Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case The Kramers- Fokker-Planck equation with a short-range potential Xue Ping WANG Université de Nantes, France Conference in honor of Johannes Sjöstrand
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case Outline Introduction. Motivation 1 The free KFP operator 2 The KFP operator with a potential 3 Low-energy spectral properties in short-range case 4
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case The Kramers-Fokker-Planck equation The Kramers-Fokker-Planck equation is the evolution equation for the distribution functions describing the Brownian motion of particles in an external field F ( x ) : ∂ W = [ − ∂ ∂ x v + ∂ ∂ v ( γ v − F ( x ) m ) − γ kT m ∆ v ] W , (1) ∂ t where W = W ( x , v ; t ) , x , v ∈ R n , t ≥ 0 and F ( x ) = − m ∇ V ( x ) is the external force. This equation is a special case of the Fokker-Planck equation.
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case The Kramers-Fokker-Planck equation After change of unknowns and suitable normalization of the physical constants, the Kramers-Fokker-Planck (KFP) equation can be written into the form ∂ t u ( x , v ; t ) + Pu ( x , v ; t ) = 0 , ( x , v ) ∈ R n × R n , n ≥ 1 , t > 0 (2) with the initial condition u ( x , v ; 0 ) = u 0 ( x , v ) (3) 4 | v | 2 − n where P = v · ∇ x − ∇ V ( x ) · ∇ v − ∆ v + 1 2 .
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case Return to the equilibrium In this talk, we are interested in the time-decay of solutions to the equation (2) in the case ∇ V ( x ) → 0 as | x | → ∞ .
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case Return to the equilibrium In this talk, we are interested in the time-decay of solutions to the equation (2) in the case ∇ V ( x ) → 0 as | x | → ∞ . The case |∇ V ( x ) | → ∞ (or at least |∇ V ( x ) | ≥ C > 0 at the infinity) has been studied by several authors: Desvilettes-Villani(CPAM, 2001), Hérau-Nier(ARMA, 2004), Helffer-Nier(LNM, 2005), , 2008 - ), · · · Hérau-Hitrik-Sjöstrand (AHP
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case Return to the equilibrium In the case |∇ V ( x ) | → ∞ and V ( x ) > 0 outside some compact set, the solutions look like u ( t ) − c ( u 0 ) m 0 = O ( e − σ t ) , σ > 0 , 2 ( v 2 2 + V ( x )) is the Maxwillian. in appropriate spaces where m 0 = e − 1 The existence of a gap between 0 and the remaining part of the spectrum is crucial for such results.
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case The question If V ( x ) ≈ a | x | µ for some a > 0 and 0 < µ < 1, then m 0 ∈ L 2 and 0 is an eigenvalue of P . If V ( x ) ≈ a ln | x | , m 0 is an eigenfunction if a > n 2 and is a resonant state if n − 2 ≤ a ≤ n 2 . But now there is no gap 2 between 0 and the remaining part of the spectrum.
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case The question If V ( x ) ≈ a | x | µ for some a > 0 and 0 < µ < 1, then m 0 ∈ L 2 and 0 is an eigenvalue of P . If V ( x ) ≈ a ln | x | , m 0 is an eigenfunction if a > n 2 and is a resonant state if n − 2 ≤ a ≤ n 2 . But now there is no gap 2 between 0 and the remaining part of the spectrum. Question. What can one say about the time-decay of solutions if V is slowly increasing or decreasing?
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case The question If V ( x ) ≈ a | x | µ for some a > 0 and 0 < µ < 1, then m 0 ∈ L 2 and 0 is an eigenvalue of P . If V ( x ) ≈ a ln | x | , m 0 is an eigenfunction if a > n 2 and is a resonant state if n − 2 ≤ a ≤ n 2 . But now there is no gap 2 between 0 and the remaining part of the spectrum. Question. What can one say about the time-decay of solutions if V is slowly increasing or decreasing? One may say that the case |∇ V | → ∞ is a non-selfadjoint eigenvalue problem, while the case |∇ V | → 0 is a non-selfadjoint scattering problem for the pair ( P 0 , P ) where P 0 is the free KFP operator P 0 = v · ∇ x − ∆ v + 1 4 | v | 2 − n 2
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case Complex harmonic oscillators 4 | v | 2 − n P 0 = v · ∇ x − ∆ v + 1 2 with the maximal demain is an accretive and hypoelliptic operator. It is unitarily equivalent with ˆ P 0 which is a direct integral of ˆ P 0 ( ξ ) , ξ ∈ R n , P 0 ( ξ ) = − ∆ v + 1 4 ( v + i 2 ξ ) 2 + ξ 2 − n ˆ 2 . One can check that σ ( ˆ P 0 ( ξ )) = { k + ξ 2 , k ∈ N } . All the eigenvalues are semisimple and the Riesz projection associated with the eigenvalue k + ξ 2 is given by Π ξ � � ψ − ξ α , ·� ψ ξ k = α . α ∈ N , | α | = k
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case Complex harmonic oscillators Here ψ ξ α ( v ) = ψ α ( v + i 2 ξ ) and ψ α , α ∈ N n , are normalized Hermite 4 v 2 − n functions: ( − ∆ v + 1 2 ) ψ α = | α | ψ α . Lemma 1 For any ξ ∈ R n and t > 0 , one has the following spectral decomposition for the semigroup: ∞ e − t ˆ P 0 ( ξ ) = e − t ( k + ξ 2 ) Π ξ � k , (4) k = 0 where the series is norm convergent as operators on L 2 ( R n v ) .
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case Complex harmonic oscillators Here ψ ξ α ( v ) = ψ α ( v + i 2 ξ ) and ψ α , α ∈ N n , are normalized Hermite 4 v 2 − n functions: ( − ∆ v + 1 2 ) ψ α = | α | ψ α . Lemma 1 For any ξ ∈ R n and t > 0 , one has the following spectral decomposition for the semigroup: ∞ e − t ˆ P 0 ( ξ ) = e − t ( k + ξ 2 ) Π ξ � k , (4) k = 0 where the series is norm convergent as operators on L 2 ( R n v ) .
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case Complex harmonic oscillators Here ψ ξ α ( v ) = ψ α ( v + i 2 ξ ) and ψ α , α ∈ N n , are normalized Hermite 4 v 2 − n functions: ( − ∆ v + 1 2 ) ψ α = | α | ψ α . Lemma 1 For any ξ ∈ R n and t > 0 , one has the following spectral decomposition for the semigroup: ∞ e − t ˆ P 0 ( ξ ) = e − t ( k + ξ 2 ) Π ξ � k , (4) k = 0 where the series is norm convergent as operators on L 2 ( R n v ) . To prove this lemma, we show that if n = 1, ∞ k � = e − ξ 2 ( t − 2 ) 4 ξ 2 e − t ( k + ξ 2 ) � Π ξ � et − 1 . 1 − e − t e (5) k = 0
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case Time-decay for free KFP operator The free KFP operator is unitarily equivalent with a direct integral of this family of complex harmonic oscillators. One deduces that σ ( P 0 ) = ∪ ξ ∈ R n σ (ˆ P 0 ( ξ )) = [ 0 , + ∞ [ . The set N is called thresholds of P 0 . The numerical range of P 0 is { z ; ℜ z ≥ 0 } .
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case Time-decay for free KFP operator The free KFP operator is unitarily equivalent with a direct integral of this family of complex harmonic oscillators. One deduces that σ ( P 0 ) = ∪ ξ ∈ R n σ (ˆ P 0 ( ξ )) = [ 0 , + ∞ [ . The set N is called thresholds of P 0 . The numerical range of P 0 is { z ; ℜ z ≥ 0 } . To study the time-decay of e − tP 0 , we introduce L 2 , s ( R 2 n ) = L 2 ( R 2 n ; � x � 2 s dxdv ) . and L p = L p ( R n x ; L 2 ( R n v )) , p ≥ 1 .
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case Time-decay for the free KFP equation Proposition 1 One has the following dispersive type estimate: ∃ C > 0 such that � e − tP 0 u � L ∞ ≤ C 2 � u � L 1 , t ≥ 3 , (6) n t for u ∈ L 1 . In particular, for any s > n 2 , one has for some C s > 0 � e − tP 0 u � L 2 , − s ≤ C s 2 � u � L 2 , s , (7) n t for t ≥ 3 and u ∈ L 2 , s .
Introduction. Motivation The free KFP operator The KFP operator with a potential Low-energy spectral properties in short-range case Time-decay for the free KFP equation Proposition 1 One has the following dispersive type estimate: ∃ C > 0 such that � e − tP 0 u � L ∞ ≤ C 2 � u � L 1 , t ≥ 3 , (6) n t for u ∈ L 1 . In particular, for any s > n 2 , one has for some C s > 0 � e − tP 0 u � L 2 , − s ≤ C s 2 � u � L 2 , s , (7) n t for t ≥ 3 and u ∈ L 2 , s .
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