Schrdingers equation with random potentials Marius Beceanu Jrg - PowerPoint PPT Presentation

Schrödinger’s equation with random potentials Marius Beceanu Jürg Fröhlich Avy Soffer Pasadena February 2015 M. Beceanu J. Fröhlich A. Soffer Random Potentials 1/15

Introduction Consider the linear Schrödinger equation in R d with random time-dependent potential i ∂ t ψ ( x , t ) − ∆ ψ ( x , t ) + V ω ( x ) ψ ( x , t ) = 0 , ψ ( 0 ) = ψ 0 . Here V ω := V ( x , X t ) . Conserved quantities for constant V : � R d | ψ ( x , t ) | 2 dx M [ ψ ] := � R d |∇ ψ ( x , t ) | 2 + V ( x , t ) | ψ ( x , t ) | 2 dx . E [ ψ ]( t ) := Dispersive inequalities (same with more derivatives): � ψ � L ∞ x + � P c ψ � L 2 � � ψ � 2 (Strichartz) t L 2 t L 2 d / ( d − 2 ) x � D 1 / 2 P c ψ � L 2 x ( Q ) � | Q | 1 / 2 d � ψ � 2 (local smoothing) . t L 2 Constant V : E and M conserved; P c ψ disperses. Time-dependent V : M is conserved. M. Beceanu J. Fröhlich A. Soffer Random Potentials 2/15

Main result Consider the equation on R 3 (or d ≥ 3) i ψ t − ∆ ψ + V ( x , X t ) ψ + ǫ ( χ ∗ | ψ | 2 ) ψ = 0 , ψ ( 0 ) = ψ 0 given . X t standard Brownian motion on bounded subset of Riemannian manifold; nontrivial V ( x , y ) ∈ C y ( L ∞ x ∩ L 1 x ) ; Hartree-type potential with small coupling constant. For any ψ 0 ∈ L 2 and | ǫ | < ǫ ( � ψ 0 � 2 ) , there is a.s. a global solution ψ s.t. E � ψ � 2 � � ψ 0 � 2 2 . t L 6 , 2 L 2 x Moreover, if A y V ( x , y ) ∈ L ∞ y ( L 1 x ∩ L ∞ x ) , then the energy remains bounded on average: E ( �∇ ψ ω ( t ) � 2 x ) ≤ �∇ ψ 0 � 2 x + C � ψ 0 � 2 x . L 2 L 2 L 2 Model problem: − ∆ + V 0 ( x ) + g ( W t ) V 1 ( x ) , V 0 ∈ C ∞ large fixed 0 potential, W t standard Brownian motion on T 1 , V 1 perturbation, coupling g ( y ) = 0 . 1 sin ( y ) . Other simple perturbations: e.g. − ∆ + V 0 ( x ) + V 1 ( x − W t ) , rotations, dilations. Can treat the general case stated above. M. Beceanu J. Fröhlich A. Soffer Random Potentials 3/15

Known results 1974 Ovchinnikov. 1985, 1986 Pillet: started the study, Feynman-Kac formula, linear L 2 wave operators. 1989, 1990 Cheremshantsev: Brownian motion over the whole space. 2009 Kang, Schenker: discrete Schrödinger, translation-invariant potential. 2010 De Roeck, Fröhlich, Pazzo: For a quantum particle interacting with infinitely many thermal reservoirs, they proved a central limit theorem, diffusive scaling for the second momentum, with no corrections, and a distribution law for finite energy states. 2013 Beceanu, Soffer: Strichartz estimates, other properties; Brownian motion on the whole space. M. Beceanu J. Fröhlich A. Soffer Random Potentials 4/15

Proof outline Consider the inhomogenous linear equation with random potential V ω := V ( x , X t ) i ∂ t ψ ω − ∆ ψ ω + V ω ψ ω = Ψ ω , ψ ω ( 0 ) := ψ 0 . Define g ( x , y , t ) := E ( ψ ( x , t ) | X t = y ) , f ( x 1 , x 2 , y , t ) := E ( ψ ( x 1 , t ) ψ ( x 2 , t ) | X t = y ) . Conditional expectations of ψ , respectively of the density matrix ψ ⊗ ψ , at time t and under the condition that X t = y . M. Beceanu J. Fröhlich A. Soffer Random Potentials 5/15

As shown by Pillet, i ∂ t g − ∆ x g − iA y g + V ( x , y ) g = G , g ( 0 ) := ψ 0 ( x ) µ 0 ( y ) , i ∂ t f − ∆ x 1 f + ∆ x 2 f − iA y f + ( V ⊗ 1 − 1 ⊗ V ) f = F , f ( 0 ) := ψ 0 ( x 1 ) ψ 0 ( x 2 ) µ 0 ( y ) , where µ 0 ( y ) is the initial distribution of X t , i.e. of X 0 , and G ( x , y , t ) := E (Ψ( x , t ) | X t = y ) , F ( x 1 , x 2 , y , t ) := E (Ψ( x 1 , t ) ψ ( x 2 , t ) − ψ ( x 1 , t )Ψ( x 2 , t ) | X t = y ) . M. Beceanu J. Fröhlich A. Soffer Random Potentials 6/15

Feynman-Kac Pillet also proved the following Feinman-Kac-type formula: � � | ψ ω ( x , t ) | 2 | V ω ( x , t ) | dx dt = f ( x , x , y , t ) | V ( x , y ) | dx dy dt . E We compute the right-hand side by using the equation of f . The left-hand side controls Strichartz estimates: � E � ψ ω � 2 � � ψ 0 � 2 | ψ ω ( x , t ) | 2 | V ω ( x , t ) | dx dt . 2 + E t L 6 , 2 L 2 x M. Beceanu J. Fröhlich A. Soffer Random Potentials 7/15

Model equation Consider the solution g to i ∂ t g − ∆ x g + iA y g + V ( x 1 , y ) g = 0 . Enough to prove that g ∈ L 2 ω L 2 t , y L 6 , 2 x . Non-triviality assumption on V : For almost every x in some open set O there exist y 1 , y 2 such that V ( x , y 1 ) � = V ( x , y 2 ) . Then − ∆ x + iA y + V has no bound states for ℑ λ ≤ 0. Proof: bound state φ ( x , y ) = ⇒ independent of y = ⇒ for each y solves ( − ∆ + V ( x , y )) φ ( x ) = λφ ( x ) . V ( x , y ) are distinct = ⇒ φ = 0 on open domain = ⇒ φ ≡ 0. M. Beceanu J. Fröhlich A. Soffer Random Potentials 8/15

Theorem Let W X := � t � − 3 / 2 L ∞ t B ( X ) . Suppose T ∈ W X is s.t. lim δ → 0 � T ( ρ ) − T ( ρ − δ ) � W X = 0 . If I + ˆ T ( λ ) is invertible in B ( X ) for every λ ∈ R , then 1 + T possesses an inverse in W X of the form 1 + S. Lemma Consider the equation, for V ∈ C y ( L ∞ x ∩ L 1 x ) and nontrivial, i ∂ t f − ∆ x f + iA y f + V ( x , y ) f = 0 , f ( 0 ) = f 0 ∈ L 1 y L 2 x . Then Strichartz: � f � L 2 � � f 0 � L 2 x . t L 6 , 2 y L 2 y L 2 x Moreover, x ) � � t � − 3 / 2 � f 0 � L p � f � L p x ) . y ( L 2 y ( L 1 x ∩ L 2 x + L ∞ M. Beceanu J. Fröhlich A. Soffer Random Potentials 9/15

Rewrite Duhamel’s identity symetrically I − χ t > 0 i | V | 1 / 2 ( x , y ) e it ( − ∆ x + V ( x , y )+ iA y ) | V | 1 / 2 sgn V ( x , y ) = � � − 1 . I + χ t > 0 i | V | 1 / 2 ( x , y ) e it ( − ∆ x + iA y ) | V | 1 / 2 sgn V ( x , y ) = Apply Wiener’s theorem. Leads to I − χ t > 0 i | V | 1 / 2 ( x , y ) e it ( − ∆ x + iA y + V ( x , y )) | V | 1 / 2 sgn V ( x , y ) being L 1 t , y L 2 x -bounded. Conclusion follows by Duhamel. M. Beceanu J. Fröhlich A. Soffer Random Potentials 10/15

Repeat for density matrix. Let V ⊗ 1 + 1 ⊗ V = V 1 V 2 , where � � | V | 1 / 2 sgn V ⊗ 1 � 1 ⊗ | V | 1 / 2 � | V | 1 / 2 ⊗ 1 V 2 := , V 1 := . 1 ⊗ | V | 1 / 2 sgn V The free resolvent is R A ( λ ) := ( − ∆ x 1 + ∆ x 2 − iA y − λ ) − 1 . Need to prove that V 2 R A ( λ ) V 1 is L 2 -bounded and, after taking the Fourier transform, I + V 2 R A ( λ ) V 1 is invertible for every λ . Matrix form � I + T 11 � T 12 I + T = . T 21 I + T 22 M. Beceanu J. Fröhlich A. Soffer Random Potentials 11/15

� � � � T 11 ) − 1 � I + i � i ( I + i � T 11 0 I T 12 I + i � T = . T 22 ) − 1 � I + i � i ( I + i � 0 T 22 T 21 I Diagonal terms are invertible by reduction to previous case: ( I + iT 11 ) − 1 = I − i χ t > 0 e it ∆ x 2 | V | 1 / 2 ( x 1 ) e it ( − ∆ x 1 + V ( x 1 , y )+ iA y ) | V | 1 / 2 sgn V ( x 1 ) . Now we need to study � � � � T 11 ) − 1 � i ( I + i � i � 0 T 12 0 S 1 =: I + i � I + =: I + S . T 22 ) − 1 � i ( I + i � i � T 21 0 S 2 0 Then � S 1 S 2 � 0 S 2 = . 0 S 2 S 1 After squaring, these off-diagonal terms are compact. One can use Fredholm’s alternative for I + S 2 = ( I − iS )( I + iS ) (same for both). Just as importantly, Wiener’s theorem applies to I + S 2 . M. Beceanu J. Fröhlich A. Soffer Random Potentials 12/15

Write ( I + iT ) − 1 in terms of ( I + S 2 ) − 1 and the diagonal terms ( I + iT kk ) − 1 . Get almost all integrable components, convolved with something explicit. Use this to evaluate f . In fact much more complicated. Conclusion: � f ( x , x , y , t ) | V ( x , y ) | dx dy dt � � ψ 0 � 2 L 2 . M. Beceanu J. Fröhlich A. Soffer Random Potentials 13/15

Well-posedness for Hartree with small initial data Proof: Contraction scheme. By L 2 norm conservation, a priori ψ ∈ L ∞ ω L ∞ t L 2 x . The inhomogenous source term is small in L 1 ω L 1 t J 1 x 1 , x 2 because � ( χ ∗ | ψ | 2 ) ψ ⊗ ψ � L 1 t J 1 x 1 , x 2 � � ψ � 2 x � ψ � 2 x . ω L 1 t L 6 , 2 t L 2 ω L 2 L 2 L ∞ ω L ∞ t L 6 , 2 and ψ belongs to L 2 ω L 2 by Strichartz. x M. Beceanu J. Fröhlich A. Soffer Random Potentials 14/15

Thank you for your attention! M. Beceanu J. Fröhlich A. Soffer Random Potentials 15/15