Observability for the Schrödinger Equation: an Optimal Transport Approach
aa r X i v : . [ m a t h . A P ] F e b OBSERVABILITY FOR THE SCHR ¨ODINGER EQUATION:AN OPTIMAL TRANSPORTATION APPROACH
FRANC¸ OIS GOLSE AND THIERRY PAUL
Abstract.
We establish an observation inequality for the Schr¨odinger equa-tion on R d , uniform in the Planck constant ̵ h ∈ [ , ] . The proof is based on thepseudometric introduced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. (2017), 57–94]. This inequality involves only effective constants which arecomputed explicitly in their dependence in ̵ h and all parameters involved. Observation inequality for the Schr¨odinger equation
Consider the Schr¨odinger equation where the (real-valued) potential V belongsto C , ( R d ) is such that the quantum Hamiltonian − ̵ h ∆ y + V ( y ) has a self-adjoint extension on H ∶= L ( R d ) :(1) i ̵ h∂ t ψ ( t, y ) = (− ̵ h ∆ y + V ( y )) ψ ( t, y ) , ψ ∣ t = = ψ in . In the equation above, ̵ h > ∥ ψ in ∥ H ≤ C ∫ T ∫ Ω ∣ ψ ( t, x )∣ dxdt , for some T >
0, where Ω is an open subset of R d , and C ≡ C [ T, Ω ] is a positiveconstant, which holds for some appropriate class of initial data ψ in (see equation(2) in [9]).Note that the r.h.s. of (2) is smaller than CT , so that (2) can be satisfied onlywhen CT ≥
1. Moreover, it is easy to check that the case CT = = R d , and reduces that way to a tautology.Therefore we will suppose in the sequel CT > . We will say that a compact subset K of R d × R d , an open set Ω of R d and T > ( x, ξ ) ∈ K , there exists t ∈ ( , T ) s.t. X ( t ; x, ξ ) ∈ Ω . Let us recall the definition of the Schr¨odinger coherent state: ∣ q, p ⟩( x ) ∶= ( π ̵ h ) − d / e −∣ x − q ∣ / ̵ h e ip ⋅ ( x − q / )/̵ h providing a decomposition of the identity on H (in a weak sense)(3) ∫ R d ∣ q, p ⟩⟨ q, p ∣ dpdq ( π ̵ h ) d = I H . Let us recall also, for any self-adjoint operator A on L ( R d ) and any ψ ∈ L ( R d ) ,the definition of the standard deviation of A in the state ψ , ∆ A ( ψ ) ∈ [ , +∞ ] :∆ A ( ψ ) = √( ψ, A ψ ) L ( R d ) − ( ψ, Aψ )) L ( R d ) We define(4) ∆ ( ψ ) ∶= ¿ÁÁÁÀ d ∑ j = ( ∆ x j ( ψ ) + ∆ − i ̵ h∂ xj ( ψ )) . Let us remark that, by the Heisenberg inequalities, for any ψ ∈ H ,(5) ∆ ( ψ ) ≥ d ̵ h. and, for any ( p, q ) ∈ R d ,(6) ∆ (∣ p, q ⟩) = √ d ̵ h. Theorem 1.1.
Assume that V belongs to C , ( R d ) and that V − ∈ L d / ( R d ) .Let T > , Ω be an open subset of R d . and K be a compact set in R d satisfyingthe Bardos-Lebeau-Rauch condition ( GC ) .Moreover, let δ > and Ω δ ∶= { x ∈ R d ∣ dist ( x, Ω ) < δ } . Then the Schr¨odinger equation (1) satisfies an observability property on [ , T ] × Ω δ of the form (2) with constant C for all vectors ψ ∈ H satisfying C [ T, K, Ω ] ( ∫ K ∣⟨ ψ ∣ p, q ⟩∣ dpdq ( π ̵ h ) d ) − D [ T, Lip ( ∇ V )] ∆ ( ψ ) δ ≥ C where C [ T, K, Ω ] = inf ( x,ξ )∈ K ∫ T Ω ( X ( t ; x, ξ )) dtD [ T, Lip ( ∇ V )] = e ( + Lip ( ∇ V ) ) T / − + Lip (∇ V ) . Moreover, the observation inequality will be satisfied for a non empty set of vectorsas soon as δ satisfies the following, non sharp, bound: δ ≥ D [ T, Lip ( ∇ V )] C [ T, K, Ω ]( − e − d K ̵ h /( π ) d ) + C − √ d ̵ h, where d K is the diameter of K . The first part of Theorem 1.1 is exactly the second part (pure state case) ofCorollary 4.2 of Theorem 4.1 in Section 4 below.
BSERVABILITY FOR SCHR ¨ODINGER 3
Controllability of the quantum dynamics has a long history in mathematics andmathematical physics. Giving an exhaustive bibliography on the subject is byfar beyond the scope of the present paper, paper, but the reader can consult thesurvey article [9] and the literature cited there, together with the important earlierreferences [3, 4], [10]For the bound on δ , we first remark that the quantity E [ ψ, δ ] ∶= C [ T, K, Ω δ ] ( ∫ K ∣⟨ ψ ∣ p, q ⟩∣ dpdq ( π ̵ h ) d ) − D [ T, Lip ( ∇ V )] ∆ ( ψ ) δ , needed to be strictly positive for the observability condition to hold true, is adifference between (a quantity proportional to) ∫ K ∣⟨ ψ ∣ p, q ⟩∣ dpdq ( π ̵ h ) d ( ≤ ,which evaluates the microlocalization of ψ on K , and (a quantity proportional to)∆ ( ψ ) ( ≥ √ d ̵ h (by (5)), which measures the spreading of ψ near its average positionin phase-space.However, this competition is balanced by the smallness of D [ T, Lip ( ∇ V )] ∆ ( ψ ) δ forlarge values of δ , namely E [ δ, ψ ] ≥ C when δ ≥ D [ T, Lip ( ∇ V )] ∆ ( ψ ) C [ T, K, Ω ]( − ∫ K ∣⟨ ψ ∣ p, q ⟩∣ dpdq ( π ̵ h ) d ) + C − . Finally, we remark that, taking ψ = ∣ p , q ⟩ for some ( p , q ) ∈ R d we have, by (6),∆ ( ψ ) = √ d ̵ h, and, when ( p , q ) belongs to the interior of K , ∫ K ∣⟨ p , q ∣ p, q ⟩∣ dpdq ( π ̵ h ) d = − ∫ R d / K e − ∣ p − p ∣ +∣ q − q ∣ ̵ h dpdq ( π ̵ h ) d ≥ − e − dist (( p ,q ) , R d / K ) ̵ h ( π ) d . We conclude by picking ( p , q ) such that, for example, dist (( p , q ) , R d / K ) ≥ d K .In the present paper, we will be working with the slightly more general Heisen-berg equation(7) i ̵ h∂ t R ( t ) = [ − ̵ h ∆ y + V ( y ) , R ( t )] , R ∣ t = = R in ≥ , trace R = , equivalent to the Schr¨odinger equation, modulo a global phase of the wave function,through the passage ψ ∈ H Ð → ∣ ψ ⟩⟨ ψ ∣ , and whose underlying classical dynamics solves the Liouville equation ∂ t f ( t, x, ξ ) + { ∣ ξ ∣ + V ( x ) , f ( t, x, ξ )} = , f ∣ t = = f in , where f in is a probability density on R d × R d having finite second moments.Corollary 4.2 contains also an equivalent statement for initial conditions whichare T¨oplitz operators. The general case of mixed states can be recovered by theinequality (12) inside the proof of Theorem 4.1.The core of the paper is Theorem 4.1 in Section 4, whose proof needs the intro-duction in Section pseudomet of a class of pseudometrics adapted to the Heisenberg Note that ∫ R d K ∣⟨ ψ ∣ p, q ⟩∣ dpdq ( π ̵ h ) d is the integral over K of the Husimi function of ψ . F. GOLSE AND T. PAUL equation (7), introduced in [6] after [5], and whose evolution under (7) is presentedin Section 3.2.
A pseudometric for comparing classical and quantum densities
This section elaborates on [6], with some marginal improvements.A density operator on H is an operator R ∈ L ( H ) such that R = R ∗ ≥ , trace ( R ) = . The set of all density operators on H will be denoted by D ( H ) . We denote by D ( H ) the set of density operators on H such that(8) trace ( R / ( − ̵ h ∆ y + ∣ y ∣ ) R / ) < ∞ . If R ∈ D ( H ) , one has(9) trace (( − ̵ h ∆ y + ∣ y ∣ ) / R ( − ̵ h ∆ y + ∣ y ∣ ) / ) = trace ( R / ( − ̵ h ∆ y + ∣ y ∣ ) R / ) < ∞ as can be seen from the lemma below (applied to A = λ ∣ y ∣ − ̵ h ∆ y and T = R ). Lemma 2.1.
Let T ∈ L ( H ) satisfy T = T ∗ ≥ , and let A be an unbounded operatoron H such that A = A ∗ ≥ . Then trace ( T / AT / ) = trace ( A / RA / ) ∈ [ , +∞ ] . Proof.
The definition of T / and A / can be found in Theorem 3.35 in chapter V, § A / and T / are self-adjoint.If trace ( T / AT / ) < ∞ , then A / T / ∈ L ( H ) and the equality holds by for-mula (1.26) in chapter X, § ( T / AT / ) = ∞ , then trace ( A / T A / ) =+∞ , for otherwise T / A / and its adjoint A / T / would belong to L ( H ) , sothat T / AT / ∈ L ( H ) , which would be in contradiction with the assumption thattrace ( T / AT / ) = ∞ . (cid:3) Let f ≡ f ( x, ξ ) be a probability density on R d × R d such that(10) ∬ R d × R d (∣ x ∣ + ∣ ξ ∣ ) f ( x, ξ ) dxdξ < ∞ . A coupling of f and R is a measurable operator-valued function ( x, ξ ) ↦ Q ( x, ξ ) such that, for a.e. ( x, ξ ) ∈ R d × R d , Q ( x, ξ ) = Q ( x, ξ ) ∗ ≥ , trace ( Q ( x, ξ )) = f ( x, ξ ) , ∬ R d × R d Q ( x, ξ ) dxdξ = R .
The second condition above implies that Q ( x, ξ ) ∈ L ( H ) for a.e. ( x, ξ ) ∈ R d × R d .Since L ( H ) is separable, the notion of strong and weak measurability are equivalentfor Q . The set of couplings of f and R is denoted by C ( f, R ) . Notice that thefunction ( x, ξ ) ↦ f ( x, ξ ) R belongs to C ( f, R ) .In [6], one considers the following “pseudometric”: for each probability density f on R d × R d and each R ∈ D ( H ) , E ̵ h,λ ( f, R ) ∶= inf Q ∈ C ( f,R ) ( ∬ R d × R d trace H ( Q ( x, ξ ) / c λ ( x, ξ, y, ̵ hD y ) Q ( x, ξ ) / ) dxdξ ) / where the quantum transportation cost is the quadratic differential operator in y ,parametrized by ( x, ξ ) ∈ R d × R d : c λ ( x, ξ, y, ̵ hD y ) ∶= λ ∣ x − y ∣ + ∣ ξ − ̵ hD y ∣ , D y ∶= − i ∇ y . BSERVABILITY FOR SCHR ¨ODINGER 5
Lemma 2.2. If R ∈ D ( H ) while f is a probability density on R d × R d with finitesecond moment (10) , one has ∬ R d × R d trace H ( Q ( x, ξ ) / c ( x, ξ, y, ̵ hD y ) Q ( x, ξ ) / ) dxdξ = ∬ R d × R d trace H ( c ( x, ξ, y, ̵ hD y ) / Q ( x, ξ ) c ( x, ξ, y, ̵ hD y ) / ) dxdξ ≤ ∬ R d × R d ( λ ∣ x ∣ + ∣ ξ ∣ ) f ( x, ξ ) dxdξ + ( R / ( − ̵ h ∆ y + λ ∣ y ∣ ) R / ) < ∞ for each Q ∈ C ( f, R ) .Proof. Notice that c λ ( x, ξ, y, ̵ hD y ) ≤ λ (∣ x ∣ + ∣ y ∣ ) + (∣ ξ ∣ − ̵ h ∆ y ) = ( λ ∣ x ∣ + ∣ ξ ∣ ) + ( λ ∣ y ∣ − ̵ h ∆ y ) so that ∬ R d × R d trace H ( Q ( x, ξ ) / c ( x, ξ, y, ̵ hD y ) Q ( x, ξ ) / ) dxdξ ≤ ∬ R d × R d trace H ( Q ( x, ξ ) / ( λ ∣ x ∣ + ∣ ξ ∣ ) Q ( x, ξ ) / ) dxdξ + ∬ R d × R d trace H ( Q ( x, ξ ) / ( λ ∣ y ∣ − ̵ h ∆ y ) Q ( x, ξ ) / ) dxdξ . First ∬ R d × R d trace H ( Q ( x, ξ ) / ( λ ∣ x ∣ + ∣ ξ ∣ ) Q ( x, ξ ) / ) dxdξ = ∬ R d × R d ( λ ∣ x ∣ + ∣ ξ ∣ ) trace H ( Q ( x, ξ )) dxdξ = ∬ R d × R d ( λ ∣ x ∣ + ∣ ξ ∣ ) f ( x, ξ ) dxdξ . Since R ∈ D ( H ) , one has trace H ( R / ( λ ∣ y ∣ − ̵ h ∆ y ) R / ) = trace H (( λ ∣ y ∣ − ̵ h ∆ y ) / R ( λ ∣ y ∣ − ̵ h ∆ y ) / ) = ∬ R d × R d trace H (( λ ∣ y ∣ − ̵ h ∆ y ) / Q ( x, ξ ) dxdξ ( λ ∣ y ∣ − ̵ h ∆ y ) / ) < ∞ , where the first equality is (9), while the second follows from the monotone conver-gence theorem (Theorem 1.27 in [11]) applied to a spectral decomposition of theharmonic oscillator λ ∣ y ∣ − ̵ h ∆ y .In particulartrace H ( λ ∣ y ∣ − ̵ h ∆ y ) / Q ( x, ξ )( λ ∣ y ∣ − ̵ h ∆ y ) / ) < ∞ for a.e. ( x, ξ ) ∈ R d × R d . Applying Lemma 2.1 to A = λ ∣ y ∣ − ̵ h ∆ y and T = Q ( x, ξ ) for a.e. ( x, ξ ) ∈ R d × R d , one hastrace H (( λ ∣ y ∣ − ̵ h ∆ y ) / Q ( x, ξ )( λ ∣ y ∣ − ̵ h ∆ y ) / ) = trace H ( Q ( x, ξ ) / ( λ ∣ y ∣ − ̵ h ∆ y ) Q ( x, ξ ) / ) for a.e. ( x, ξ ) ∈ R d × R d . Integrating both sides of this equality over R d × R d , onefinds that ∬ R d × R d trace H ( Q ( x, ξ ) / ( λ ∣ y ∣ − ̵ h ∆ y ) Q ( x, ξ ) / ) dxdξ = trace H (( λ ∣ y ∣ − ̵ h ∆ y ) / R ( λ ∣ y ∣ − ̵ h ∆ y ) / ) < ∞ . F. GOLSE AND T. PAUL
In particular trace H ( Q ( x, ξ ) / c ( x, ξ, y, ̵ hD y ) Q ( x, ξ ) / ) < ∞ for a.e. ( x, ξ ) ∈ R d × R d . Applying again Lemma 2.1 with A = c ( x, ξ, y, ̵ hD y ) and T = Q ( x, ξ ) for all such ( x, ξ ) shows thattrace H ( Q ( x, ξ ) / c ( x, ξ, y, ̵ hD y ) Q ( x, ξ ) / ) = trace H ( c ( x, ξ, y, ̵ hD y ) / Q ( x, ξ ) c ( x, ξ, y, ̵ hD y ) / ) for a.e. ( x, ξ ) ∈ R d , and the equality in the lemma follows from integrating bothsides of this last identity over R d × R d . (cid:3) The main properties of this pseudo-metric are recalled in the following theorem.Before stating it, we recall some fundamental notions and introduce some notations.The Wigner transform of R ∈ D ( H ) is W ̵ h [ R ]( x, ξ ) = ( π ) d ∫ R d r ( x + ̵ hy, x − ̵ hy ) e − iξ ⋅ y dy where r is the integral kernel of R . Obviously W ̵ h [ R ] is real-valued, but in general W ̵ h [ R ] is not a.e. nonnegtive in general.Instead of the Wigner transform, one can consider a mollified variant thereof,the Husimi transform of R , that is ̃ W ̵ h [ R ]( x, ξ ) = ( e ̵ h ∆ x,ξ / W ̵ h [ R ])( x, ξ ) ≥ ( x, ξ ) ∈ R d × R d . The Schr¨odinger coherent state is ∣ q, p ⟩( x ) ∶= ( π ̵ h ) − d / e − ∣ x − q ∣ / ̵ h e ip ⋅ ( x − q / )/̵ h . For each Borel probability measure µ on R d × R d , one defines the T¨oplitz operatorwith symbol ( π ̵ h ) d µ :OP T ̵ h [( π ̵ h ) d µ ] ∶= ∬ R d × R d ∣ q, p ⟩⟨ q, p ∣ µ ( dqdp ) ∈ D ( H ) . Proposition 2.3.
For each probability density f and each Borel probability measure µ on R d × R d with finite second order moment (10) . Then OP T ̵ h [( π ̵ h ) d µ ] ∈ D ( H ) , and one has E ̵ h,λ ( f, OP T ̵ h [( π ̵ h ) d µ ]) ≤ max ( , λ ) dist MK , ( f, µ ) + ( λ + ) d ̵ h . Proof.
Let P ( x, ξ, dqdp ) be an optimal coupling of f ( x, ξ ) and µ ( dqdp ) for dist MK , .Set Q ( x, ξ ) ∶= OP T ̵ h [( π ̵ h ) d P ( x, ξ, ⋅ )] . Then Q ∈ C ( f, OP T ̵ h [( π ̵ h ) d µ ]) according toLemma 3.1 in [6]), so that E ̵ h,λ ( f, OP T ̵ h [( π ̵ h ) d µ ]) ≤ ∬ R d × R d trace H ( Q ( x, ξ ) / c λ ( x, ξ, y, ̵ hD y ) Q ( x, ξ ) / ) dxdξ . For each p, q ∈ R d , one hastrace H ( c λ ( x, ξ, y, ̵ hD y ) / ∣ q, p ⟩⟨ q, p ∣ c λ ( x, ξ, y, ̵ hD y ) / ) = ⟨ q, p ∣ c λ ( x, ξ, y, ̵ hD y )∣ q, p ⟩ = λ ∣ x − q ∣ + ∣ ξ − p ∣ + ( λ + )̵ h BSERVABILITY FOR SCHR ¨ODINGER 7 according to fla. (55) in [5]. For each finite positive Borel measure m on R d × R d ,one has trace H ( c λ ( x, ξ, y, ̵ hD y ) / OP T ̵ h [( π ̵ h ) d m ] c λ ( x, ξ, y, ̵ hD y ) / ) = ∬ R d × R d ( λ ∣ x − q ∣ + ∣ ξ − p ∣ + ( λ + )̵ h ) m ( dpdq ) . by the monotone convergence theorem (Theorem 1.27 in [11]) applied to a spectraldecomposition of the transportation cost operator c λ ( x, ξ, y, ̵ hD y ) , which is a shiftedharmonic oscillator.Specializing this formula to the case x = ξ = m = µ shows that the operatorOP T ̵ h [( π ̵ h ) d µ ] ∈ D ( H ) .Specializing this formula to the case m = P ( x, ξ, dqdp ) and integrating in ( x, ξ ) shows that ∬ R d × R d trace H ( c λ ( x, ξ, y, ̵ hD y ) / OP T ̵ h [( π ̵ h ) d P ( x, ξ, ⋅ )] c λ ( x, ξ, y, ̵ hD y ) / ) dxdξ = ∬ R d × R d ∬ R d × R d ( λ ∣ x − q ∣ + ∣ ξ − p ∣ ) P ( x, ξ, dqdp ) + ( λ + ) = dist MK , ( f, µ ) + ( λ + )̵ h and since Q ∶ ( x, ξ ) ↦ OP T ̵ h [( π ̵ h ) d P ( x, ξ, ⋅ )] belongs to C ( f, OP T ̵ h [( π ̵ h ) d µ ]) , ∬ R d × R d trace H ( Q ( x, ξ ) / c λ ( x, ξ, y, ̵ hD y ) Q ( x, ξ ) / ) dxdξ = ∬ R d × R d trace H ( c λ ( x, ξ, y, ̵ hD y ) / Q ( x, ξ ) c λ ( x, ξ, y, ̵ hD y ) / ) dxdξ . With the previous equality and the inequality above, the proof is complete. (cid:3) Evolution of the pseudo-metric under the Schr¨odinger dynamics
Denote by t ↦ ( X ( t ; x, ξ ) , Ξ ( t ; x, ξ )) the solution of the Cauchy problem for theHamiltonian system˙ X = Ξ , ˙Ξ = −∇ V ( X ) , ( X ( x, ξ ) , Ξ ( x, ξ )) = ( x, ξ ) . Since V ∈ C , ( R d ) , this solution is defined for all t ∈ R , for all x, ξ ∈ R d . Hence-forth, we denote by Φ t the map ( x, ξ ) ↦ Φ t ( x, ξ ) ∶ = ( X ( t ; x, ξ ) , Ξ ( t ; x, ξ )) , and by H ≡ H ( x, ξ ) ∶ = ∣ ξ ∣ + V ( x ) the Hamiltonian.On the other hand, assume that V − ∈ L d / ( R d ) , so that H ∶ = − ̵ h ∆ + V isself-adjoint on H by Lemma 4.8b in chapter VI, § U ( t ) ∶ = exp ( it H /̵ h ) is a unitary group on H . Theorem 3.1.
Let f in be a probability density on R d × R d which satisfies (10) ,and let R in ∈ D ( H ) . For each t ≥ , set R ( t ) ∶ = U ( t ) ∗ R in U ( t ) , f ( t, X, Ξ ) ∶ = f in ( Φ − t ( X, Ξ )) for a.e. ( X, Ξ ) ∈ R d × R d . Then, for each λ > and each t ≥ , one has E ̵ h,λ ( f ( t, ⋅ , ⋅ ) , R ( t )) ≤ E ̵ h,λ ( f in , R in ) exp ( t ( λ + Lip ( ∇ V ) λ ) t ) . This theorem is a slight improvement of Theorem 2.7 in [6] in the special case N =
1. For the sake of being complete, we recall the argument in [6], with theappropriate modifications.
F. GOLSE AND T. PAUL
Proof.
Let Q in ∈ C ( f in , R in ) . Set Q ( t, X, Ξ ) ∶ = U ( t ) ∗ Q in ○ Φ − t ( X, Ξ ) U ( t ) for all t ∈ R and a.e. ( x, ξ ) ∈ R d × R d , and E ( t ) ∶ = ∬ R d trace H ( Q ( t, X, Ξ ) / c λ ( X, Ξ , y, ̵ hD y ) Q ( t, X, Ξ ) / ) dXd Ξ . Since Φ t leaves the phase space volume element dxdξ invariant E ( t ) =∬ R d trace H (√ Q in ( x, ξ ) U ( t ) c λ ( Φ t ( x, ξ ) , y, ̵ hD y ) U ( t ) ∗ √ Q in ( x, ξ )) dxξ . By construction, Q ( t, ⋅ , ⋅ ) ∈ C ( f ( t, ⋅ , ⋅ ) , R ( t )) . Indeed, for a.e. ( X, Ξ ) ∈ R d ,0 ≤ Q in ( Φ − t ( X, Ξ )) = Q in ( Φ − t ( X, Ξ )) ∗ ∈ L ( H ) so that Q ( t, X, Ξ ) ∈ L ( H ) satisfies Q ( t, X, Ξ ) = U ( t ) Q in ( Φ − t ( X, Ξ )) U ( t ) ∗ = U ( t ) Q in ( Φ − t ( X, Ξ )) U ( t ) ∗ = Q ( t, X, Ξ ) ∗ ≥ . Besidestrace H ( Q ( t, X, Ξ )) = trace H ( Q in ( Φ − t ( X, Ξ ))) = f in ( Φ − t ( X, Ξ )) = f ( t, X, Ξ ) while ∬ R d × R d Q ( t, X, Ξ ) dXd Ξ = U ( t ) ( ∬ R d × R d Q in ( Φ − t ( X, Ξ )) dXd Ξ ) U ( t ) ∗ = U ( t ) ( ∬ R d × R d Q in ( x, ξ ) dxdξ ) U ( t ) ∗ = U ( t ) R in U ( t ) ∗ = R ( t ) . In particular E ( t ) ≥ E ̵ h,λ ( f ( t ) , R ( t )) , for each t ≥ . Let e j ( x, ξ, ⋅ ) for j ∈ N be a H -complete orthonormal system of eigenvectors of Q in ( x, ξ ) for a.e. x, ξ ∈ R d . Hencetrace H (√ Q in ( x, ξ ) U ( t ) c λ ( Φ t ( x, ξ ) , y, ̵ hD y ) U ( t ) ∗ √ Q in ( x, ξ )) = ∑ j ∈ N ρ j ( x, ξ )⟨ U ( t ) e j ( x, ξ )∣ c λ ( Φ t ( x, ξ ) , y, ̵ hD y )∣ U ( t ) e j ( x, ξ )⟩ where ρ j ( x, ξ ) is the eigenvalue of Q in ( x, ξ ) defined by Q in ( x, ξ ) e j ( x, ξ ) = ρ j ( x, ξ ) e j ( x, ξ ) , for a.e. ( x, ξ ) ∈ R d × R d . If φ ≡ φ ( y ) ∈ C ∞ c ( R d ) , the map t ↦ ⟨ U ( t ) φ ∣ c λ ( Φ t ( x, ξ ) , y, ̵ hD y )∣ U ( t ) φ ⟩ is of class C on R , and one has ddt ⟨ U ( t ) φ ∣ c λ ( Φ t ( x, ξ ) , y, ̵ hD y )∣ U ( t ) φ ⟩ = ⟨ i ̵ h H U ( t ) φ ∣ c λ ( Φ t ( x, ξ ) , y, ̵ hD y )∣ U ( t ) φ ⟩ + ⟨ U ( t ) φ ∣ c λ ( Φ t ( x, ξ ) , y, ̵ hD y )∣ i ̵ h H U ( t ) φ ⟩ + ⟨ U ( t ) φ ∣{ H ( Φ t ( x, ξ )) , c λ ( Φ t ( x, ξ ) , y, ̵ hD y )}∣ U ( t ) φ ⟩ . BSERVABILITY FOR SCHR ¨ODINGER 9
In other words ddt ⟨ U ( t ) φ ∣ c λ ( Φ t ( x, ξ ) , y, ̵ hD y )∣ U ( t ) φ ⟩ = ⟨ U ( t ) φ ∣ i ̵ h [ H , c λ ( Φ t ( x, ξ ) , y, ̵ hD y )]∣ U ( t ) φ ⟩ + ⟨ U ( t ) φ ∣{ H ( Φ t ( x, ξ )) , c λ ( Φ t ( x, ξ ) , y, ̵ hD y )}∣ U ( t ) φ ⟩ . A straightforward computation shows that { H ( Φ t ( x, ξ )) , c λ ( Φ t ( x, ξ ) , y, ̵ hD y )} + i ̵ h [ H , c λ ( Φ t ( x, ξ ) , y, ̵ hD y )] = λ d ∑ k = (( X k − y k )( Ξ k − ̵ hD y k ) + ( Ξ k − ̵ hD y k )( X k − y k )) − d ∑ k = (( ∂ k V ( X ) − ∂ k V ( y ))( Ξ k − ̵ hD y k ) + ( Ξ k − ̵ hD y k )( ∂ k V ( X ) − ∂ k V ( y ))) ≤ λ d ∑ k = ( λ ∣ X k − y k ∣ + ∣ Ξ k − ̵ hD y k ∣ ) + λ d ∑ k = ( λ ∣ ∂ k V ( X ) − ∂ k V ( y )∣ + ∣ Ξ k − ̵ hD y k ∣ ) ≤ λ d ∑ k = ( λ ∣ X k − y k ∣ + ∣ Ξ k − ̵ hD y k ∣ ) + Lip ( ∇ V ) λ d ∑ k = ( λ ∣ X k − y ∣ + ∣ Ξ k − ̵ hD y k ∣ ) ≤ ( λ + Lip ( ∇ V ) λ ) c λ ( X, Ξ , y, ̵ hD y ) . Hence ⟨ U ( t ) φ ∣ c λ ( Φ t ( x, ξ ) , y, ̵ hD y )∣ U ( t ) φ ⟩ ≤ ⟨ φ ∣ c λ ( x, ξ, y, ̵ hD y )∣ φ ⟩ + ( λ + Lip ( ∇ V ) λ ) ∫ t ⟨ U ( s ) φ ∣ c λ ( Φ s ( x, ξ ) , y, ̵ hD y )∣ U ( s ) φ ⟩ ds so that ⟨ U ( t ) φ ∣ c λ ( Φ t ( x, ξ ) , y, ̵ hD y )∣ U ( t ) φ ⟩ ≤ ⟨ φ ∣ c λ ( x, ξ, y, ̵ hD y )∣ φ ⟩ exp (( λ + Lip ( ∇ V ) λ ) t ) for each φ ∈ C ∞ c ( R d ) . By density of C ∞ c ( R d ) in the form domain of c λ ( x, ξ, y, ̵ hD y ) ≤ ⟨ U ( t ) e j ( x, ξ )∣ c λ ( Φ t ( x, ξ ) , y, ̵ hD y )∣ U ( t ) e j ( x, ξ )⟩ ≤ ⟨ e j ( x, ξ )∣ c λ ( x, ξ, y, ̵ hD y )∣ e j ( x, ξ )⟩ exp (( λ + Lip ( ∇ V ) λ ) t ) for a.e. ( x, ξ ) ∈ R d × R d , so thattrace H (√ Q in ( x, ξ ) U ( t ) c λ ( Φ t ( x, ξ ) , y, ̵ hD y ) U ( t ) ∗ √ Q in ( x, ξ )) = ∑ j ∈ N ρ j ( x, ξ )⟨ U ( t ) e j ( x, ξ )∣ c λ ( Φ t ( x, ξ ) , y, ̵ hD y )∣ U ( t ) e j ( x, ξ )⟩ ≤ exp (( λ + Lip ( ∇ V ) λ ) t ) ∑ j ∈ N ρ j ( x, ξ )⟨ e j ( x, ξ )∣ c λ ( x, ξ, y, ̵ hD y )∣ e j ( x, ξ )⟩ = exp (( λ + Lip ( ∇ V ) λ ) t ) trace H (√ Q in ( x, ξ ) c λ ( x, ξ, y, ̵ hD y )√ Q in ( x, ξ )) . Integrating both side of this inequality over R d × R d shows that E ( t ) ≤ E ( ) exp (( λ + Lip ( ∇ V ) λ ) t ) . Hence, for each t ≥ Q in ∈ C ( f, R ) , one has E ̵ h,λ ( f ( t ) , R ( t )) ≤ E ( ) exp (( λ + Lip ( ∇ V ) λ ) t ) . Minimizing the right hand side of this inequality as Q in runs through C ( f in , R in ) ,one arrives at the inequality E ̵ h,λ ( f ( t ) , R ( t )) ≤ E ̵ h,λ ( f in , R in ) exp ( ( λ + Lip ( ∇ V ) λ ) t ) . (cid:3) The observation inequality
In this section, we state and prove an observation inequality for the Schr¨odingerequation.Let K be a compact subset of R d × R d , let Ω be an open set of R d and let T > ( x, ξ ) ∈ K , there exists t ∈ ( , T ) s.t. X ( t ; x, ξ ) ∈ Ω . Theorem 4.1.
Assume that V belongs to C , ( R d ) and that V − ∈ L d / ( R d ) . Let T > , let K ⊂ R d × R d be compact and let Ω ⊂ R d be an open set of R d satisfying(GC). Let χ ∈ Lip ( R d ) be such that χ ( x ) > for each x ∈ Ω .For each t ≥ , set R ( t ) ∶ = U ( t ) ∗ R in U ( t ) , f ( t, X, Ξ ) ∶ = f in ( Φ − t ( X, Ξ )) for a.e. ( X, Ξ ) ∈ R d × R d . Then, when R in is a pure state ∣ ψ in ⟩⟨ ψ in ∣ , ∫ T ∫ R d χ ( x )∣ ψ ( t, x )∣ dx ) dt ≥ inf ( x,ξ )∈ K ∫ T χ ( X ( t ; x, ξ )) dt ∬ ( x,ξ )∈ K ̃ W ̵ h [ ψ in ]( x, ξ ) dxdξ − ( χ ) exp ( ( + Lip ( ∇ V ) ) T ) − ( + Lip ( ∇ V ) ) ∆ ( ψ in ) . When R in ∶ = OP T [( π ̵ h ) d f in ] is a T¨oplitz operator of symbol a probability density f in on R d × R d with support in K , ∫ T trace ( χR ( t )) dt ≥ inf ( x,ξ )∈ K ∫ T χ ( X ( t ; x, ξ )) dt − Lip ( χ ) C ( T, Lip ( ∇ V ))√ d ̵ h where C ( T, L ) = inf λ > exp ( ( λ + L λ ) T ) − ( λ + L λ ) √ + λ . In particular, setting λ = L C ( T, L ) ≤ e LT − L √ + L . BSERVABILITY FOR SCHR ¨ODINGER 11
In fact, one can eliminate all mention of the cutoff function χ in the final state-ment, as follows. Corollary 4.2.
Under the same assumptions as in Theorem 4.1, one has C [ T, K, Ω ] ∶ = inf ( x,ξ )∈ K ∫ T Ω ( X ( t ; x, ξ )) dt > , and for each δ > , denoting Ω δ ∶ = { x ∈ R d ∣ dist ( x, Ω ) < δ } . , ∫ T trace ( Ω δ R ( t )) dt ≥ C [ T, K, Ω ] − C ( T, Lip ( ∇ V )) √ d ̵ hδ in the T¨oplitz case, and ∫ T ∫ Ω δ ∣ ψ ( t, x )∣ dx ) dt ≥ inf ( x,ξ )∈ K ∫ T Ω ( X ( t ; x, ξ )) dt ∬ ( x,ξ )∈ K ̃ W ̵ h [ ψ in ]( x, ξ ) dxdξ − ( ( + Lip ( ∇ V ) ) T ) − ( + Lip ( ∇ V ) ) ∆ ( ψ in ) δ in the pure state case. The corollary can be used to obtain an observation inequality for T¨oplitz op-erators as “test observables” as follows: let T > K ⊂ R d × R d be a compact subset of the phase-space supporting the initial data,and let Ω ⊂ R d be the open set where one observes the solution of the Schr¨odingerequation on the time interval [ , T ] . Assume that T, K,
Ω satisfies the geomet-ric condition (GC). With these data, one computes C [ T, K, Ω ] >
0. Choose then ̵ h, δ > ̵ hδ < C [ T, K, Ω ] dC ( T, Lip ( ∇ V )) . Then the Heisenberg equation (7) satisfies the observability property on [ , T ] × Ω δ for all T¨oplitz initial density operators whose symbol is supported in K . Proof of the corollary.
Since Ω is open, the function Ω is lower semicontinuous.According to condition (GC), for each ( x, ξ ) ∈ K , there exists t x,ξ ∈ ( , T ) such that Ω ( X ( t x,ξ ; x, ξ )) =
1. Since the set { t ∈ ( , T ) ∣ Ω ( X ( t ; x, ξ )) > / } is open, there exists η x,ξ > [ t x,ξ − η x,ξ , [ t x,ξ + η x,ξ ] ⊂ ( , T ) and then ∫ T Ω ( X ( t ; x, ξ )) dt ≥ η x,ξ > , for each ( x, ξ ) ∈ K .
By Fatou’s lemma, the function ( x, ξ ) ↦ ∫ T Ω ( X ( t ; x, ξ )) dt is lower semicontinuous, and positive on K . Hence C [ T, K, Ω ] ∶ = inf ( x,ξ )∈ K ∫ T Ω ( X ( t ; x, ξ )) dt > . Apply Theorem 4.1 with χ defined as follows: χ δ ( x ) = ( − dist ( x, Ω ) δ ) + , in which case Lip ( χ ) = δ . One concludes by observing that ∫ T trace ( χ δ R ( t )) dt ∫ T trace ( Ω δ R ( t )) dt , whereas ∫ T Ω ( X ( t ; x, ξ )) dt ≤ ∫ T χ δ ( X ( t ; x, ξ )) dt . (cid:3) Proof.
Notice that trace ( χ ( R ( t )) − ∬ R d × R d χ ( x ) f ( t, x, ξ ) dxdξ = ∬ R d × R d trace H (( χ ( y ) − χ ( x )) Q ( t, x, ξ )) dxdξ for each Q ≡ Q ( t, x, ξ ) ∈ C ( f ( t ) , R ( t )) . Hence ∣ trace ( χR ( t )) − ∬ R d × R d χ ( x ) f ( t, x, ξ ) dxdξ ∣ = ∣ ∬ R d × R d trace H (( χ ( y ) − χ ( x )) Q ( t, x, ξ )) dxdξ ∣ ≤ ∬ R d × R d ∣ trace H (( χ ( y ) − χ ( x )) Q ( t, x, ξ ))∣ dxdξ = ∬ R d × R d ∣ trace H ( Q ( t, x, ξ ) / ( χ ( y ) − χ ( x )) Q ( t, x, ξ ) / )∣ dxdξ ≤ ∬ R d × R d trace H ( Q ( t, x, ξ ) / ∣ χ ( y ) − χ ( x )∣ Q ( t, x, ξ ) / ) dxdξ ≤ Lip ( χ ) ∬ R d × R d trace H ( Q ( t, x, ξ ) / ∣ x − y ∣ Q ( t, x, ξ ) / ) dxdξ ≤ Lip ( χ ) ∬ R d × R d trace H ( Q ( t, x, ξ ) / ( ǫ ∣ x − y ∣ + ǫ ) Q ( t, x, ξ ) / ) ∣ dxdξ . Minimizing in ǫ > ∣ trace H ( χR ( t )) − ∬ R d × R d χ ( x ) f ( t, x, ξ ) dxdξ ∣ ≤ Lip ( χ ) ( ∬ R d × R d trace H ( Q ( t, x, ξ ) / ∣ x − y ∣ Q ( t, x, ξ ) / ) ∣ dxdξ ) / ≤ Lip ( χ ) λ ( ∬ R d × R d trace H ( Q ( t, x, ξ ) / c λ ( x, ξ, y, ̵ hD y ) Q ( t, x, ξ ) / ) ∣ dxdξ ) / . This holds for each Q ( t ) ∈ C ( f ( t ) , R ( t )) ; minimizing in Q ( t ) ∈ C ( f ( t ) , R ( t )) leadsto the bound ∣ trace H ( χR ( t )) − ∬ R d × R d χ ( x ) f ( t, x, ξ ) dxdξ ∣ ≤ Lip ( χ ) λ E ̵ h,λ ( f ( t ) , R ( t )) . BSERVABILITY FOR SCHR ¨ODINGER 13
By Theorem 3.1 ∣ trace H ( χR ( t )) − ∬ R d × R d χ ( x ) f ( t, x, ξ ) dxdξ ∣ ≤ Lip ( χ ) λ E ̵ h,λ ( f in , R in ) exp ( ( λ + Lip ( ∇ V ) λ ) t ) . On the other hand ∬ R d × R d χ ( x ) f ( t, x, ξ ) dxdξ = ∬ R d × R d χ ( x ) f in ( X ( t ; x, ξ ) , Ξ ( t ; x, ξ )) dxdξ = ∬ R d × R d χ ( X ( t ; x, ξ )) f in ( x, ξ ) dxdξ . Hence ∫ T trace ( χR ( t )) dt ≥ ∬ R d × R d ( ∫ T χ ( X t ( x, ξ )) dt ) f in ( x, ξ ) dxdξ − Lip ( χ ) λ E ̵ h,λ ( f in , R in ) ∫ T exp ( ( λ + Lip ( ∇ V ) λ ) t ) dt ≥ ∬ R d × R d ( ∫ T χ ( X t ( x, ξ )) dt ) f in ( x, ξ ) dxdξ − Lip ( χ ) λ exp ( ( λ + Lip ( ∇ V ) λ ) T ) − ( λ + Lip ( ∇ V ) λ ) E ̵ h,λ ( f in , R in ) . ≥ inf ( x,ξ )∈ K ∫ T χ ( X ( t ; x, ξ )) dt ∬ ( x,ξ )∈ K f in ( x, ξ ) dxdξ (11) − Lip ( χ ) λ exp ( ( λ + Lip ( ∇ V ) λ ) T ) − ( λ + Lip ( ∇ V ) λ ) E ̵ h,λ ( f in , R in ) . In particular, putting f in = ̃ W ̵ h [ R in ] and λ =
1, one obtains ∫ T trace ( χR ( t )) dt ≥ inf ( x,ξ )∈ K ∫ T χ ( X ( t ; x, ξ )) dt ∬ ( x,ξ )∈ K (̃ W ̵ h [ R in ]( x, ξ ) dxdξ − Lip ( χ ) λ exp ( ( λ + Lip ( ∇ V ) λ ) T ) − ( λ + Lip ( ∇ V ) λ ) E ̵ h,λ ((̃ W ̵ h [ R in ] , R in ) . (12)For R in = ∣ ψ in ⟩⟨ ψ in ∣ , we know by Proposition 9.1. in [7] that E ̵ h, (̃ W ̵ h [ R in ] , R in ) ≤ ( R in ) and we get the conclusion of Theorem 4.1 in the pure state case.If f in is any compactly supported probability density, the inequality (11) that ∫ T trace ( χR ( t )) dt ≥ inf ( x,ξ )∈ supp ( f in ) ∫ T χ ( X ( t ; x, ξ )) dt − Lip ( χ ) λ exp ( ( λ + Lip ( ∇ V ) λ ) T ) − ( λ + Lip ( ∇ V ) λ ) E ̵ h,λ ( f in , R in ) . Now, if R in is the T¨oplitz operator with symbol ( π ̵ h ) d µ in , where µ in is a Borelprobability measure on R d × R d , ∫ T trace ( χR ( t )) dt ≥ inf ( x,ξ )∈ supp ( f in ) ∫ T χ ( X ( t ; x, ξ )) dt − Lip ( χ ) λ exp ( ( λ + Lip ( ∇ V ) λ ) T ) − ( λ + Lip ( ∇ V ) λ ) √ max ( , λ ) dist MK , ( f in , µ in ) + ( λ + ) d ̵ h . In particular, if R in = OP T ̵ h [( π ̵ h ) d f in ] , one has ∫ T trace ( χR ( t )) dt ≥ inf ( x,ξ )∈ supp ( f in ) ∫ T χ ( X ( t ; x, ξ )) dt − Lip ( χ ) λ exp ( ( λ + Lip ( ∇ V ) λ ) T ) − ( λ + Lip ( ∇ V ) λ ) √ ( λ + ) d ̵ h . Maximizing the right hand side as λ runs through ( , +∞ ) , one finds that ∫ T trace ( χR ( t )) dt ≥ inf ( x,ξ )∈ supp ( f in ) ∫ T χ ( X ( t ; x, ξ )) dt − Lip ( χ ) C ( T, Lip ( ∇ V ))√ d ̵ h , where C ( T, L ) ∶ = inf λ > exp ( ( λ + L λ ) T ) − λ + L √ λ + . If L >
0, one can take λ = L so that C ( T, L ) ≤ e LT − L √ + L . (cid:3) Notice that, in the case where L =
0, one can choose λ = r / T with re r = ( e r − ) , r > , λ = r / T , and find that C ( T, ) ≤ e r − r T √ + r T . Acknowledgments.
We would like to thank warmly Claude Bardos for havingread the first version of this paper and mentioned several references.
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Email address : [email protected] (T.P.) Laboratoire J.-L. Lions, Sorbonne Universit´e & CNRS, boˆıte courrier 187,75252 Paris Cedex 05, France
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