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

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¨ ohlich, Antti Knowles, Vedran Sohinger 1

I. Hartree theory Energy : the Hartree functional is given by � � |∇ φ ( x ) | 2 + v ( x ) | φ ( x ) | 2 � E H ( φ ) = dx � + 1 w ( x − y ) | φ ( x ) | 2 | φ ( y ) | 2 dxdy 2 and acts on L 2 ( R d ) (we will consider d = 1 , 2 , 3). We assume v is confining and w ∈ L ∞ ( R d ) pointwise non- negative if d = 1 or of positive type if d = 2 , 3. Evolution : the time-dependent Hartree equation is given by i∂ t φ t = [ − ∆ + v ( x )] φ t + ( w ∗ | φ t | 2 ) φ t 2

Invariant measure : formally given by Z e − � � dµ H = 1 E H ( φ )+ κ � φ � 2 dφ 2 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, . . . 3

Free functional : let � � |∇ φ ( x ) | 2 + v ( x ) | φ ( x ) | 2 + κ | φ ( x ) | 2 � E 0 ( φ ) = dx = � φ, hφ � with � h = − ∆ + v ( x ) + κ = λ n | u n �� u n | n We assume  = � Tr h − 1 n ∈ N λ − 1  < ∞ for d = 1  n  = �  Tr h − 2 n ∈ N λ − 2 < ∞ for d = 2 , 3 n Free measure : to define dµ 0 ∼ exp( −E 0 ( φ )) dφ , we expand � � ω n | ω n | 2 √ λ n ⇒ E 0 ( φ ) = � φ, hφ � = φ ( x ) = u n ( x ) n ∈ N n ∈ N Hence we define µ 0 on C N = {{ ω n } n ∈ N : ω n ∈ C } as product of iid Gaussian measures with densities 1 πe −| ω n | 2 dω n dω ∗ n 4

Expected L 2 norm: observe that | ω n | 2 � � 1 E µ 0 � φ � 2 = Tr h − 1 2 = E µ 0 = λ n λ n n ∈ N n ∈ N is finite for d = 1, but it is infinite for d = 2 , 3. Hartree invariant measure : for d = 1, we can define µ H = 1 Z e − W µ 0 with interaction � W ( φ ) = 1 w ( x − y ) | φ ( x ) | 2 | φ ( y ) | 2 dxdy ≤ � w � ∞ � φ � 4 2 2 2 For d = 2 , 3, on the other hand, W = ∞ almost surely . 5

Wick ordering : for K > 0 we introduce cutoff fields � ω n √ λ n φ K ( x ) = u n ( x ) n ≤ K and we define � ρ K ( x ) = E µ 0 | φ K ( x ) | 2 = λ − 1 | u n ( x ) | 2 n n ≤ K and the cutoff renormalized interaction � � � � � W K = 1 | φ K ( x ) | 2 − ρ K ( x ) | φ K ( y ) | 2 − ρ K ( y ) w ( x − y ) dxdy 2 Lemma : W K is Cauchy sequence in L p ( C N , dµ 0 ) for all p < ∞ . We denote by W r its limit (independent of p ). For d = 2 , 3, we define renormalized Gibbs measure 1 � e − W r ( φ ) dµ 0 ( φ ) e − W r µ 0 µ r H = Note that µ r H is invariant with respect to the Hartree flow. 6

II. Mean field quantum systems Hamilton operator : has form N N � � � � + 1 on L 2 s ( R dN ) H N = − ∆ x j + v ( x j ) w ( x i − x j ) N j =1 i<j Ground state : ψ N ≃ φ ⊗ N , where φ 0 is minimizer of E H . 0 Dynamics : governed by the many-body Schr¨ odinger equation i∂ t ψ N,t = H N ψ N,t if ψ N, 0 ≃ φ ⊗ N , then ψ N,t ≃ φ ⊗ N Convergence to Hartree : t where φ t solves time-dependent Hartree equation . Rigorous works : Hepp, Ginibre-Velo, Spohn, Erd˝ os-Yau, Bardos- Golse-Mauser, Fr¨ ohlich-Knowles-Schwarz, Rodnianski-S., Knowles- Pickl, Fr¨ ohlich-Knowles-Pizzo, Grillakis-Machedon-Margetis, T.Chen- Pavlovic, X.Chen-Holmer, Ammari-Nier, Lewin-Nam-S., . . . 7

Question : what corresponds to Hartree invariant measure in many-body setting? Thermal equilibrium : at temperature β − 1 , it is described by E β A = Tr A̺ β with density matrix ̺ β = 1 e − βH N , Z β = Tr e − βH N Z β Remark 1 : if β > 0 fixed, ̺ β still exhibits condensation . At one-particle level this leads to trivial measure δ φ 0 . To recover invariant measure, need to take β = 1 /N . Remark 2 : number of particles at many-body level corresponds to L 2 -norm at Hartree level. To recover invariant measure, need fluctuations of number of particles. 8

III. Fock space and grand canonical ensemble Fock space : we define � � L 2 ( R d ) ⊗ s m = L 2 s ( R md ) F = m ≥ 0 m ≥ 0 Creation and annihilation operators : for f ∈ L 2 ( R d ), let m � 1 ( a ∗ ( f )Ψ) ( m ) ( x 1 , . . . , x m ) = f ( x j )Ψ ( m − 1) ( x 1 , . . . , � x j , . . . , x m ) √ m j =1 � � ( a ( f )Ψ) ( m ) ( x 1 , . . . , x m ) = dx f ( x )Ψ ( m +1) ( x, x 1 , . . . , x m ) m + 1 They satisfy canonical commutation relations � = � f, g � , � = 0 � a ( f ) , a ∗ ( g ) � a ∗ ( f ) , a ∗ ( g ) [ a ( f ) , a ( g )] = 9

We define operator valued distributions a ( x ) , a ∗ ( x ) such that � � a ∗ ( f ) = f ( x ) a ∗ ( x ) dx, and a ( f ) = f ( x ) a ( x ) dx Number of particles operator : is given by � a ∗ ( x ) a ( x ) dx N = Hamilton operator : is defined through � � a ∗ ( x ) [ − ∆ x + v ( x )] a ( x ) + 1 w ( x − y ) a ∗ ( x ) a ∗ ( y ) a ( y ) a ( x ) H N = 2 N Notice that [ H N , N ] = 0 and m m � � � � + 1 H N | F m = − ∆ x j + v ( x j ) w ( x i − x j ) N j =1 i<j 10

Grand canonical ensemble : at inverse temperature β = N − 1 and chemical potential κ , equilibrium is described by 1 e − 1 Z N = Tr e − 1 N ( H N + κ N ) , N ( H N + κ N ) ̺ N = with Z N Rescaled operators : it is useful to define 1 1 a ∗ a ∗ ( x ) √ √ a N ( x ) = a ( x ) , N ( x ) = N N Expressed in terms of the rescaled fields , we find � � ̺ N = Z − 1 a ∗ exp − N ( x )( − ∆ x + v ( x ) + κ ) a N ( x ) dx N � � +1 w ( x − y ) a ∗ N ( x ) a ∗ N ( y ) a N ( y ) a N ( x ) dxdy 2 Notice that N ( y )] = 1 [ a N ( x ) , a ∗ [ a N ( x ) , a N ( y )] = [ a ∗ N ( x ) , a ∗ N δ ( x − y ) , N ( y )] = 0 are almost commuting operators. 11

IV. Non-interacting Gibbs states and Wick ordering Non-interacting Gibbs state : we diagonalize � � � � a ∗ λ j a ∗ N ( x ) − ∆ x j + v ( x j ) + κ a N ( x ) dx = N ( u j ) a N ( u j ) j which leads to e − � 1 j λ j a ∗ ̺ (0) N ( u j ) a N ( u j ) = N Z (0) N Expectation of rescaled number of particles N ( u i ) a N ( u i ) e − λ i a ∗ N ( u i ) a N ( u i ) N ( u i ) a N ( u i ) = Tr a ∗ = 1 1 E (0) a ∗ Tr e − λ i a ∗ e λ i /N − 1 N N ( u i ) a N ( u i ) N Hence � � � 1 N ( u i ) a N ( u i ) = 1 1 O (1) , for d = 1 E (0) a ∗ = e λ i /N − 1 N → ∞ , for d = 2 , 3 N N i i ∈ N 12

Interaction : expectation of � W N = 1 w ( x − y ) a ∗ N ( x ) a ∗ N ( y ) a N ( y ) a N ( x ) dxdy 2 is finite but, for d = 2 , 3, it diverges, as N → ∞ . Wick ordering : replace W N with the Wick ordered interaction � N = 1 � dxdy � a ∗ � � a ∗ W r w ( x − y ) N ( x ) a N ( x ) − ρ N ( x ) N ( y ) a N ( y ) − ρ N ( y ) 2 with | u j ( x ) | 2 � N ( x ) a N ( x ) = 1 ρ N ( x ) = E (0) a ∗ N e λ j /N − 1 N j ∈ N We write the resulting grand canonical density matrix 1 1 e −H r e − ( H N, 0 + W r N ) ̺ r N = N = Z r Z r N N with � a ∗ H N, 0 = N ( x ) [ − ∆ x + v ( x ) + κ ] a N ( x ) dx 13

V. Comparison with invariant measure for Hartree Correlation functions : for k ∈ N , define correlation function γ ( k ) as non-negative trace class operator on L 2 ( R kd ) with kernel N γ ( k ) N ( x 1 , . . . , x k ; y 1 , . . . , y k ) N a ∗ N ( x 1 ) . . . a ∗ = E r N ( x k ) a N ( y k ) . . . a N ( y 1 ) = Tr a ∗ N ( x 1 ) . . . a ∗ N ( x k ) a N ( y k ) . . . a N ( y 1 ) ̺ r N Joint moments : define γ ( k ) of invariant measure through H γ ( k ) H ( x 1 , . . . , x k ; y 1 , . . . , y k ) = E r H φ ( x 1 ) . . . φ ( x k ) φ ( y k ) . . . φ ( y 1 ) � φ ( x 1 ) . . . φ ( x k ) φ ( y k ) . . . φ ( y 1 ) e − W r ( φ ) dµ 0 ( φ ) = � e − W r ( φ ) dµ 0 ( φ ) 14

Conjecture : we expect that, for all fixed k ∈ N , � � � � � γ ( k ) − γ ( k ) � � lim = 0 � HS N H N →∞ Theorem [Lewin-Nam-Rougerie, 2016] : let d = 1. Then conjecture holds true, with no need for renormalization. In [Fr¨ ohlich-Knowles-S.-Sohinger, 2017] we give different proof of this theorem. Very recently, [Lewin-Nam-Rougerie, 2018] announced proof of conjecture for d = 2 (renormalization needed). In most interesting case d = 3, conjecture remains open. We prove it, but only for slightly modified many-body Gibbs states. 15

Modification : for fixed η > 0, we consider 1 e − η H N, 0 e − [ (1 − 2 η ) H N, 0 + W r N ] e − η H N, 0 ̺ r N,η = Z r N,η We denote by γ ( k ) η,N the correlation functions associated to ̺ r N,η . Remark : ̺ r N,η is still density matrix of a quantum state. Theorem [Fr¨ ohlich-Knowles-S.-Sohinger, 2017] : let d = 2 , 3, h = − ∆ + v ( x ) + κ with Tr h − 2 < ∞ , w ∈ L ∞ ( R d ) positive definite. Then, for all fixed η > 0 and k ∈ N , we have � � � � � γ ( k ) N,η − γ ( k ) � � lim = 0 � HS H N →∞ 16