 2 ℑ 2i −
Clearly j x = x and j x = x. Thus, by (iii), we have a unique decomposition
ℜ ℜ ℑ ℑ
x = ( x) ( x) with ( x) ,( x) = 0. Similar property holds for x. Now
+ − + −
ℜ ℜ − ℜ h ℜ ℜ i ℑ
one has
x = ( x) ( x) +i ( x) ( x) . (2.2)
+ − + −
ℜ − ℜ { ℑ − ℑ }
Of course, ( x) ,( x) p. This is the meaning of (iv).
± ±
ℜ ℑ ∈ ♦
If x y p, then we will write x y (or y x) w.r.t. p. Let A and B be densely
− ∈ ≥ ≤
defined linear operators on h. If Ax Bx w.r.t. p for all x dom(A) dom(B) p,
≥ ∈ ∩ ∩
then we will write A☎B (or B ✂A) w.r.t. p. Especially if A satisfies 0✂A w.r.t. p,
then we say that A preserves positivity with respect to p. The symbol “☎” was first
introduced by Miura [27].
An element x in p is called to be strictly positive if x,y 0 for all y p 0 . We
h i ≥ ∈ \{ }
will write this as x > 0 w.r.t. p. Of course, an inequality x > y w.r.t. p means x y is
−
strictly positive w.r.t. p. If bounded operators A and B satsify Ax > Bx w.r.t. p for
all x p 0 , then we will express this as A✄B (or B✁A) w.r.t. p. Clearly if A✄B
∈ \{ }
w.r.t. p, then A☎B w.r.t. p. We say that A improves positivity w.r.t. p if A✄0 w.r.t.
p.
3
2.2 Monotonically decreasing self-adjoint operators
Let p bea self-dual cone in the Hilbert space h. Let H bea family of self-adjoint
λ λ≥0
{ }
operators on h. In this section we always assume the following.
(A. 1) There exists a constant M > , independent of λ, such that H M for all
λ
−∞ ≥
λ 0.
≥
(A. 2) For all λ 0 and s 0, e−sHλ ☎0 w.r.t. p.
≥ ≥
(A. 3) Each H has the common domain D, i.e., dom(H ) = D for all λ 0.
λ λ
≥
Under these assumptions, we can show the existence of the limiting hamiltonian as
follows.
Theorem 2.3 Weassume (A.1), (A.2)and(A.3). Suppose thatH ismonotonically
λ
decreasing in λ in the sense that
λ λ = H ☎H w.r.t. p. (2.3)
1 ≤ 2 ⇒ λ1 λ2
Then, there exists a unique self-adjoint operator H, bounded from below by M, with
following properties:
(i) H converges to H in strong resolvent sense as λ ,
λ
→ ∞
(ii) For all λ 0 and s 0, e−sH ☎e−sHλ w.r.t. p. In particular, e−sH ☎0 w.r.t. p
≥ ≥
for all s 0.
≥
In the above theorem, we proved the limiting hamiltonian denoted by H satisfies
e−sH ☎ 0 w.r.t. p for all s 0. Our next problem is to show e−sH ✄ 0 w.r.t. p
≥
for all s > 0. This problem is important because if we can solve it, then by the
Perron-Frobenius-Faris theorem(Theorem A.3), the ground state of H (if it exsits) is
automatically unique and strictly positive w.r.t. p. Below we will give two answers
about this problem.
Theorem 2.4 Inaddition totheassumptions inTheorem 2.3, assume that, there exists
a λ 0 such that
≥
e−sHλ ✄0 (2.4)
w.r.t. p for all s > 0. Let H be the self-adjoint operator obtained in Theorem 2.3. Then
one has
e−sH ✄0 (2.5)
w.r.t. p for all s > 0. Consequently, if infspec(H) is an eigenvalue, then it is unique
and the corresponding eigenvector can be chosen as strictly positive w.r.t. p.
Theorem 2.4 was applied toshow theuniquenessof thegroundstate for thepolaron
and bipolaron Hamiltonians without cutoffs in [28, 29]. Instead, the following theorem
is convenient for concrete applications in this paper.
4
Theorem 2.5 In addition to the assumptions in Theorem 2.3, assume that, for some
µ 0, a family of self-adjoint operators (H +µ)−1 is ergodic in the sense that,
λ λ≥0
≥ { }
for any x,y p 0 , there exists a λ 0 so that x,(H +µ)−1y > 0. Let H be the
λ
∈ \{ } ≥ h i
self-adjoint operator in Theorem 2.3. Then one has
e−sH ✄0 (2.6)
w.r.t. p for all s > 0. Consequently, if infspec(H) is an eigenvalue, then it is unique
and the corresponding eigenvector can be chosen as strictly positive w.r.t. p.
The operator monotonicity (2.3) further gives useful information concerning spec-
trum of the Hamiltonians. Indeed, in our previous work [31], the following theorem
was proven.
Theorem 2.6 [31] Assume (A. 1), (A. 2), (A. 3), (2.3). Set E = infspec(H ) and
λ λ
E = infspec(H). Then E is monotonically decreasing in λ and lim E = E.
λ λ
λ→∞
3 Proofs of Theorems 2.3, 2.4, 2.5
3.1 Proof of Theorem 2.3
STEP 1. Let x,y p. Define a function F (λ) by F (λ) = x,e−Hλy . Then, by
x,y x,y
∈ h i
(2.3), the functiom F (λ) is monotonically increasing in λ and, by the assumption
x,y
(A. 1), it is bounded. Hence lim F (λ) exists. Then, since h is linearly spanned by
x,y
λ→∞
p(Remark 2.2), the limit lim x,e−sHλy exists for all x,y h. Now one can define a
λ→∞h i ∈
sesquilinear form B (, ) by
s
· ·
B (x,y) = lim x,e−sHλy . (3.7)
s
λ→∞h i
By the assumption (A. 1), one sees B (x,y) x y e−sM. Thus, by the Riesz’s
s
| | ≤ k kk k
representation theorem, there exists a unique positive operator T such that
s
B (x,y) = x,T y . (3.8)
s s
h i
Then, by definition, one has
T e−sM, T = w- lim e−sHλ, (3.9)
s s
k k ≤ λ→∞
where w-lim means the weak limit. Moreover, by the monotonicity (2.3), one has
e−sHλ ✂T (3.10)
s
for all λ 0. [Proof: By Proposition A.1, one has e−sHλ2 ✂e−sHλ2 whenever λ2 λ1.
≥ ≥
Then taking λ , one concludes (3.10). Remark that to apply Proposition A.1, we
2
→ ∞
used the assumptions (A. 2) and (A. 3). ]
STEP 2. In this step we will show
s- lim e−sHλ = T , (3.11)
s
λ→∞
5
where s-lim means the strong limit.
Since h is linearly spanned by p(Remark 2.2), it sufficies to show that
s- lim e−sHλx = T x (3.12)
s
λ→∞
for all x p. To this end, observe that
∈
(T e−sHλ)x 2 = x, T2 e−sHλT T e−sHλ +e−sHλe−sHλ x . (3.13)
k s − k s − s− s
(cid:10) (cid:0) (cid:1) (cid:11)
Note that, by (3.10), one has e−sHλe−sHλ✂T e−sHλ which implies x,e−sHλe−sHλx
s
h i ≤
x,T e−sHλx . Hence
s
h i
RHS of (3.13) x, T2 e−sHλT T e−sHλ +T e−sHλ x
≤ s − s− s s
(cid:10) (cid:0) (cid:1) (cid:11)
= x,(T e−sHλ)T x
s s
h − i
0 (3.14)
→
as λ by (3.9). This proves (3.12).
→ ∞
STEP3. Since e−sHλ isaone-parametersemigroup, T isalsoone-parameter
s≥0 s s≥0
{ } { }
semigroup by STEP 2. If one can show the strong continuity of T in s 0, one sees
s
≥
that there exists a unique self-adjoint operator H such that T = e−sH by Propositon
s
B.2. By our construction, it is clear that
e−sH = s- lim e−sHλ, e−sH ☎e−sHλ, H M. (3.15)
λ→∞ ≥
This is the assertion in the theorem.
Let prove the strong continuity of T in s. Without loss of generality, we may
s
assume M > 0, i.e., H M > 0. (Indeed, if M > 0, there is nothing to do. On the
λ
≥
other hand, if M 0, then we simply study H˜ = H M +ε (ε > 0) instead of H
λ λ λ
≤ −
itself. Obviously H˜ ε for all λ. ) Hence 1l T 0 holds for all s 0. Fix λ 0
λ s
≥ − ≥ ≥ ≥
arbitrarily. One has, by (3.10),
e−sHλ ✂T (3.16)
s
w.r.t. p. Thus 1l e−sHλ ☎1l T w.r.t. p. Hence, for all x p, we have
s
− − ∈
0 x,(1l T )x x,(1l e−sHλ)x (3.17)
s
≤ h − i ≤ h − i
for all s 0. Taking s +0,
≥ →
lim x,(1l T )x = 0 (3.18)
s
s→+0h − i
holds for all x p. Next observe that, for any x p,
∈ ∈
(1l T )x 2 = x,(1l 2T +T )x 0 (3.19)
s s 2s
k − k h − i →
as s +0 by (3.18). Then since h is linearly spanned by p(Remark 2.2), one sees
→
(T 1l)x 0 as s +0 for all x h. ✷
s
k − k → → ∈
6
3.2 Proof of Theorem 2.4
The proof of this theorem was already given in [29]. Here we will repeat it for reader’s
convenience.
By Theorem 2.3 (ii),
e−sH ☎e−sHλ (3.20)
w.r.t. p. Since e−sHλ ✄0, one arrives at
e−sH ☎e−sHλ ✄0 (3.21)
w.r.t. p for all s > 0. Remainder assertions comes from Theorem A.3. ✷
3.3 Proof of Theorem 2.5
By (2.3) and Proposition A.1, one has
(H +µ)−1☎(H +µ)−1 (3.22)
λ
w.r.t. p for all λ 0. Choose x,y p 0 arbitrarily. Then, by the ergodicity of
≥ ∈ \{ }
(H +µ)−1 , there is a λ 0 such that x,(H +µ)−1y > 0. Combining this with
λ λ≥0 λ
{ } ≥ h i
(3.22), we have
x,(H +µ)−1y x,(H +µ)−1y > 0. (3.23)
λ
h i ≥h i
This means (H + µ)−1 ✄ 0 w.r.t. p. Now we apply Theorem A.3 and conclude the
assertion. ✷
4 Second quantization
Before we study the Fro¨hlich polaron, we briefly summarize necessary results of the
second quantization. Many of these have already been stated in the previous work [31].
4.1 Basic definitions
The bosonic Fock space over h is defined by
⊕
F(h) = F(n)(h), F(n)(h) = h⊗sn, (4.1)
Xn≥0
where h⊗sn is the n-fold symmetric tensor product of h with convention h⊗s0 = C.
F(n)(h) is called the n-boson subspace.
We denote by a(f)(f h) the annihilation operator on F(h), its adjoint a(f)∗,
∈
called the creation operator, is defined by
⊕
a(f)∗ϕ = √nS (f ϕ(n−1)) (4.2)
n
Xn≥1 ⊗
for ϕ = ⊕ ϕ(n) dom(a(f)∗), where S is the symmetrizer on F(n)(h) = h⊗sn.
n≥0 ∈ n
The annihPilation- and creation operators satisfy the cannonical commutation relations
(CCRs)
[a(f),a(g)∗]= f,g , [a(f),a(g)] = 0= [a(f)∗,a(g)∗] (4.3)
h i
7
on a suitable dense subspace in F(h).
LetC beacontractionoperatoronh,thatis, C 1. Thenwedefineacontraction
k k ≤
operator Γ(C) on F(h) by
⊕
Γ(C)= C⊗n (4.4)
Xn≥0
withC⊗0 = 1l,theidentity operator. Foraself-adjointoperatorAonh, letusintroduce
⊕
dΓ(A) = 0 1l⊗(k−1) A 1l⊗(n−k) (4.5)
⊕ Xn≥1 Xn≥k≥1 ⊗ ⊗
acting in F(h). Then dΓ(A) is essentially self-adjoint. We denote its closure by the
same symbol. A typical example is the bosonic number operator N = dΓ(1l). We
f
remark the following relation between Γ() and dΓ():
· ·
Γ(eitA) =eitdΓ(A). (4.6)
In particular if A is positive, then one has
Γ(e−tA) =e−tdΓ(A). (4.7)
Proposition 4.1 Let A be a positive self-adjoint operator. Then we have the following
operator inequalities:
a(f)∗a(f) A−1/2f 2(dΓ(A)+1l), (4.8)
≤ k k
dΓ(A)+a(f)+a(f)∗ A−1/2f 2. (4.9)
≥ −k k
4.2 Fock space over L2-space
In this paper, the bosonic Fock space over L2(R3) = L2(R3,dk) will often appear and
k
we simply denote as
F= F(L2(R3)). (4.10)
k
The n-boson subspace F(n) = L2(R3)⊗sn is naturally identified with L2 (R3n) = ϕ
k sym ∈
L2(R3n) ϕ(k ,...,k ) = ϕ(k ,...,k ) a.e. σ S , where S is the perm(cid:8)uta-
k | 1 n σ(1) σ(n) ∀ ∈ n n
tion group on a set 1,2,...,n . Hence (cid:9)
{ }
⊕
F = C L2 (R3n). (4.11)
⊕ Xn≥1 sym k
The annihilation- and creation operators are symbolically expressed as
a(f)= dkf(k)a(k), a(f)∗ = dkf(k)a(k)∗. (4.12)
Z Z
R3 R3
If ω is a multipilication operator by the function ω(k), then dΓ(ω) is formally written
as
dΓ(ω)= dkω(k)a(k)∗a(k). (4.13)
ZR3
k
8
4.3 The Fro¨hlich cone
In order to discuss the inequalities introduced in 2, we have to determine a self-dual
§
cone in F. Here we will intoduce a natural self-dual cone in F which is suitable for our
analysis in later sections.
Under the natural identification L2(R3)⊗sn = L2 (R3n), set
k sym k
F(n) = ϕ L2 (R3n) ϕ(k ,...,k ) 0 a.e. (4.14)
+ ∈ sym | 1 n ≥
(cid:8) (cid:9)
with F(0) = R = r R r 0 . Then each F(n) is a self-dual cone in F(n) =
+ + { ∈ | ≥ } +
L2 (R3n).
sym k
Definition 4.2 The Fr¨ohlich cone is defined by
⊕ (n)
F = F . (4.15)
+ +
Xn≥0
Again F is a self-dual cone in F.
+
♦
We summarize properties of operators in F below. All propositions were proven in
[31].
Proposition 4.3 Let C be a contraction on L2(R3). Then if C ☎0 w.r.t. L2(R3) ,
k k +
one has Γ(C)☎0 w.r.t. F , where L2(R3) = f L2(R3) f(k) 0 a.e. . Especially
+ k { ∈ k | ≥ }
one has the following.
(i) For a self-adjoint operator A, if eitA☎0 w.r.t. L2(R3) , then one has Γ(eitA)☎0
+
w.r.t. F .
+
(ii) For a positive self-adjoint operator B, if e−tB ☎0 w.r.t. L2(R3) , then one has
+
Γ(e−tB)☎0 w.r.t. F .
+
Proposition 4.4 If f 0 w.r.t. L2(R3) , then a(f)∗☎0 and a(f)☎0 w.r.t. F .
≥ k + +
Proposition 4.5 (Ergodicity) For each f L2(R3), let φ(f) be a linear operator
∈ k
defined by
φ(f)= a(f)+a(f)∗. (4.16)
If f > 0 w.r.t. L2(R3) , that is, f(k)> 0 a.e. k, then φ(f) is ergodic in the sense that,
k +
for any x,y (F F ) 0 , there exists an n 0 N such that x,φ(f)ny > 0.
+ fin
∈ ∩ \{ } ∈ { }∪ h i
4.4 Local properties
Let B be a ball of radius Λ in R3 and let χ bea function on R3 defined by χ (k) = 1
Λ k Λ Λ
if k B and χ (k) = 0 otherwise. Then as a multiplication operator, χ is an
Λ Λ Λ
∈
orthogonal projection on L2(R3) and Q = Γ(χ ) is also an orthogonal projection on
k Λ Λ
F. Now let us define the local Fock space by
F = Q F. (4.17)
Λ Λ
Clearly F = F . Since χ L2(R3) = L2(B ), F can be identified with F(L2(B )).
∞ Λ k Λ Λ Λ
The following proposition was unstated in [31].
9
Proposition 4.6 For each Λ 0, put Q⊥ = 1l Q . Then one obtains the following.
≥ Λ − Λ
(i) Q ☎0 w.r.t. F .
Λ +
(ii) Q⊥ ☎0 w.r.t. F .
Λ +
Proof. (i) immediately follows from Proposition 4.3.
(ii) Under the identification (4.11), we see
n
(Q ϕ )(k ,...,k )= χ (k )ϕ (k ,...,k ) (4.18)
Λ n 1 n Λ j n 1 n
Y
j=1
for each ϕ L2 (R3n). Hence
n ∈ sym k
n
(Q⊥ϕ )(k ,...,k ) = 1 χ (k ) ϕ (k ,...,k ). (4.19)
Λ n 1 n n −Y Λ j o n 1 n
j=1
Ifϕ (k ,...,k ) 0a.e., thentherighthandsideof(4.19)ispositivefora.e. k ,...,k
n 1 n 1 n
because 1 n ≥χ (k ) 0. This means Q⊥☎0 w.r.t. F . ✷
− j=1 Λ j ≥ Λ +
Q
As to the annihilation- and creation operators, we remark the following properties:
a(f)Q = a(χ f)= dkf(k)a(k), (4.20)
Λ Λ Z
|k|≤Λ
Q a(f)∗ = a(χ f)∗ = dkf(k)a(k)∗, (4.21)
Λ Λ Z
|k|≤Λ
dΓ(ω)Q = dΓ(χ ω)= dkω(k)a(k)∗a(k). (4.22)
Λ Λ Z
|k|≤Λ
Next let us introduce a natural self-dual cone in F . To this end, define
Λ
F(n) = ϕ L2 (B×n) ϕ(k ,...k ) 0 a.e. (4.23)
Λ,+ ∈ sym Λ | 1 n ≥
(cid:8) (cid:9)
with F(0) = R+. Each F(n) is a self-dual cone in L2(B )⊗sn = L2 (B×n).
Λ,+ Λ,+ Λ sym Λ
Definition 4.7 The local Fr¨ohlich cone is defined by
⊕ (n)
F = F . (4.24)
Λ,+ Λ,+
Xn≥0
F is a self-dual cone in F .
Λ,+ Λ
♦
Proposition 4.8 Propositions 4.3, 4.4 and 4.5 are still true even if one replaces
L2(R3) and F by L2(B ) and F respectively.
k + + Λ + Λ,+
As to the proof of the above proposition, see [31].
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