Derivation of the Nonlinear Schr odinger Equation from Many Body - PowerPoint PPT Presentation

Derivation of the Nonlinear Schr¨ odinger Equation from Many Body Quantum Dynamics Benjamin Schlein Rutgers University, December 5, 2005 math-ph/0508010: Joint work with L. Erd˝ os and H.-T. Yau math-ph/0504051: Joint work with A. Elgart math-ph/0410038: Joint work with L. Erd˝ os and H.-T. Yau math-ph/0410005: Joint work with A. Elgart, L. Erd˝ os, and H.-T. Yau Summary 1. Introduction 2. The nonlinear Hartree equation 3. Bose-Einstein Condensates 4. The nonlinear Schr¨ odinger equation 5. The uniqueness problem 6. The Gross-Pitaevskii Limit

1. Introduction N-Boson System: Quantum mechanical N -boson systems are described by a wave function ψ N ∈ L 2 s ( R 3 N ) , symmetric w.r.t. permutations . The dynamics is governed by the Schr¨ odinger equation i∂ t ψ N,t = H N ψ N,t . H N is the Hamiltonian of the system, N L 2 s ( R 3 N ) . � � H N = − ∆ x j + V ( x i − x j ) acts on j =1 i<j The expectation of H N � ( ψ N , Hψ N ) = d x ψ N ( x )( Hψ N )( x ) is the energy of the system.

Macroscopic Dynamics: In typical physical systems, N ≃ 10 23 . ⇒ Impossible to solve the Schr¨ odinger equation. For practical purposes ⇒ describe the macroscopic evolution. ⇒ Mean-field systems: very weak interaction. The Hamiltonian is given by N N ∆ x j + 1 � � H N = − V ( x i − x j ) . N j =1 i<j ⇒ Kinetic and potential energy are = O ( N ); in mean field sys- tems the macroscopic evolution can be described by nonlinear one-particle equations. ⇒ Application in atomic physics (dynamics of BEC), condensed matter physics (BCS superconductors), plasma physics, cosmol- ogy (dynamics of bosonic stars),...

Consider evolution of a condensate, N � ψ N, 0 ( x ) = ϕ ( x j ) ( x = ( x 1 , . . . , x N )) . j =1 If the factorization is preserved in time, N � ψ N,t ( x ) = ϕ t ( x j ) j =1 ⇒ replace interaction by effective one-particle potential N N 1 V ( x i − x j ) ≃ 1 � d x i V ( x i − x j ) | ϕ t ( x i ) | 2 ≃ ( V ∗ | ϕ t | 2 )( x j ) � � N N i � = j i � = j Conjecture: if ψ N, 0 ( x ) = � N j =1 ϕ ( x j ), then, as N → ∞ , N � ψ N,t ( x ) ≃ ϕ t ( x j ) j =1 with ϕ t being a solution of the nonlinear Hartree equation i∂ t ϕ t = − ∆ ϕ t + ( V ∗ | ϕ t | 2 ) ϕ t .

Previous Results: • Hepp, 1974: Derivation of the Hartree equation for smooth potentials. • Spohn, 1980: Generalization to bounded potentials. • Erd˝ os and Yau, 2001: Derivation for the Coulomb potential V ( x ) = ± 1 / | x | . • Elgart and S., 2005: Derivation of the relativistic nonlinear Hartree equation (application in cosmology: boson stars) i∂ t ϕ t = (1 − ∆) 1 / 2 ϕ t + ( V ∗ | ϕ t | 2 ) ϕ t for Coulomb potential V ( x ) = λ/ | x | , with λ > λ crit = − 4 /π .

2. General Strategy for Derivation of the Hartree equation Marginal Densities: • Density matrix ⇒ γ N,t ( x ; x ′ ) = ψ N,t ( x ) ψ N,t ( x ′ ) γ N,t = | ψ N,t �� ψ N,t | satisfies Heisenberg equation i∂ t γ N,t = [ H N , γ N,t ] , Tr γ N,t = 1 . • For k = 1 , . . . , N , define the k -particle marginal density � γ ( k ) N,t ( x k ; x ′ d x N − k γ N,t ( x k , x N − k ; x ′ k ) = k , x N − k ) . Here x k = ( x 1 , . . . , x k ), x N − k = ( x k +1 , . . . , x N ), Tr γ ( k ) N,t = 1. For every k -particle observable J ( k ) : � ψ N,t , ( J ( k ) ⊗ 1 ( N − k ) ) ψ N,t � = Tr( J ( k ) ⊗ 1 ( N − k ) ) γ N,t = Tr J ( k ) γ ( k ) N,t

The BBGKY Hierarchy: The family { γ ( k ) N,t } N k =1 satisfies k k + 1 � � � � i∂ t γ ( k ) − ∆ x j , γ ( k ) V ( x i − x j ) , γ ( k ) � � N,t = N,t N,t N j =1 i<j k 1 − k � � � � V ( x j − x k +1 ) , γ ( k +1) � + Tr k +1 . N,t N j =1 Written in terms of the kernels, k � � i∂ t γ ( k ) γ ( k ) N,t ( x k ; x ′ N,t ( x k ; x ′ � k ) = − ∆ x j + ∆ x ′ k ) j j =1 + 1 i − x j )) γ ( k ) ( V ( x i − x j ) − V ( x ′ N,t ( x k ; x ′ � k ) N 1 ≤ i<j ≤ k k 1 − k � � � � V ( x j − x k +1 ) − V ( x ′ � � + d x k +1 j − x k +1 ) N j =1 × γ ( k +1) ( x k , x k +1 ; x ′ k , x k +1 ) . N,t

The Hartree Hierarchy: As N → ∞ , the BBGKY hierarchy formally converges to the Hartree hierarchy k k � � � � i∂ t γ ( k ) − ∆ x j , γ ( k ) V ( x j − x k +1 ) , γ ( k +1) � � ∞ ,t = + Tr k +1 ∞ ,t ∞ ,t j =1 j =1 ⇒ Infinite system of coupled equations. Remark: the factorized family of densities { γ ( k ) } k ≥ 1 with t k � � γ ( k ) γ ( k ) ( x k ; x ′ ϕ t ( x j ) ϕ t ( x ′ = | ϕ t �� ϕ t | ⊗ k � k ) = j ) t t j =1 is a solution of the Hartree hierarchy if ϕ t satisfies i∂ t ϕ t = − ∆ ϕ t + ( V ∗ | ϕ t | 2 ) ϕ t .

Strategy for Rigorous Derivation: • Prove the compactness of { γ ( k ) N,t } N k =1 with respect to some weak topology ⇒ there exists at least one limit point { γ ( k ) ∞ ,t } k ≥ 1 of { γ ( k ) N,t } N k =1 . • Prove that the limit point { γ ( k ) ∞ ,t } k ≥ 1 is a solution of the infi- nite Hartree equation. • Prove the uniqueness of the solution of the infinite Hartree hierarchy. γ ( k ) N,t → γ ( k ) = | ϕ t �� ϕ t | ⊗ k . ⇒ for every k ≥ 1, t ∈ R , t ⇒ � ψ N,t , ( J ( k ) ⊗ 1 ( N − k ) ) ψ N,t � → � ϕ ⊗ k , J ( k ) ϕ ⊗ k � as N → ∞ . t t ψ N,t ( x ) ≃ � N In this sense j =1 ϕ t ( x j ) for large N .

3. Bose-Einstein Condensation. Definition: BEC exists if max σ ( γ (1) N ) = O (1) as N → ∞ ( In general γ (1) j λ j = 1 ) = � j λ j | φ j �� φ j | , with 0 < λ j ≤ 1, � N Interpretation: a macroscopic number of particles occupies the same one-particle state. Recently, Lieb-Seiringer proved γ (1) N ( x ; x ′ ) → φ ( x ) φ ( x ′ ) as N → ∞ for the ground state of a trapped Bose gas. ⇒ Complete condensation into φ , the minimizer of the Gross- Pitaevskii energy functional � |∇ ϕ ( x ) | 2 + V ext ( x ) | ϕ ( x ) | 2 + 4 πa 0 | ϕ ( x ) | 4 � � E GP ( ϕ ) = d x , where a 0 = scattering length of pair potential.

Experiments on BEC: in 2001, Cornell-Ketterle-Wieman re- ceived Nobel prize in physics for experiments which first proved the existence of BEC for trapped Bose gas. In the experiments gases are trapped in small volumes by strong magnetic fields, and cooled down at very low temperatures. Then one observes the dynamical evolution of the condensate when the trap is removed.

To interpret these experiments one need an accurate descrip- tion of the dynamics of the trapped condensate. To this end, physicists use the time dependent Gross-Pitaevskii equation i∂ t ϕ t ( x ) = − ∆ ϕ t ( x ) + V ext ( x ) ϕ t ( x ) + 8 πa 0 | ϕ t ( x ) | 2 ϕ t ( x ) .

4. The Nonlinear Schr¨ odinger Equation. Delta-Potential: we choose smooth V ( x ) ≥ 0 and define V N ( x ) = N 3 β V ( N β x ) , with β > 0 . We consider the system described by the Hamiltonian N N ∆ x j + 1 � � H N = − V N ( x i − x j ) . N j =1 i<j As N → ∞ , � V N ( x ) → b 0 δ ( x ) with b 0 = d x V ( x ) . We expect that the macroscopic dynamics is described by the one-particle nonlinear Schr¨ odinger equation i∂ t ϕ t = − ∆ ϕ t + b 0 | ϕ t | 2 ϕ t � � b 0 δ ∗ | ϕ t | 2 � � = − ∆ ϕ t + ϕ t . Problem much more difficult because, in 3 dim, δ �≤ − const · ∆.

Main Result: Consider the initial data N with ϕ ∈ H 1 ( R 3 ) . � ψ N, 0 ( x ) = ϕ ( x j ) , j =1 Let V N ( x ) = N 3 β V ( N β x ) , with 0 < β < 1 / 2 , and V ≥ 0 . Then, for all t ∈ R , k ≥ 1, γ ( k ) N,t → γ ( k ) = | ϕ t �� ϕ t | ⊗ k as N → ∞ . t Here ϕ t is the solution of � i∂ t ϕ t = − ∆ ϕ t + b 0 | ϕ t | 2 ϕ t , ϕ t =0 = ϕ, b 0 = d x V ( x ) The convergence is in the weak* topology of L 1 ( L 2 ( R 3 k )); for every compact operator J ( k ) on L 2 ( R 3 k ), we have J ( k ) ⊗ 1 ( N − k ) � ψ N,t � = Tr J ( k ) γ ( k ) � � ψ N,t , N,t → Tr J ( k ) γ ( k ) = � ϕ ⊗ k , J ( k ) ϕ ⊗ k � . t t t

the densities { γ ( k ) N,t } N The BBGKY Hierarchy: k =1 satisfy the BBGKY hierarchy k k + 1 � � � � i∂ t γ ( k ) − ∆ x j , γ ( k ) V N ( x i − x j ) , γ ( k ) � � N,t = N,t N,t N j =1 i<j � N 1 − k � � � V N ( x j − x k +1 ) , γ ( k +1) � + Tr k +1 . N,t N j =1 As N → ∞ ⇒ infinite hierarchy k N � � � � i∂ t γ ( k ) − ∆ x j , γ ( k ) δ ( x j − x k +1 ) , γ ( k +1) � � ∞ ,t = + b 0 Tr k +1 . ∞ ,t ∞ ,t j =1 j =1 Strategy to derive nonlinear Schr¨ odinger equation as before: first prove compactness, then convergence, and finally uniqueness.

A-priori Estimates: Let 0 < β < 3 / 5. Then ∃ C : � ψ N,t , (1 − ∆ x 1 ) . . . (1 − ∆ x k ) ψ N,t � 2 � � � � (1 − ∆ x 1 ) 1 / 2 . . . (1 − ∆ x k ) 1 / 2 ψ N,t ( x ) ≤ C k . � � = d x � � � Tr (1 − ∆ x 1 ) . . . (1 − ∆ x k ) γ ( k ) N,t ≤ C k ⇒ A-priori estimates follow from energy estimates, ( H N + N ) k ≥ C k N k (1 − ∆ x 1 ) . . . (1 − ∆ x k ) and from conservation of the energy: � ψ N,t , (1 − ∆ x 1 ) . . . (1 − ∆ x k ) ψ N,t � ≤ C k N − k � ψ N,t , ( H N + N ) k ψ N,t � = C k N − k � ψ N, 0 , ( H N + N ) k ψ N, 0 � ≤ C k .