Mean-field evolution of fermionswith Coulomb interaction
Chiara Saffirio
Universitat Zurich
in collaboration withM. Porta, S. Rademacher and B. Schlein
Munich, March 30-April 1, 2017.
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 1 / 11
SettingConsider a system of N interacting fermions with wave function ψN ∈ L2
a(R3N)
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 2 / 11
SettingConsider a system of N interacting fermions with wave function ψN ∈ L2
a(R3N)
Interaction: V (x) = 1|x|
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 2 / 11
SettingConsider a system of N interacting fermions with wave function ψN ∈ L2
a(R3N)
Interaction: V (x) = 1|x|
Motivation: dynamics of large atoms (e.g. electrically neutral atom)
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 2 / 11
SettingConsider a system of N interacting fermions with wave function ψN ∈ L2
a(R3N)
Interaction: V (x) = 1|x|
Motivation: dynamics of large atoms (e.g. electrically neutral atom)
HN =N∑
j=1
[−∆Xj −
N|Xj |
]+
N∑i<j
1|Xi − Xj |
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 2 / 11
SettingConsider a system of N interacting fermions with wave function ψN ∈ L2
a(R3N)
Interaction: V (x) = 1|x|
Motivation: dynamics of large atoms (e.g. electrically neutral atom)
HN =N∑
j=1
[−∆Xj −
N|Xj |
]+
N∑i<j
1|Xi − Xj |
General goal: study the dynamics of the low energy states
i∂tψN,t = HNψN,t ,ψN,0 = ψN .
N is large =⇒ look for scaling regimes in which the evolution can be wellapproximated by an effective dynamics.
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 2 / 11
SettingConsider a system of N interacting fermions with wave function ψN ∈ L2
a(R3N)
Interaction: V (x) = 1|x|
Motivation: dynamics of large atoms (e.g. electrically neutral atom)
HN =N∑
j=1
[−∆Xj −
N|Xj |
]+
N∑i<j
1|Xi − Xj |
Scaling?
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 2 / 11
SettingConsider a system of N interacting fermions with wave function ψN ∈ L2
a(R3N)
Interaction: V (x) = 1|x|
Motivation: dynamics of large atoms (e.g. electrically neutral atom)
HN =N∑
j=1
[−∆Xj −
N|Xj |
]+
N∑i<j
1|Xi − Xj |
Space variable scaling: Xj = N13 xj
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 2 / 11
SettingConsider a system of N interacting fermions with wave function ψN ∈ L2
a(R3N)
Interaction: V (x) = 1|x|
Motivation: dynamics of large atoms (e.g. electrically neutral atom)
HN =N∑
j=1
[−∆Xj −
N|Xj |
]+
N∑i<j
1|Xi − Xj |
Space variable scaling: Xj = N13 xj
HN =N∑
j=1
[−N
23 ∆xj −
N43
|xj |
]+ N
13
N∑i<j
1|xi − xj |
= N43
N∑
j=1
[−N−
23 ∆xj −
1|xj |
]+
1N
N∑i<j
1|xi − xj |
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 2 / 11
SettingConsider a system of N interacting fermions with wave function ψN ∈ L2
a(R3N)
Interaction: V (x) = 1|x|
Motivation: dynamics of large atoms (e.g. electrically neutral atom)
HN =N∑
j=1
[−∆Xj −
N|Xj |
]+
N∑i<j
1|Xi − Xj |
Space variable scaling: Xj = N13 xj
HN =N∑
j=1
[−N
23 ∆xj −
N43
|xj |
]+ N
13
N∑i<j
1|xi − xj |
= N43
N∑
j=1
[−ε2∆xj −
1|xj |
]+
1N
N∑i<j
1|xi − xj |
ε = N−
13
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 2 / 11
SettingConsider a system of N interacting fermions with wave function ψN ∈ L2
a(R3N)
Interaction: V (x) = 1|x|
Motivation: dynamics of large atoms (e.g. electrically neutral atom)
HN =N∑
j=1
[−∆Xj −
N|Xj |
]+
N∑i<j
1|Xi − Xj |
Space variable scaling: Xj = N13 xj
i∂tψN,t = N43
N∑
j=1
[−ε2∆xj −
1|xj |
]+
1N
N∑i<j
1|xi − xj |
ψN,t
ε = N−13
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 2 / 11
SettingConsider a system of N interacting fermions with wave function ψN ∈ L2
a(R3N)
Interaction: V (x) = 1|x|
Motivation: dynamics of large atoms (e.g. electrically neutral atom)
HN =N∑
j=1
[−∆Xj −
N|Xj |
]+
N∑i<j
1|Xi − Xj |
Space variable scaling: Xj = N13 xj
i∂tψN,t = N43
N∑
j=1
[−ε2∆xj −
1|xj |
]+
1N
N∑i<j
1|xi − xj |
ψN,t
Time scaling:
iε∂tψN,t =
N∑
j=1
[−ε2∆xj −
1|xj |
]+
1N
N∑i<j
1|xi − xj |
ψN,t ε = N−13
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 2 / 11
SettingConsider a system of N interacting fermions with wave function ψN ∈ L2
a(R3N)
Interaction: V (x) = 1|x|
Motivation: dynamics of large atoms (e.g. electrically neutral atom)
HN =N∑
j=1
[−∆Xj −
N|Xj |
]+
N∑i<j
1|Xi − Xj |
Space variable scaling: Xj = N13 xj
i∂tψN,t = N43
N∑
j=1
[−ε2∆xj −
1|xj |
]+
1N
N∑i<j
1|xi − xj |
ψN,t
Time scaling:
iε∂tψN,t =
N∑
j=1
[−ε2∆xj −
1|xj |
]+
1N
N∑i<j
1|xi − xj |
ψN,t ε = N−13
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 2 / 11
Effective evolution equation?
Ground state ' Slater determinant (Bach ’92, Graf & Solovej ’94)
ψSlater(x1, . . . , xN) =1√N!
det(fj (xi ))i,j≤N , fjj≤N o.n.s. in L2(R3).
Reduced one-particle density matrix:
ωN = N tr2...N |ψSlater〉〈ψSlater| =∑N
j=1 |fj〉〈fj | minimiser of the HF energy
EHF(ωN) = Tr (−∆ + Vext)ωN +1
2N1/3
∫dx dy
1|x − y |
[ρ(x)ρ(y)− |ωN(x ; y)|2]
where ρ(x) = 1NωN(x ; x).
Dynamics =⇒ γ(1)N,t = N tr2...N |ψN,t〉〈ψN,t | ' ωN,t
ωN,t solution to the time dependent Hartree-Fock eqn
iε∂tωN,t = [−ε2∆xj + 1|·| ∗ ρt − Xt , ωN,t ]
with Xt (x ; y) = 1N
1|x−y|ωN,t (x ; y).
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 3 / 11
Effective evolution equation?
Ground state ' Slater determinant (Bach ’92, Graf & Solovej ’94)
ψSlater(x1, . . . , xN) =1√N!
det(fj (xi ))i,j≤N , fjj≤N o.n.s. in L2(R3).
Reduced one-particle density matrix:
ωN = N tr2...N |ψSlater〉〈ψSlater| =∑N
j=1 |fj〉〈fj |
minimiser of the HF energy
EHF(ωN) = Tr (−∆ + Vext)ωN +1
2N1/3
∫dx dy
1|x − y |
[ρ(x)ρ(y)− |ωN(x ; y)|2]
where ρ(x) = 1NωN(x ; x).
Dynamics =⇒ γ(1)N,t = N tr2...N |ψN,t〉〈ψN,t | ' ωN,t
ωN,t solution to the time dependent Hartree-Fock eqn
iε∂tωN,t = [−ε2∆xj + 1|·| ∗ ρt − Xt , ωN,t ]
with Xt (x ; y) = 1N
1|x−y|ωN,t (x ; y).
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 3 / 11
Effective evolution equation?
Ground state ' Slater determinant (Bach ’92, Graf & Solovej ’94)
ψSlater(x1, . . . , xN) =1√N!
det(fj (xi ))i,j≤N , fjj≤N o.n.s. in L2(R3).
Reduced one-particle density matrix:
ωN = N tr2...N |ψSlater〉〈ψSlater| =∑N
j=1 |fj〉〈fj | minimiser of the HF energy
EHF(ωN) = Tr (−∆ + Vext)ωN +1
2N1/3
∫dx dy
1|x − y |
[ρ(x)ρ(y)− |ωN(x ; y)|2]
where ρ(x) = 1NωN(x ; x).
Dynamics =⇒ γ(1)N,t = N tr2...N |ψN,t〉〈ψN,t | ' ωN,t
ωN,t solution to the time dependent Hartree-Fock eqn
iε∂tωN,t = [−ε2∆xj + 1|·| ∗ ρt − Xt , ωN,t ]
with Xt (x ; y) = 1N
1|x−y|ωN,t (x ; y).
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 3 / 11
Effective evolution equation?
Ground state ' Slater determinant (Bach ’92, Graf & Solovej ’94)
ψSlater(x1, . . . , xN) =1√N!
det(fj (xi ))i,j≤N , fjj≤N o.n.s. in L2(R3).
Reduced one-particle density matrix:
ωN = N tr2...N |ψSlater〉〈ψSlater| =∑N
j=1 |fj〉〈fj | minimiser of the HF energy
EHF(ωN) = Tr (−∆ + Vext)ωN +1
2N1/3
∫dx dy
1|x − y |
[ρ(x)ρ(y)− |ωN(x ; y)|2]
where ρ(x) = 1NωN(x ; x).
Dynamics =⇒ γ(1)N,t = N tr2...N |ψN,t〉〈ψN,t | ' ωN,t
ωN,t solution to the time dependent Hartree-Fock eqn
iε∂tωN,t = [−ε2∆xj + 1|·| ∗ ρt − Xt , ωN,t ]
with Xt (x ; y) = 1N
1|x−y|ωN,t (x ; y).
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 3 / 11
Effective evolution equation?
Ground state ' Slater determinant (Bach ’92, Graf & Solovej ’94)
ψSlater(x1, . . . , xN) =1√N!
det(fj (xi ))i,j≤N , fjj≤N o.n.s. in L2(R3).
Reduced one-particle density matrix:
ωN = N tr2...N |ψSlater〉〈ψSlater| =∑N
j=1 |fj〉〈fj | minimiser of the HF energy
EHF(ωN) = Tr (−∆ + Vext)ωN +1
2N1/3
∫dx dy
1|x − y |
[ρ(x)ρ(y)− |ωN(x ; y)|2]
where ρ(x) = 1NωN(x ; x).
Dynamics =⇒ γ(1)N,t = N tr2...N |ψN,t〉〈ψN,t | ' ωN,t
ωN,t solution to the time dependent Hartree-Fock eqn
iε∂tωN,t = [−ε2∆xj + 1|·| ∗ ρt − Xt , ωN,t ]
with Xt (x ; y) = 1N
1|x−y|ωN,t (x ; y).
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 3 / 11
Effective evolution equation?
Ground state ' Slater determinant (Bach ’92, Graf & Solovej ’94)
ψSlater(x1, . . . , xN) =1√N!
det(fj (xi ))i,j≤N , fjj≤N o.n.s. in L2(R3).
Reduced one-particle density matrix:
ωN = N tr2...N |ψSlater〉〈ψSlater| =∑N
j=1 |fj〉〈fj | minimiser of the HF energy
EHF(ωN) = Tr (−∆ + Vext)ωN +1
2N1/3
∫dx dy
1|x − y |
[ρ(x)ρ(y)− |ωN(x ; y)|2]
where ρ(x) = 1NωN(x ; x).
Dynamics =⇒ γ(1)N,t = N tr2...N |ψN,t〉〈ψN,t | ' ωN,t
ωN,t solution to the time dependent Hartree-Fock eqn
iε∂tωN,t = [−ε2∆xj + 1|·| ∗ ρt − Xt , ωN,t ]
with Xt (x ; y) = 1N
1|x−y|ωN,t (x ; y).
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 3 / 11
Our result
TheoremLet ωN be a sequence of orthogonal projections on L2(R3) with tr ωN = N andtr (−ε2∆)ωN ≤ CN.
Let ωN,t be a solution to the TDHF equation with initial data ωN .Assume there exist T > 0, p > 5, C > 0 such that ∀ t ∈ [0,T ]
‖ρ|[x,ωN,t ]|‖L1 ≤ CNε , ‖ρ|[x,ωN,t ]|‖Lp ≤ CNε .
Let ψN ∈ L2a(R3N) be s.t. its one-particle reduced density matrix γ(1)
N satisfies
tr |γ(1)N − ωN | ≤ CNα , α ∈ [0,1) .
Consider the evolution ψN,t = e−itHN/εψN and γ(1)N,t the corresponding
one-particle reduced density matrix.Then for every δ > 0, there exists C > 0 s.t.
tr |γ(1)N,t − ωN,t | ≤ C(t)(Nα + N11/12+δ) .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 4 / 11
Our result
TheoremLet ωN be a sequence of orthogonal projections on L2(R3) with tr ωN = N andtr (−ε2∆)ωN ≤ CN.Let ωN,t be a solution to the TDHF equation with initial data ωN .
Assume there exist T > 0, p > 5, C > 0 such that ∀ t ∈ [0,T ]
‖ρ|[x,ωN,t ]|‖L1 ≤ CNε , ‖ρ|[x,ωN,t ]|‖Lp ≤ CNε .
Let ψN ∈ L2a(R3N) be s.t. its one-particle reduced density matrix γ(1)
N satisfies
tr |γ(1)N − ωN | ≤ CNα , α ∈ [0,1) .
Consider the evolution ψN,t = e−itHN/εψN and γ(1)N,t the corresponding
one-particle reduced density matrix.Then for every δ > 0, there exists C > 0 s.t.
tr |γ(1)N,t − ωN,t | ≤ C(t)(Nα + N11/12+δ) .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 4 / 11
Our result
TheoremLet ωN be a sequence of orthogonal projections on L2(R3) with tr ωN = N andtr (−ε2∆)ωN ≤ CN.Let ωN,t be a solution to the TDHF equation with initial data ωN .Assume there exist T > 0, p > 5, C > 0 such that ∀ t ∈ [0,T ]
‖ρ|[x,ωN,t ]|‖L1 ≤ CNε , ‖ρ|[x,ωN,t ]|‖Lp ≤ CNε .
Let ψN ∈ L2a(R3N) be s.t. its one-particle reduced density matrix γ(1)
N satisfies
tr |γ(1)N − ωN | ≤ CNα , α ∈ [0,1) .
Consider the evolution ψN,t = e−itHN/εψN and γ(1)N,t the corresponding
one-particle reduced density matrix.Then for every δ > 0, there exists C > 0 s.t.
tr |γ(1)N,t − ωN,t | ≤ C(t)(Nα + N11/12+δ) .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 4 / 11
Our result
TheoremLet ωN be a sequence of orthogonal projections on L2(R3) with tr ωN = N andtr (−ε2∆)ωN ≤ CN.Let ωN,t be a solution to the TDHF equation with initial data ωN .Assume there exist T > 0, p > 5, C > 0 such that ∀ t ∈ [0,T ]
‖ρ|[x,ωN,t ]|‖L1 ≤ CNε , ‖ρ|[x,ωN,t ]|‖Lp ≤ CNε .
Let ψN ∈ L2a(R3N) be s.t. its one-particle reduced density matrix γ(1)
N satisfies
tr |γ(1)N − ωN | ≤ CNα , α ∈ [0,1) .
Consider the evolution ψN,t = e−itHN/εψN and γ(1)N,t the corresponding
one-particle reduced density matrix.Then for every δ > 0, there exists C > 0 s.t.
tr |γ(1)N,t − ωN,t | ≤ C(t)(Nα + N11/12+δ) .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 4 / 11
Our result
TheoremLet ωN be a sequence of orthogonal projections on L2(R3) with tr ωN = N andtr (−ε2∆)ωN ≤ CN.Let ωN,t be a solution to the TDHF equation with initial data ωN .Assume there exist T > 0, p > 5, C > 0 such that ∀ t ∈ [0,T ]
‖ρ|[x,ωN,t ]|‖L1 ≤ CNε , ‖ρ|[x,ωN,t ]|‖Lp ≤ CNε .
Let ψN ∈ L2a(R3N) be s.t. its one-particle reduced density matrix γ(1)
N satisfies
tr |γ(1)N − ωN | ≤ CNα , α ∈ [0,1) .
Consider the evolution ψN,t = e−itHN/εψN and γ(1)N,t the corresponding
one-particle reduced density matrix.
Then for every δ > 0, there exists C > 0 s.t.
tr |γ(1)N,t − ωN,t | ≤ C(t)(Nα + N11/12+δ) .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 4 / 11
Our result
TheoremLet ωN be a sequence of orthogonal projections on L2(R3) with tr ωN = N andtr (−ε2∆)ωN ≤ CN.Let ωN,t be a solution to the TDHF equation with initial data ωN .Assume there exist T > 0, p > 5, C > 0 such that ∀ t ∈ [0,T ]
‖ρ|[x,ωN,t ]|‖L1 ≤ CNε , ‖ρ|[x,ωN,t ]|‖Lp ≤ CNε .
Let ψN ∈ L2a(R3N) be s.t. its one-particle reduced density matrix γ(1)
N satisfies
tr |γ(1)N − ωN | ≤ CNα , α ∈ [0,1) .
Consider the evolution ψN,t = e−itHN/εψN and γ(1)N,t the corresponding
one-particle reduced density matrix.Then for every δ > 0, there exists C > 0 s.t.
tr |γ(1)N,t − ωN,t | ≤ C(t)(Nα + N11/12+δ) .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 4 / 11
Our result
TheoremLet ωN be a sequence of orthogonal projections on L2(R3) with tr ωN = N andtr (−ε2∆)ωN ≤ CN.Let ωN,t be a solution to the TDHF equation with initial data ωN .Assume there exist T > 0, p > 5, C > 0 such that ∀ t ∈ [0,T ]
‖ρ|[x,ωN,t ]|‖L1 ≤ CNε , ‖ρ|[x,ωN,t ]|‖Lp ≤ CNε .
Let ψN ∈ L2a(R3N) be s.t. its one-particle reduced density matrix γ(1)
N satisfies
tr |γ(1)N − ωN | ≤ CNα , α ∈ [0,1) .
Consider the evolution ψN,t = e−itHN/εψN and γ(1)N,t the corresponding
one-particle reduced density matrix.Then for every δ > 0, there exists C > 0 s.t.
tr |γ(1)N,t − ωN,t | ≤ C(t)(Nα + N11/12+δ) .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 5 / 11
Our result
TheoremLet ωN be a sequence of orthogonal projections on L2(R3) with tr ωN = N andtr (−ε2∆)ωN ≤ CN.Let ωN,t be a solution to the TDHF equation with initial data ωN .Assume there exist T > 0, p > 5, C > 0 such that ∀ t ∈ [0,T ]
‖ρ|[x,ωN,t ]|‖L1 ≤ CNε , ‖ρ|[x,ωN,t ]|‖Lp ≤ CNε .
Let ψN ∈ L2a(R3N) be s.t. its one-particle reduced density matrix γ(1)
N satisfies
tr |γ(1)N − ωN | ≤ CNα , α ∈ [0,1) .
Consider the evolution ψN,t = e−itHN/εψN and γ(1)N,t the corresponding
one-particle reduced density matrix.Then for every δ > 0, there exists C > 0 s.t.
tr |γ(1)N,t − ωN,t | ≤ C(t)(Nα + N11/12+δ) .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 5 / 11
Our result
TheoremLet ωN be a sequence of orthogonal projections on L2(R3) with tr ωN = N andtr (−ε2∆)ωN ≤ CN.Let ωN,t be a solution to the TDHF equation with initial data ωN .Assume there exist T > 0, p > 5, C > 0 such that ∀ t ∈ [0,T ]
‖ρ|[x,ωN,t ]|‖L1 ≤ CNε , ‖ρ|[x,ωN,t ]|‖Lp ≤ CNε .
Let ψN ∈ L2a(R3N) be s.t. its one-particle reduced density matrix γ(1)
N satisfies
tr |γ(1)N − ωN | ≤ CNα , α ∈ [0,1) .
Consider the evolution ψN,t = e−itHN/εψN and γ(1)N,t the corresponding
one-particle reduced density matrix.Then for every δ > 0, there exists C > 0 s.t.
tr |γ(1)N,t − ωN,t | ≤ C(t)(Nα + N11/12+δ) .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 5 / 11
Remarks
Reduce the many-body problem to a PDE problem
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 6 / 11
Remarks
Reduce the many-body problem to a PDE problem
Translation invariant states
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 6 / 11
Remarks
Reduce the many-body problem to a PDE problem
Translation invariant states
Λ box with periodic b.c. =⇒ ωN,t (x ; y) ' ωN,t (x − y)
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 6 / 11
Remarks
Reduce the many-body problem to a PDE problem
Translation invariant states
Λ box with periodic b.c. =⇒ ωN,t (x ; y) ' ωN,t (x − y)
[x , ωN,t ] and |[x , ωN,t ]| translation invariant
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 6 / 11
Remarks
Reduce the many-body problem to a PDE problem
Translation invariant states
Λ box with periodic b.c. =⇒ ωN,t (x ; y) ' ωN,t (x − y)
[x , ωN,t ] and |[x , ωN,t ]| translation invariant =⇒ ρ|[x,ωN,t ]| ≡ const
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 6 / 11
Remarks
Reduce the many-body problem to a PDE problem
Translation invariant states
Λ box with periodic b.c. =⇒ ωN,t (x ; y) ' ωN,t (x − y)
[x , ωN,t ] and |[x , ωN,t ]| translation invariant =⇒ ρ|[x,ωN,t ]| ≡ const
ρ|[x,ωN ]| = O(Nε)
ωN function of (x − y) decaying at distance |x − y | ε.
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 6 / 11
Remarks
Reduce the many-body problem to a PDE problem
Translation invariant states
Λ box with periodic b.c. =⇒ ωN,t (x ; y) ' ωN,t (x − y)
[x , ωN,t ] and |[x , ωN,t ]| translation invariant =⇒ ρ|[x,ωN,t ]| ≡ const
ρ|[x,ωN ]| = O(Nε)
ωN function of (x − y) decaying at distance |x − y | ε.
N dependence in the HF equation
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 6 / 11
Remarks
Reduce the many-body problem to a PDE problem
Translation invariant states
Λ box with periodic b.c. =⇒ ωN,t (x ; y) ' ωN,t (x − y)
[x , ωN,t ] and |[x , ωN,t ]| translation invariant =⇒ ρ|[x,ωN,t ]| ≡ const
ρ|[x,ωN ]| = O(Nε)
ωN function of (x − y) decaying at distance |x − y | ε.
N dependence in the HF equation: N →∞?
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 6 / 11
Remarks
Reduce the many-body problem to a PDE problem
Translation invariant states
Λ box with periodic b.c. =⇒ ωN,t (x ; y) ' ωN,t (x − y)
[x , ωN,t ] and |[x , ωN,t ]| translation invariant =⇒ ρ|[x,ωN,t ]| ≡ const
ρ|[x,ωN ]| = O(Nε)
ωN function of (x − y) decaying at distance |x − y | ε.
N dependence in the HF equation: N →∞?The Vlasov equation is the next degree of approximation.
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 6 / 11
Remarks
Reduce the many-body problem to a PDE problem
Translation invariant states
Λ box with periodic b.c. =⇒ ωN,t (x ; y) ' ωN,t (x − y)
[x , ωN,t ] and |[x , ωN,t ]| translation invariant =⇒ ρ|[x,ωN,t ]| ≡ const
ρ|[x,ωN ]| = O(Nε)
ωN function of (x − y) decaying at distance |x − y | ε.
N dependence in the HF equation: N →∞?The Vlasov equation is the next degree of approximation.The Wigner transform of ωN,t for N →∞ solves the Vlasov equation
∂tWt (x , v) + v · ∇xWt (x , v) = ∇(V ∗ ρt )(x) · ∇v Wt (x , v)
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 6 / 11
Previous results
1980: Narnhofer and Sewell (convergence to Vlasov for analyticpotentials);1982: Spohn (extension to C2 potentials);
2003: Elgart, Erdos, Schlein, Yau (TDHF short times, analytic pot.);other scalings:
I 2002: Bardos, Golse, Gottlieb, Mauser;I 2010: Frohlich, Knowles;
2013: Benedikter, Porta, Schlein (larger class of pot., all times);2015: Benedikter, Jaksic, Porta, S., Schlein (mixed states);2016 Coulomb potential with different scalings:
I Bach, Breteaux, Petrat, Pickl, Tzaneteas;I Petrat, Pickl.
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 7 / 11
Previous results
1980: Narnhofer and Sewell (convergence to Vlasov for analyticpotentials);1982: Spohn (extension to C2 potentials);2003: Elgart, Erdos, Schlein, Yau (TDHF short times, analytic pot.);
other scalings:I 2002: Bardos, Golse, Gottlieb, Mauser;I 2010: Frohlich, Knowles;
2013: Benedikter, Porta, Schlein (larger class of pot., all times);2015: Benedikter, Jaksic, Porta, S., Schlein (mixed states);2016 Coulomb potential with different scalings:
I Bach, Breteaux, Petrat, Pickl, Tzaneteas;I Petrat, Pickl.
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 7 / 11
Previous results
1980: Narnhofer and Sewell (convergence to Vlasov for analyticpotentials);1982: Spohn (extension to C2 potentials);2003: Elgart, Erdos, Schlein, Yau (TDHF short times, analytic pot.);other scalings:
I 2002: Bardos, Golse, Gottlieb, Mauser;I 2010: Frohlich, Knowles;
2013: Benedikter, Porta, Schlein (larger class of pot., all times);2015: Benedikter, Jaksic, Porta, S., Schlein (mixed states);2016 Coulomb potential with different scalings:
I Bach, Breteaux, Petrat, Pickl, Tzaneteas;I Petrat, Pickl.
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 7 / 11
Previous results
1980: Narnhofer and Sewell (convergence to Vlasov for analyticpotentials);1982: Spohn (extension to C2 potentials);2003: Elgart, Erdos, Schlein, Yau (TDHF short times, analytic pot.);other scalings:
I 2002: Bardos, Golse, Gottlieb, Mauser;I 2010: Frohlich, Knowles;
2013: Benedikter, Porta, Schlein (larger class of pot., all times);2015: Benedikter, Jaksic, Porta, S., Schlein (mixed states);
2016 Coulomb potential with different scalings:I Bach, Breteaux, Petrat, Pickl, Tzaneteas;I Petrat, Pickl.
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 7 / 11
Previous results
1980: Narnhofer and Sewell (convergence to Vlasov for analyticpotentials);1982: Spohn (extension to C2 potentials);2003: Elgart, Erdos, Schlein, Yau (TDHF short times, analytic pot.);other scalings:
I 2002: Bardos, Golse, Gottlieb, Mauser;I 2010: Frohlich, Knowles;
2013: Benedikter, Porta, Schlein (larger class of pot., all times);2015: Benedikter, Jaksic, Porta, S., Schlein (mixed states);2016 Coulomb potential with different scalings:
I Bach, Breteaux, Petrat, Pickl, Tzaneteas;I Petrat, Pickl.
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 7 / 11
Idea of the proofFock space representation:
F =⊕n≥0
L2a(R3n)
creation and annihilation operators: ∀ f ∈ L2(R3)
a∗(f ) =
∫dx f (x) a∗x a(f ) =
∫dx f (x) ax
number of particle operator:
N =
∫dx a∗x ax
second quantisation of J, operator on L2(R3)
dΓ(J) =
∫∫dx dy J(x ; y) a∗x ay
Let Ψ ∈ F ,
γ(1)Ψ (x ; y) = 〈Ψ, a∗x ay Ψ〉 =⇒ tr γ
(1)Ψ = 〈Ψ, NΨ〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 8 / 11
Idea of the proofFock space representation:
F =⊕n≥0
L2a(R3n)
creation and annihilation operators:
∀ f ∈ L2(R3)
a∗(f ) =
∫dx f (x) a∗x a(f ) =
∫dx f (x) ax
number of particle operator:
N =
∫dx a∗x ax
second quantisation of J, operator on L2(R3)
dΓ(J) =
∫∫dx dy J(x ; y) a∗x ay
Let Ψ ∈ F ,
γ(1)Ψ (x ; y) = 〈Ψ, a∗x ay Ψ〉 =⇒ tr γ
(1)Ψ = 〈Ψ, NΨ〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 8 / 11
Idea of the proofFock space representation:
F =⊕n≥0
L2a(R3n)
creation and annihilation operators: ∀ f ∈ L2(R3)
a∗(f ) =
∫dx f (x) a∗x a(f ) =
∫dx f (x) ax
number of particle operator:
N =
∫dx a∗x ax
second quantisation of J, operator on L2(R3)
dΓ(J) =
∫∫dx dy J(x ; y) a∗x ay
Let Ψ ∈ F ,
γ(1)Ψ (x ; y) = 〈Ψ, a∗x ay Ψ〉 =⇒ tr γ
(1)Ψ = 〈Ψ, NΨ〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 8 / 11
Idea of the proofFock space representation:
F =⊕n≥0
L2a(R3n)
creation and annihilation operators: ∀ f ∈ L2(R3)
a∗(f ) =
∫dx f (x) a∗x a(f ) =
∫dx f (x) ax
number of particle operator:
N =
∫dx a∗x ax
second quantisation of J, operator on L2(R3)
dΓ(J) =
∫∫dx dy J(x ; y) a∗x ay
Let Ψ ∈ F ,
γ(1)Ψ (x ; y) = 〈Ψ, a∗x ay Ψ〉 =⇒ tr γ
(1)Ψ = 〈Ψ, NΨ〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 8 / 11
Idea of the proofFock space representation:
F =⊕n≥0
L2a(R3n)
creation and annihilation operators: ∀ f ∈ L2(R3)
a∗(f ) =
∫dx f (x) a∗x a(f ) =
∫dx f (x) ax
number of particle operator:
N =
∫dx a∗x ax
second quantisation of J, operator on L2(R3)
dΓ(J) =
∫∫dx dy J(x ; y) a∗x ay
Let Ψ ∈ F ,
γ(1)Ψ (x ; y) = 〈Ψ, a∗x ay Ψ〉 =⇒ tr γ
(1)Ψ = 〈Ψ, NΨ〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 8 / 11
Idea of the proofFock space representation:
F =⊕n≥0
L2a(R3n)
creation and annihilation operators: ∀ f ∈ L2(R3)
a∗(f ) =
∫dx f (x) a∗x a(f ) =
∫dx f (x) ax
number of particle operator:
N =
∫dx a∗x ax
second quantisation of J, operator on L2(R3)
dΓ(J) =
∫∫dx dy J(x ; y) a∗x ay
Let Ψ ∈ F ,
γ(1)Ψ (x ; y) = 〈Ψ, a∗x ay Ψ〉 =⇒ tr γ
(1)Ψ = 〈Ψ, NΨ〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 8 / 11
Idea of the proofFock space representation:
F =⊕n≥0
L2a(R3n)
creation and annihilation operators: ∀ f ∈ L2(R3)
a∗(f ) =
∫dx f (x) a∗x a(f ) =
∫dx f (x) ax
number of particle operator:
N =
∫dx a∗x ax
second quantisation of J, operator on L2(R3)
dΓ(J) =
∫∫dx dy J(x ; y) a∗x ay
Let Ψ ∈ F ,
γ(1)Ψ (x ; y) = 〈Ψ, a∗x ay Ψ〉
=⇒ tr γ(1)Ψ = 〈Ψ, NΨ〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 8 / 11
Idea of the proofFock space representation:
F =⊕n≥0
L2a(R3n)
creation and annihilation operators: ∀ f ∈ L2(R3)
a∗(f ) =
∫dx f (x) a∗x a(f ) =
∫dx f (x) ax
number of particle operator:
N =
∫dx a∗x ax
second quantisation of J, operator on L2(R3)
dΓ(J) =
∫∫dx dy J(x ; y) a∗x ay
Let Ψ ∈ F ,
γ(1)Ψ (x ; y) = 〈Ψ, a∗x ay Ψ〉 =⇒ tr γ
(1)Ψ = 〈Ψ, NΨ〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 8 / 11
Idea of the proofBogoliubov transformation:ωN =
∑Nj=1 |fj〉〈fj |, fjj≤N o.n.s. in L2(R3).
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 9 / 11
Idea of the proofBogoliubov transformation:ωN =
∑Nj=1 |fj〉〈fj |, fjj≤N o.n.s. in L2(R3).
RωN : F → F unitary implementor of a Bolgoliubov transformation
RωN Ω = a∗(f1) . . . a∗(fN) Ω
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 9 / 11
Idea of the proofBogoliubov transformation:ωN =
∑Nj=1 |fj〉〈fj |, fjj≤N o.n.s. in L2(R3).
RωN : F → F unitary implementor of a Bolgoliubov transformation
RωN Ω = a∗(f1) . . . a∗(fN) Ω
s.t. ∀ g ∈ L2(R3), R∗ωNa∗(g)RωN = a∗((1− ωN) g) + a(ωN g) .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 9 / 11
Idea of the proofBogoliubov transformation:ωN =
∑Nj=1 |fj〉〈fj |, fjj≤N o.n.s. in L2(R3).
RωN : F → F unitary implementor of a Bolgoliubov transformation
RωN Ω = a∗(f1) . . . a∗(fN) Ω
s.t. ∀ g ∈ L2(R3), R∗ωNa∗(g)RωN = a∗((1− ωN) g) + a(ωN g) .
Compute γ(1)N,t :
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 9 / 11
Idea of the proofBogoliubov transformation:ωN =
∑Nj=1 |fj〉〈fj |, fjj≤N o.n.s. in L2(R3).
RωN : F → F unitary implementor of a Bolgoliubov transformation
RωN Ω = a∗(f1) . . . a∗(fN) Ω
s.t. ∀ g ∈ L2(R3), R∗ωNa∗(g)RωN = a∗((1− ωN) g) + a(ωN g) .
Compute γ(1)N,t :
γ(1)N,t (x ; y) = 〈 e−iHN t/εRωN Ω , a∗x ay e−iHN t/εRωN Ω〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 9 / 11
Idea of the proofBogoliubov transformation:ωN =
∑Nj=1 |fj〉〈fj |, fjj≤N o.n.s. in L2(R3).
RωN : F → F unitary implementor of a Bolgoliubov transformation
RωN Ω = a∗(f1) . . . a∗(fN) Ω
s.t. ∀ g ∈ L2(R3), R∗ωNa∗(g)RωN = a∗((1− ωN) g) + a(ωN g) .
Compute γ(1)N,t :
γ(1)N,t (x ; y) = 〈RωN,t R
∗ωN,t
e−iHN t/εRωN Ω , a∗x ay RωN,t R∗ωN,t
e−iHN t/εRωN Ω〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 9 / 11
Idea of the proofBogoliubov transformation:ωN =
∑Nj=1 |fj〉〈fj |, fjj≤N o.n.s. in L2(R3).
RωN : F → F unitary implementor of a Bolgoliubov transformation
RωN Ω = a∗(f1) . . . a∗(fN) Ω
s.t. ∀ g ∈ L2(R3), R∗ωNa∗(g)RωN = a∗((1− ωN) g) + a(ωN g) .
Compute γ(1)N,t :
γ(1)N,t (x ; y) = 〈RωN,t R
∗ωN,t
e−iHN t/εRωN Ω , a∗x ay RωN,t R∗ωN,t
e−iHN t/εRωN Ω〉= 〈UN(t) Ω , R∗ωN,t
a∗x ay RωN ,t UN(t) Ω〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 9 / 11
Idea of the proofBogoliubov transformation:ωN =
∑Nj=1 |fj〉〈fj |, fjj≤N o.n.s. in L2(R3).
RωN : F → F unitary implementor of a Bolgoliubov transformation
RωN Ω = a∗(f1) . . . a∗(fN) Ω
s.t. ∀ g ∈ L2(R3), R∗ωNa∗(g)RωN = a∗((1− ωN) g) + a(ωN g) .
Compute γ(1)N,t :
γ(1)N,t (x ; y) = 〈RωN,t R
∗ωN,t
e−iHN t/εRωN Ω , a∗x ay RωN,t R∗ωN,t
e−iHN t/εRωN Ω〉= 〈UN(t) Ω , R∗ωN,t
a∗x RωN ,t R∗ωN,tay RωN ,t UN(t) Ω〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 9 / 11
Idea of the proofBogoliubov transformation:ωN =
∑Nj=1 |fj〉〈fj |, fjj≤N o.n.s. in L2(R3).
RωN : F → F unitary implementor of a Bolgoliubov transformation
RωN Ω = a∗(f1) . . . a∗(fN) Ω
s.t. ∀ g ∈ L2(R3), R∗ωNa∗(g)RωN = a∗((1− ωN) g) + a(ωN g) .
Compute γ(1)N,t :
γ(1)N,t (x ; y) = 〈RωN,t R
∗ωN,t
e−iHN t/εRωN Ω , a∗x ay RωN,t R∗ωN,t
e−iHN t/εRωN Ω〉= 〈UN(t) Ω , R∗ωN,t
a∗x RωN ,t R∗ωN,tay RωN ,t UN(t) Ω〉
= 〈UN(t)Ω, [a∗(1− ωt,x ) + a(ωt,x )][a(1− ωt,y ) + a∗(ωt,y )]UN(t)Ω〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 9 / 11
Idea of the proofBogoliubov transformation:ωN =
∑Nj=1 |fj〉〈fj |, fjj≤N o.n.s. in L2(R3).
RωN : F → F unitary implementor of a Bolgoliubov transformation
RωN Ω = a∗(f1) . . . a∗(fN) Ω
s.t. ∀ g ∈ L2(R3), R∗ωNa∗(g)RωN = a∗((1− ωN) g) + a(ωN g) .
Compute γ(1)N,t :
γ(1)N,t (x ; y) = 〈RωN,t R
∗ωN,t
e−iHN t/εRωN Ω , a∗x ay RωN,t R∗ωN,t
e−iHN t/εRωN Ω〉= 〈UN(t) Ω , R∗ωN,t
a∗x RωN ,t R∗ωN,tay RωN ,t UN(t) Ω〉
= 〈UN(t)Ω, [a∗(1− ωt,x ) + a(ωt,x )][a(1− ωt,y ) + a∗(ωt,y )]UN(t)Ω〉= ωN,t (x ; y) + terms in normal order
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 9 / 11
Idea of the proofBogoliubov transformation:ωN =
∑Nj=1 |fj〉〈fj |, fjj≤N o.n.s. in L2(R3).
RωN : F → F unitary implementor of a Bolgoliubov transformation
RωN Ω = a∗(f1) . . . a∗(fN) Ω
s.t. ∀ g ∈ L2(R3), R∗ωNa∗(g)RωN = a∗((1− ωN) g) + a(ωN g) .
Compute γ(1)N,t :
γ(1)N,t (x ; y) = 〈RωN,t R
∗ωN,t
e−iHN t/εRωN Ω , a∗x ay RωN,t R∗ωN,t
e−iHN t/εRωN Ω〉= 〈UN(t) Ω , R∗ωN,t
a∗x RωN ,t R∗ωN,tay RωN ,t UN(t) Ω〉
= 〈UN(t)Ω, [a∗(1− ωt,x ) + a(ωt,x )][a(1− ωt,y ) + a∗(ωt,y )]UN(t)Ω〉= ωN,t (x ; y) + terms in normal order
tr |γ(1)N,t − ωN,t | . 〈UN(t) Ω , N UN(t) Ω〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 9 / 11
Idea of the proof
Control on the fluctuations
Bound on the expected number of excitations of the Slater determinant
〈UN(t)Ω , N UN(t)Ω〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 10 / 11
Idea of the proof
Control on the fluctuations
Bound on the expected number of excitations of the Slater determinant
〈UN(t)Ω , N UN(t)Ω〉
To control fluctuation: Gronwall type estimate
iεddt〈UN(t)Ω,NUN(t)Ω〉 = . . .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 10 / 11
Idea of the proof
Control on the fluctuations
Bound on the expected number of excitations of the Slater determinant
〈UN(t)Ω , N UN(t)Ω〉
To control fluctuation: Gronwall type estimate
iεddt〈UN(t)Ω,NUN(t)Ω〉
=4i Im1N
∫∫dx dy
1|x − y |
〈UN(t)Ω, a(ωt,x )a(ωt,y )a(1− ωt,y )a(1− ωt,x )UN(t)Ω〉
+ 〈UN(t)Ω, a∗(1− ωt,y )a∗(ωt,y )a∗(ωt,x )a(ωt,x )UN(t)Ω〉
+ 〈UN(t)Ω, a∗(1− ωt,x )a(ωt,y )a(1− ωt,y )a(1− ωt,x )UN(t)Ω〉
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 10 / 11
Fefferman – de la Llave representation
(?)1N
∫dx dy
1|x − y |
〈UN(t)Ω, a(ωt,x )a(ωt,y )a(1− ωt,y )a(1− ωt,x )UN(t)Ω〉
Smooth version of Fefferman–de la Llave representation for the Coulombpotential:
1|x − y |
= C∫ ∞
0
drr5
∫dz χ(r ,z)(x)χ(r ,z)(y) , χ(r ,z)(x) = e−|x−z|2/r2
Insert it in (?):
CN
∫dx dy
∫ ∞0
drr5
∫dz χ(r ,z)(x)χ(r ,z)(y)
× 〈UN(t)Ω, a(ωt,x )a(ωt,y )a(1− ωt,y )a(1− ωt,x )UN(t)Ω〉
=CN
∫ k
0dr∫
dx dy dz · · ·+ CN
∫ ∞k
dr∫
dx dy dz . . .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 11 / 11
Fefferman – de la Llave representation
(?)1N
∫dx dy
1|x − y |
〈UN(t)Ω, a(ωt,x )a(ωt,y )a(1− ωt,y )a(1− ωt,x )UN(t)Ω〉
Smooth version of Fefferman–de la Llave representation for the Coulombpotential:
1|x − y |
= C∫ ∞
0
drr5
∫dz χ(r ,z)(x)χ(r ,z)(y) , χ(r ,z)(x) = e−|x−z|2/r2
Insert it in (?):
CN
∫dx dy
∫ ∞0
drr5
∫dz χ(r ,z)(x)χ(r ,z)(y)
× 〈UN(t)Ω, a(ωt,x )a(ωt,y )a(1− ωt,y )a(1− ωt,x )UN(t)Ω〉
=CN
∫ k
0dr∫
dx dy dz · · ·+ CN
∫ ∞k
dr∫
dx dy dz . . .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 11 / 11
Fefferman – de la Llave representation
(?)1N
∫dx dy
1|x − y |
〈UN(t)Ω, a(ωt,x )a(ωt,y )a(1− ωt,y )a(1− ωt,x )UN(t)Ω〉
Smooth version of Fefferman–de la Llave representation for the Coulombpotential:
1|x − y |
= C∫ ∞
0
drr5
∫dz χ(r ,z)(x)χ(r ,z)(y) , χ(r ,z)(x) = e−|x−z|2/r2
Insert it in (?):
CN
∫dx dy
∫ ∞0
drr5
∫dz χ(r ,z)(x)χ(r ,z)(y)
× 〈UN(t)Ω, a(ωt,x )a(ωt,y )a(1− ωt,y )a(1− ωt,x )UN(t)Ω〉
=CN
∫ k
0dr∫
dx dy dz · · ·+ CN
∫ ∞k
dr∫
dx dy dz . . .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 11 / 11
Fefferman – de la Llave representation
(?)1N
∫dx dy
1|x − y |
〈UN(t)Ω, a(ωt,x )a(ωt,y )a(1− ωt,y )a(1− ωt,x )UN(t)Ω〉
Smooth version of Fefferman–de la Llave representation for the Coulombpotential:
1|x − y |
= C∫ ∞
0
drr5
∫dz χ(r ,z)(x)χ(r ,z)(y) , χ(r ,z)(x) = e−|x−z|2/r2
Insert it in (?):
CN
∫dx dy
∫ ∞0
drr5
∫dz χ(r ,z)(x)χ(r ,z)(y)
× 〈UN(t)Ω, a(ωt,x )a(ωt,y )a(1− ωt,y )a(1− ωt,x )UN(t)Ω〉
=CN
∫ k
0dr∫
dx dy dz · · ·+ CN
∫ ∞k
dr∫
dx dy dz . . .
Chiara Saffirio (UZH) Macroscopic Limits of Quantum Systems Munich, March 30-April 1, 2017. 11 / 11