[PPT] - Invariant measures for NLS equations as limit of many-body quantum PowerPoint Presentation

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Invariant measures for NLS equations as limit of many-body quantum states

Benjamin Schlein, University of Zurich ICMP 2018, PDE Session Montreal, July 25, 2018 Joint with J¨ urg Fr¨

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Energy: the Hartree functional is given by EH(φ) = |∇φ(x)|2 + v(x)|φ(x)|2 dx + 1 2

and acts on L2(Rd) (we will consider d = 1, 2, 3). We assume v is confining and w ∈ L∞(Rd) pointwise non- negative if d = 1 or of positive type if d = 2, 3. Evolution: the time-dependent Hartree equation is given by i∂tφt = [−∆ + v(x)] φt + (w ∗ |φt|2)φt

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Invariant measure: formally given by dµH = 1 Z e−

Constructive QFT in ’70s: Nelson, Glimm-Jaffe, Simon, . . . Recently, problem awoke interest of dispersive pde’s community. Important application of this line of research is the almost sure well-posedness for rough initial data. Results by: Lebowitz-Rose-Speer, Bourgain, Zhidkov, Bourgain- Bulut, Burq-Tzvetkov, Burq-Thomann-Tzvetkov, Nahmod-Oh- Rey-Bellet-Sheffield-Staffilani, Oh-Popovnicu, Oh-Quastel, Deng- Tzvetkov-Visciglia, Oh-Tzvetkov-Wang, Cacciafesta-de Suzzoni, Genovese-Luc´ a-Valeri, . . .

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Free functional: let E0(φ) = |∇φ(x)|2 + v(x)|φ(x)|2 + κ|φ(x)|2 dx = φ, hφ with h = −∆ + v(x) + κ =

λn|unun| We assume

Tr h−1 =

< ∞ for d = 1 Tr h−2 =

< ∞ for d = 2, 3 Free measure: to define dµ0 ∼ exp(−E0(φ))dφ, we expand φ(x) =

ωn √λn un(x) ⇒ E0(φ) = φ, hφ =

|ωn|2 Hence we define µ0 on CN = {{ωn}n∈N : ωn ∈ C} as product of iid Gaussian measures with densities 1 πe−|ωn|2dωndω∗

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Expected L2 norm: observe that Eµ0 φ2

|ωn|2 λn =

1 λn = Tr h−1 is finite for d = 1, but it is infinite for d = 2, 3. Hartree invariant measure: for d = 1, we can define µH = 1 Z e−Wµ0 with interaction W(φ) = 1 2

2 φ4

For d = 2, 3, on the other hand, W = ∞ almost surely.

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Wick ordering: for K > 0 we introduce cutoff fields φK(x) =

ωn √λn un(x) and we define ρK(x) = Eµ0|φK(x)|2 =

λ−1

|un(x)|2 and the cutoff renormalized interaction WK = 1 2

|φK(y)|2 − ρK(y)

Lemma: WK is Cauchy sequence in Lp(CN, dµ0) for all p < ∞. We denote by W r its limit (independent of p). For d = 2, 3, we define renormalized Gibbs measure µr

1

Note that µr

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Hamilton operator: has form HN =

N

w(xi − xj)

Ground state: ψN ≃ φ⊗N , where φ0 is minimizer of EH. Dynamics: governed by the many-body Schr¨

i∂tψN,t = HNψN,t Convergence to Hartree: if ψN,0 ≃ φ⊗N, then ψN,t ≃ φ⊗N

where φt solves time-dependent Hartree equation. Rigorous works: Hepp, Ginibre-Velo, Spohn, Erd˝

Golse-Mauser, Fr¨

Pickl, Fr¨

Pavlovic, X.Chen-Holmer, Ammari-Nier, Lewin-Nam-S., . . .

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Question: what corresponds to Hartree invariant measure in many-body setting? Thermal equilibrium: at temperature β−1, it is described by EβA = Tr A̺β with density matrix ̺β = 1 Zβ e−βHN, Zβ = Tr e−βHN Remark 1: if β > 0 fixed, ̺β still exhibits condensation. At

To recover invariant measure, need to take β = 1/N. Remark 2: number of particles at many-body level corresponds to L2-norm at Hartree level. To recover invariant measure, need fluctuations of number of particles.

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Fock space: we define F =

L2(Rd)⊗sm =

L2

Creation and annihilation operators: for f ∈ L2(Rd), let (a∗(f)Ψ)(m)(x1, . . . , xm) = 1 √m

f(xj)Ψ(m−1)(x1, . . . , xj, . . . , xm) (a(f)Ψ)(m)(x1, . . . , xm) =

They satisfy canonical commutation relations

[a(f), a(g)] =

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We define operator valued distributions a(x), a∗(x) such that a∗(f) =

and a(f) =

Number of particles operator: is given by N =

Hamilton operator: is defined through HN =

2N

Notice that [HN, N] = 0 and HN|Fm =

N

w(xi − xj)

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Grand canonical ensemble: at inverse temperature β = N−1 and chemical potential κ, equilibrium is described by ̺N = 1 ZN e− 1

with ZN = Tr e− 1

Rescaled operators: it is useful to define aN(x) = 1 √ N a(x), a∗

1 √ N a∗(x) Expressed in terms of the rescaled fields, we find ̺N = Z−1

exp

+1 2

[aN(x), a∗

N δ(x−y), [aN(x), aN(y)] = [a∗

are almost commuting operators.

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Non-interacting Gibbs state: we diagonalize

λja∗

which leads to ̺(0)

= 1 Z(0)

e−

Expectation of rescaled number of particles E(0)

a∗

Tre−λia∗

= 1 N 1 eλi/N − 1 Hence E(0)

1 N

a∗

N

1 eλi/N − 1 =

for d = 1 → ∞, for d = 2, 3

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Interaction: expectation of WN = 1 2

is finite but, for d = 2, 3, it diverges, as N → ∞. Wick ordering: replace WN with the Wick ordered interaction W r

2

with ρN(x) = E(0)

a∗

N

|uj(x)|2 eλj/N − 1 We write the resulting grand canonical density matrix ̺r

1 Zr

e−Hr

1 Zr

e−(HN,0+W r

with HN,0 =

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Correlation functions: for k ∈ N, define correlation function γ(k)

as non-negative trace class operator on L2(Rkd) with kernel γ(k)

= Er

= Tr a∗

Joint moments: define γ(k)

γ(k)

= Er

=

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Conjecture: we expect that, for all fixed k ∈ N, lim

− γ(k)

= 0 Theorem [Lewin-Nam-Rougerie, 2016]: let d = 1. Then conjecture holds true, with no need for renormalization. In [Fr¨

Very recently, [Lewin-Nam-Rougerie, 2018] announced proof

In most interesting case d = 3, conjecture remains open. We prove it, but only for slightly modified many-body Gibbs states.

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Modification: for fixed η > 0, we consider ̺r

1 Zr

e−ηHN,0 e−[(1−2η)HN,0+W r

We denote by γ(k)

Remark: ̺r

Theorem [Fr¨

d = 2, 3, h = −∆ + v(x) + κ with Tr h−2 < ∞, w ∈ L∞(Rd) positive definite. Then, for all fixed η > 0 and k ∈ N, we have lim

= 0

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Observables: for ξ ∈ L(L2(Rk)), we define random variable Θ(ξ) =

× φ(x1) . . . φ(xk)φ(yk) . . . φ(y1) and quantum observable (on F) ΘN(ξ) =

× a∗

Dynamics: let St be nonlinear Hartree flow. We define Ψt [Θ(ξ)] =

× Stφ(x1) . . . Stφ(xk)Stφ(yk) . . . Stφ(y1) and quantum evolution Ψt

× e−iHNta∗

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Theorem [Fr¨

Let w ∈ L∞(R), non-negative. Given k ∈ N, ξj ∈ L(L2(Rpj)) and times tj, for j = 1, . . . , k, we have ENΨt1

→ EH Ψt1 [Θ(ξ1)] . . . Ψtk [Θ(ξk)] as N → ∞. Remark: taking k = 1 and using invariance of quantum state, Theorem implies in particular invariance of nonlinear Gibbs measure w.r.t. the Hartree flow.

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Duhamel expansion: start from e−[(1−2η)HN,0+W r

= e

1 1 − 2η

dt e−(1−2η−t)HN,0 W r

e−ηHN,0e−(1−2η)HNe−ηHN,0 = e−HN,0 +

1 (1 − 2η)m

dt1· · ·

dtm × e−(1−t1)HN,0 W r

+ 1 (1 − 2η)n

dt1· · ·

dtn × e−(1−t1)HN,0 W r

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Evolved fields operator: remark that etH0,Na∗

Fully expanded terms: need to compute free expectations! Wick theorem: we have E(0)

a♯1

=

E(0)

E(0)

N 1 eh/N − 1 (x; y) E(0)

N 1 eh/N − 1 (x; y) + 1 N δ(x − y)

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Diagrammatic expansion: recall that W r

2

Pairings are encoded in Feynman diagrams Bound: using diagrammatic representation and assumption Tr h−2 < ∞, we conclude that each pairing is bounded, uniformly in N. Convergence: as N → ∞, each pairing tends to corresponding term in expansion of Hartree invariant measure.

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Error term: use Cauchy-Schwarz to get rid of interacting term. Here, for d = 2, 3, we need modification to avoid interacting exponential carrying full time. Final obstacle: number of pairing ∼ (2n)!, time integral ∼ 1/n! Hence, series does not converge! Borel resummation: given formal power series representation A(z) =

amzm

B(z) =

am m!zm Formally, we can then reconstruct A through A(z) =

e−tB(tz)dt

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Theorem [Sokal, 1980]: Let A(z) and (AN(z))N∈N be analytic

CR =

for some R > 0. For n ∈ N suppose A(z) =

amzm + Rn(z), AN(z) =

am,Nzm + Rn,N(z) with |am| + sup

|am,N| ≤ Cmm!, |Rm(z)| + sup

|Rm,N(z)| ≤ Cm|z|m m! for all m ∈ N, z ∈ CR. Suppose moreover that, for all m ∈ N: limN→∞ am,N = am. Then AN(z) → A(z) for all z ∈ CR.

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Wick-ordering of many-body Hamiltonian: given HN =

+ 1 2

we rewrite it as HN =

+ 1 2

Subtracting constant and shifting chemical potential, we obtain

ρN))(x) + κ] aN(x)dx + 1 2

with ¯ ρN = E(0)

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Fix point problem: theorem can be applied to HN if we find

ρN)) s.t. ρN(x) = E(0)

Theorem [Fr¨

such that v(x + y) ≤ Cv(x)v(y) and Tr (−∆ + v + κ)−2 < ∞. Then for every N ∈ N there exists vN solving the counterterm

v such that lim

vN + κ)−1 − (−∆ + v + κ)−1