Proof of Theorem 4.3.1

Proof of Theorem 4.3.1. (A) Non-zero eigenvalues. The non-zero eigenvalues of Kθ(0, λ) are simple, given by εa,b(0, λ) = λ²δa,b, for a 6= b. We denote by ϕa,b⊗ X_a,b

By analytic perturbation theory, K_θ(σ, λ) has a simple eigenvalue in the vicinity of λ²δ_a,b, for small σ. It is given by

ε_a,b(σ, λ) = λ²δ_a,b+ σε⁽¹⁾_a,b+ σ²ε⁽²⁾_a,b+ O_λ(σ³), (A.25)

where (see [20, Sect. II.2.2] and also [34, Thm. XII.12])

ε⁽¹⁾_a,b = Tr(LSPa,b) = [HS]a,a− [HS]b,b. (A.26)

Here, we have set [H_S]_a,b = hϕ_a, H_Sϕ_bi. The second order correction is

ε⁽²⁾_a,b = −Tr L_S(K_θ(0, λ) − λ²δ_a,b)⁻¹P¯_a,bL_SP_a,b
. (A.27)

We write ¯P for 1l − P for general projections P . We set P_a,b^S = |ϕ_a,bihϕ_a,b| and P_a,b^R = |X_a,bihX_a,b^∗ |. Then P_a,b= P_a,b^S ⊗ P_a,b^R and ¯P_a,b = ¯P_a,b^S ⊗ 1l_R+ P_a,b^S ⊗ ¯P_a,b^R. It follows that ¯Pa,bLS (ϕa,b⊗ Xa,b) = ( ¯P_a,b^S LSϕa,b) ⊗ Xa,b. Using this and ¯P_a,b^S =P

(c,d)6=(a,b)P_c,d^S in expression (A.27) yields

ε⁽²⁾_a,b = − X

(c,d)6=(a,b)

ϕ_a,b⊗ X_a,b^∗ , L_S(K_θ(0, λ) − λ²δ_a,b)⁻¹ϕ_c,d⊗ X_a,b hϕ_c,d, L_Sϕ_a,bi .

‘Replacing’ ϕ_c,d⊗ X_a,b by the eigenvector ϕ_c,d⊗ X_c,d, we obtain

ε⁽²⁾_a,b = − X

(c,d)6=(a,b)

1

λ²(δ_c,d− δ_a,b)| hϕa,b, LSϕc,di |²X_a,b^∗ , Xc,d + ξ, (A.28)

where

ξ = X

(c,d)6=(a,b)

ϕa,b⊗ X_a,b^∗ , L_S(K_θ(0, λ) − λ²δ_a,b)⁻¹ϕ_c,d⊗ (X_c,d− X_a,b) hϕc,d, L_Sϕ_a,bi . (A.29) By perturbation theory, we have Xa,b= Ω_R+O(λ). Therefore, X_c,d−X_a,b= O(λ) and X_a,b^∗ , X_c,d = 1 + O(λ). Together with the bound (A.33) of Corollary A.2.2 below, we obtain

|ξ| ≤ C

|λ|. (A.30)

Finally,

hϕ_a,b, L_Sϕ_c,di = χ_b=d[H_S]_a,c− χ_a=c[H_S]_d,b. (A.31)

Relation (2.20) for a 6= b follows from (A.28), (A.30) and (A.31) and a little algebra.

Proposition A.2.1 (Bound on the resolvent). There are constants C and λ₀

where C₁ is a constant depending only on Imθ.

Knowing the bound on the resolvent we can obtain a bound on the reduced resol-vent.

for some constant C₂ depending on Imθ.

Proof of Corollary A.2.2. The reduced resolvent has the representation

(Kθ(0, λ) − λ²δa,b)⁻¹P¯a,b = −1 (independent of λ) such that Γ_a,b(λ) encircles only the eigenvalue λ²δ_a,band such that Γ_a,b(λ) lies within the region of z for which the bound (A.32) holds, according to Proposition A.2.1. Then dist(E , z) is a constant times λ². It follows that

k(K_θ(0, λ) − λ²δ_a,b)⁻¹P¯_a,bk ≤ C

Proof of Proposition A.2.1. Let P_R = |Ω_RihΩ_R|, ¯P_R = 1l − P_R, and R(z) = (Kθ(0, λ) − z)⁻¹.

Step 1. For any ψ ∈ H we have

ψ, ¯P_R(K_θ(0, λ) − z) ¯P_Rψ

≥ Imψ, ¯P_R(K_θ(0, λ) − z) ¯P_Rψ

= ψ, ¯P_R N^1/2{Imθ + λImN^−1/2I_θN^−1/2}N^1/2− Imz
P¯_Rψ

≥ (Imθ − C|λ| − Imz)k ¯P_Rψk²

≥ ¹₂Imθ k ¯P_Rψk².

By the Cauchy-Schwartz inequality, it follows that k ¯PR(Kθ(0, λ)−z) ¯PRψk ≥ ¹₂Imθ k ¯PRψk and therefore

k ¯P_RR(z) ¯P_Rk ≤ 2

Imθ. (A.34)

Step 2. Consider the Feshbach map

F_z = P_R(−z − λ²I_θP¯_RR(z) ¯P_RI_θ)P_R

= P_R(−z − λ²I_θP¯_RR(0) ¯P_RI_θ)P_R+ O(λ²|z|). (A.35)

Let

Gz = −λ²PRIθP¯RR(z) ¯PRIθPR. (A.36)

By the isospectrality property of the Feshbach map (see e.g. [5, Theorem IV.1]) we know that

G_λ²_δa,bϕ_a,b⊗ Ω_R = λ²δ_a,bϕ_a,b⊗ Ω_R,

for all a, b = 1, . . . , N . We also have G_z − G_ζ = O(λ²|z − ζ|), as long as Imz, Imζ <

1

4Imθ. It follows that G₀ϕ_a,b⊗ Ω_R = λ²δ_a,bϕ_a,b⊗ Ω_R+ O(λ⁴), for all a, b = 1, . . . , N .

Therefore, G₀ =PN

a,b=1λ²δ_a,b|ϕ_a,bihϕ_a,b| ⊗ P_R+ O(λ⁴), and so

G_z =

N

X

a,b=1

λ²δ_a,b|ϕ_a,bihϕ_a,b| ⊗ P_R + O(λ⁴+ λ²|z|). (A.37)

Using (A.37) and (A.36) in (A.35) shows that

Fz =

N

X

a,b=1

(λ²δa,b− z)|ϕa,bihϕa,b| ⊗ PR+ O(λ⁴+ λ²|z|). (A.38)

The sum on the right side is an invertible operator, the norm of the inverse being

a,b=1,...,Nmax |λ²δ_a,b− z|⁻¹ = [dist(E , z)]⁻¹.

Therefore, there is a constant C s.t. if

λ⁴+ λ²|z| < C dist(E, z), (A.39)

then Fz is invertible and

kF_z⁻¹k ≤ 2

dist(E , z). (A.40)

Let α > 0 be fixed, and take z s.t. dist(E , z) ≥ αλ². Then (A.39) is satisfied provided λ is small enough and |z| < Cα.

Step 3. The resolvent R(z) is related to ¯P_RR(z) ¯P_R and F_z⁻¹ by (see e.g. [5, Eqn.

(IV.14)])

R(z) = P_R− ¯P_RR(z) ¯P_RK_θ(0, λ)P_R
F_z⁻¹ P_R− P_RK_θ(0, λ) ¯P_RR(z) ¯P_R
+ ¯P_RR(z) ¯P_R.

We combine this equation with the bounds k ¯P_RK_θ(0, λ)P_Rk, kP_RK_θ(0, λ) ¯P_Rk ≤ C|λ|

and (A.34), (A.40) to arrive at the estimate (A.32). This completes the proof of

Proposition A.2.1.  (B) Zero eigenvalue. Let P (σ) be the group projection associated to the eigen-values of Kθ(σ, λ) bifurcating out of the origin as σ 6= 0. Here, we consider λ fixed and σ small. The null space of K_θ(0, λ) is known exactly, see (4.16). Let X_a,a^∗ ∈ H_R be the vector satisfying K_a,a^∗ X_a,a^∗ = 0 andΩ_R, X_a,a^∗  = 1. We have X_a,a^∗ = Ω_R+ O(λ).

Then P (0) = PN

a=1|ϕ_a,aihϕ_a,a| ⊗ |Ω_RihX_a,a^∗ |. Note that P (0)L_SP (0) = 0. Analytic perturbation theory gives

K_θ(σ, λ)P (σ) = σ²T₂+ O_λ(σ³)

T₂ = −P (0)L_SK_θ(0, λ)⁻¹L_SP (0). (A.41)

We have L_SP (0) =PN a=1

P

c,d=1,...,N ;c6=d|ϕ_c,dihϕ_a,a| ⊗ P_Rhϕ_c,d, L_Sϕ_a,ai + O(λ). Next,

K_θ(0, λ)⁻¹ϕ_c,d⊗ Ω_R = K_θ(0, λ)⁻¹ϕ_c,d⊗ (X_c,d+ Ω_R− X_c,d)

= 1

λ²δc,d

ϕ_c,d⊗ Ω_R+ O(λ⁻¹), (A.42)

where we use Corollary A.2.2 in the last step. Starting from (A.41) and using (A.42), we arrive at

T2 = 2i

λ²T + O(λ⁻¹), (A.43)

where the operator T has matrix elements [T ]a,b = hϕ_a,a⊗ Ω_R, T ϕ_b,b⊗ Ω_Ri given by (2.19). In this derivation, we also use that δ_b,a = −δ_a,b, see (2.15). Note that T is a real symmetric matrix, [T ]_a,b < 0 for a 6= b, and [T ]_a,a = −P

b6=a[T ]_a,b. These properties imply that for x = (x₁, . . . , x_N) ∈ C^N, hx, T xi =PN

a,b=1|[T ]_a,b| |x_a− x_b|² ≥ 0. Therefore, if [T ]_a,b 6= 0 for all a 6= b, then zero is a simple eigenvalue of T , with eigenvector proportional to (1, . . . , 1) and all other eigenvalues of T are strictly positive. This completes the proof of Theorem 4.3.1. 

B.1 Tomita-Takesaki theory

We present a brief overview of Tomita-Takesaki modular theory in this section. We refer to [3, 6] for a more detailed exposition.

Let M be a von Neumann algebra on a Hilbert space H containing a unit vector Ω which is cyclic and separating for M and M⁰. Hence we may define operators S₀ : MΩ → MΩ and F₀ : M⁰Ω → M⁰Ω as the following:

S0AΩ =A^∗Ω, ∀A ∈ M F₀BΩ =B^∗Ω, ∀B ∈ M⁰.

(B.1)

Since Ω is cyclic and separating, S₀ and F₀ are well-defined on dense domains. Now we state the following well known result [3, 6].

Proposition B.1.1. The operators S₀ and F₀ are closable and F₀ = S₀^∗, S₀ = F₀^∗. To simplify the notations, let S = S0 and F = F0. It follows from the polar decomposition theorem that there exits a unique positive self-adjoint operator ∆ and a unique anti-unitary operator J such that

S = J ∆^1/2 = ∆^−1/2J. (B.2)

80

Here, ∆ is the so-called modular operator and J the modular conjugation associated with (M, Ω). The following theorem states many important equalities related to the polar decomposition which can be found in [6].

Theorem B.1.2.

∆ =F S, ∆⁻¹ = SF, J² = I, S =J ∆^1/2= ∆^−1/2J, F =J ∆^−1/2= ∆^1/2J, J ∆^it =∆^itJ,

J Ω =∆Ω = Ω.

(B.3)

Note that the modular operator ∆ may be unbounded. The following remarkable Tomita-Takesaki theorem plays an important role in operator algebra and mathemat-ical physics.

Theorem B.1.3.

J MJ =M⁰,

∆^itM∆^−it =M ∀t ∈ R.

(B.4)

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