Time Dependent Quantum Perturbations Uniform in the Semiclassical Regime
aa r X i v : . [ m a t h . A P ] F e b TIME DEPENDENT QUANTUM PERTURBATIONS UNIFORMIN THE SEMICLASSICAL REGIME
FRANC¸ OIS GOLSE AND THIERRY PAUL
Abstract.
We present a time dependent quantum perturbation result, uni-form in the Planck constant for potential whose gradient is bounded a.e..Weshow also that the classical limit of the perturbed quantum dynamics remainsin a tubular neighborhood of the classical unperturbed one, the size of thisneighborhood being of the order of the square root of the size of the pertur-bation. We treat both Schr¨odinger and von Neumann-Heisenberg equations. in memory of
Arthur Wightman
Contents
1. Introduction 12. Main result 33. Applications to the classical limit 54. The case of mixed states 95. Proof of Theorems 2.1 and 4.1, and Proposition 3.2 106. Proof of Corollaries 2.2 and 4.2 157. Proof of Proposition 2.4 16References 161.
Introduction
Perturbation theory has a very special status in Quantum Mechanics. On oneside, it is responsible to most of its more spectacular success, from atomic to nu-clear physics. On the other side, it has a very peculiar epistemological status: itwas while he was working with Max Born [B25] on the Bohr-Sommerfeld quantiza-tion of celestial perturbations series, as explicitly stated by Poincar´e in his famous“M´emoires” [P1892], that Heisenberg went to the idea of replacing the commuta-tive algebra of convolution — corresponding to multiple multiplications of Fourierseries appearing in computations on action-angles variables — by the famous non-commutative algebra of matrices [H25].After quantum mechanics was truly settled, perturbation theory took a com-pletely different form, in the paradigm of functional analysis “`a la Kato” and ap-peared then mostly in the framework of the so-called Rayleigh-Schr¨odinger series.A kind of paradox is that it took a long time to link back the Rayleigh-Schr¨odingerseries to the “original” formalism of quantization of, say, Birkhoff series [B28],
Date : February 11, 2021. though, in the mean time, the latter continued to be extensively used for appliedpurpose e.g. in heavy chemical computations.It seems that Arthur Wightman proposed to several PhD students to work onthis problem. One of the difficulty is that, starting with the second term of theRayleigh-Schr¨odinger expansion, E i = ∑ k ⟨ ψ i , V ψ k ⟩⟨ ψ k , V ψ i ⟩ E i − E k , appear poles in the Planck constant, for example when the unperturbed eigenvaluesare the one of the harmonic oscillator E i = ( i + )̵ h . Although this pole disappearsat the classical limit ̵ h → ∑ k ⟨ ψ i ,V ψ k ⟩⟨ ψ k ,V ψ i ⟩ i − k vanishes in thislimit for parity reasons, controlling all the terms of the series remained for years atask considered as unachievable.To our knowledge, the first proof on the convergence term by term of theRayleigh-Schr¨odinger expansion to the quantized Birkhoff one, for perturbationsof non-resonant harmonic oscillators, was given in [G87], by implementing the per-turbation procedure in the so-called Bargman representation (see also [D91] for animplementation in the framework of the Lie method). The reader interested in thissubject can also consult [P16, P162] for a proof (also for general non harmonic un-perturbed Hamiltonians) in a generalization of ´Ecalle’s mould theory and [NPST18]for a link between Rayleigh-Schr¨odinger expansion and Hopf algebras.When one considers time-dependent perturbation theory, i.e. comparison be-tween two quantum evolutions associated to two “close” Hamiltonians H and H ′ ,the situation is more difficult. The simple Duhamel formula e − i tH ̵ h − e − i tH ′̵ h = i ̵ h ∫ t e − i ( t − s ) H ̵ h ( H ′ − H ) e − i sH ′̵ h ds clearly shows that a pole in the Planck constant is again involved. But to ourknowledge, no combinatorics or normal form can help to remove it in general andone si usually reduced to the trivial estimate ∥ e − i tH ̵ h − e − i tH ′̵ h ∥ ≤ t ∥ H ′ − H ∥̵ h valid for, e.g. any Schatten norm, the operator, Hilbert-Schmidt or trace norm forexample.In the present paper, we will get rid of this pole in ̵ h phenomenon by estimatingthe difference between two quantum evolutions (in a weak topology consisting intracing against a set of test observables) in two forms:- one linear in the norm of the difference of the Hamiltonians plus a term van-ishing with ̵ h - the other proportional to the norm of the difference of the Hamiltonians to thepower 1 / ̵ h . UANTUM PERTURBATIONS 3
The proofs of our results, Sections 5 and 6, will be using the framework of thevon Neumann-Heisenberg equation for density operators D , ∂ t D = i ̵ h [ D, H ] , but our results, Theorem 2.1 and Corollary 2.2, will be first presented for purestates, Section 2, that is when D = ∣ ψ ⟩⟨ ψ ∣ , in which case it reduces to the usualSchr¨odinger one (modulo a global phase of the wave function) i ̵ h∂ t ψ = Hψ.
The mixed state situation will be treated in Section 4, Theorem 4.1.Our results will need very low regularity of the perturbed potential, namely theboundness of its gradient, and of the unperturbed one, Lipschitz continuity of itsgradient. In this situation, the classical underlying dynamics is well posed for theunperturbed Hamiltonian, but not for the perturbed one. To our knowledge, theclassical limit for pure state in this perturbed situation is unknown. We show inSection 3 that the limit as ̵ h → t is close to the one of the initial state pushed forward by the unperturbedclassical flow, Theorem 3.1. 2. Main result
For λ, µ ∈ [ , ] , let us consider the quantum Hamiltonian H λ,µ = H ∶= − ̵ h ∆ x + λ ∣ x ∣ + µV on H ∶= L ( R d ) . Here V ≡ V ( x ) ∈ R such that V ∈ C , ( R d ) . For any other realpotential U ∈ W , ∞ ( R d ) , we define, for ǫ ∈ [ , ] H λ,µǫ = H ǫ ∶= − ̵ h ∆ x + λ ∣ x ∣ + µV + ǫU. Henceforth we denote H ∶= H , = − ̵ h ∆ x + ∣ x ∣ (harmonic oscillator) D ( H ) ∶= { R ∈ L ( H ) s.t. R = R ∗ ≥ H ( R ) = } (density operators) , D ( H ) ∶= { R ∈ D ( H ) s.t. trace H ( R / H R / ) < ∞ } (finite second moments) . For ψ ∈ H , we define(1) ∆ ( ψ ) ∶= √( ψ, ( x − ( ψ, xψ )) + ( − i ̵ h ∇ x − ( ψ, − i ̵ h ∇ x ψ )) ψ ) Note that the Heisenberg inequalities √( ψ, ( x k − ( ψ, x k ψ )) , ψ ) . √( ψ, ( − i ̵ h ∇ x k − ( ψ, − i ̵ h ∇ x k ψ )) ψ ) ≥ ̵ h / , k = , . . . , d, imply that ∆ ( ψ ) ≥ √ d ̵ h. On D ( H ) we define the following distance(2) d ( R, S ) ∶= sup max ∣ α ∣ , ∣ β ∣≤ [ d ]+ ∥D α − i ̵ h ∇ D βx F ∥ ≤ ∣ trace ( F ( R − S ))∣ , where D A = i ̵ h [ A, ⋅ ] for each (possibly unbounded) self-adjoint operator A on H and ∥ . ∥ is the trace norm on D ( H ) . The fact that d is a distance has been provedin [GJP20, Appendix A]. F. GOLSE AND T. PAUL
By abuse of notation, we will call for ψ, ϕ ∈ H d ( ψ, ϕ ) ∶ = d (∣ ψ ⟩⟨ ψ ∣ , ∣ ϕ ⟩⟨ ϕ ∣) Consider the family of Schr¨odinger equations, for ǫ ∈ [ , ] ,(3) i ̵ h∂ t ψ ǫ ( t ) = H ǫ , ψ ǫ ( t ) ψ ǫ ( ) = ψ inǫ ∈ H ( R d ) , Theorem 2.1.
Let ψ inǫ satisfy the following hypothesis: (4) ∆ ( ψ in ) = O (√̵ h ) . Then, for every t , d ( ψ ( t ) , ψ ǫ ( t )) ≤ C ( t ) ǫ + D ( t )̵ h, where C ( t ) , D ( t ) , given by (19) , (20) , satisfy C ( t ) = e ∣ t ∣( − λ + µ Lip (∇ V )) − − λ + µ Lip ( ∇ V ) C ( ψ inǫ , H ψ inǫ ) , ∥ V ∥ ∞ , ∥ U ∥ ∞ , ∥∇ U ∥ ∞ < ∞ D ( t ) = e ∣ t ∣( − λ + µ Lip (∇ V )) D d < ∞ The following result gives an upper bound independent of ̵ h . Corollary 2.2. d ( ψ ( t ) , ψ ǫ ( t )) ≤ E ( t ) ǫ , with E ( t ) = min (√ C ( t ) + D ( t ) , ∣ t ∣∥ U ∥ ∞ ) . Note that when λ = , Lip ( ∇ V ) = C ( t ) increases linearly in time and D ( t ) is independent of time and C ( t ) , E ( t ) increase linearly in time. Remark 2.3.
Other choices than the hypothesis 4 are possible, that we didn’t men-tion for sake of clarity of the main statements. For example4’ ∆ ( ψ in ) = O ( ǫ ) :in this case the statement of both Theorem 2.1 and Corollary 2.2 remain the samewith a slight chance of the constants C ( t ) , D ( t ) .4” ∆ ( ψ in ) = O (̵ h α ) , ≤ α < :in this case the statement of both Theorem 2.1 and Corollary 2.2 become d ( ψ ( t ) , ψ ǫ ( t )) ≤ C ( t ) ǫ + D ′ ( t )̵ h α d ( ψ ( t ) , ψ ǫ ( t )) ≤ E ′ ( t ) ǫ αα + for constants C ( t , D ( t ) , E ( t ) easily computable form the proofs of Section 5. Let us finish this section by some topological remarks, inspired by [GP18, Section4]. The distance d defines a weak topology, very different a priori of the usual strongtopologies associated to Hilbert spaces in quantum mechanics. Nevertheless, itseems to us better adapted to the semiclassical approximation for the followingreason. UANTUM PERTURBATIONS 5
Let us consider two coherent states pined up at two points z = ( p , q ) , z = ( p , q ) of phase -space T ∗ R d : ψ z j ( x ) = ( π ̵ h ) − d / e − i p j .x ̵ h e − ( x − q j ) ̵ h , j = , ∥ ψ z − ψ z ∥ L ( R d ) = ∥∣ ψ z ⟩⟨ ψ z ∣ − ∣ ψ z ⟩⟨ ψ z ∣∣ Hilbert − Schmidt = − e − ( z − z ∣ ̵ h so that, as ̵ h → ∥ ψ z − ψ z ∥ L ( R d ) = ∥∣ ψ z ⟩⟨ ψ z ∣ − ∣ ψ z ⟩⟨ ψ z ∣∣ Hilbert − Schmidt = z = z → ∀ z ≠ z . In other words, the Lebesgue or Schatten norms behave for small values of ̵ h as thediscrete topology, the one which only discriminate points.At the contrary, d is much more sensitive to the localization on phase space asshows our next result, proven in Section 7 below. Proposition 2.4.
For any bounded convex domain Ω ⊂ R d , there exists C Ω > such that, for any z , z ∈ Ω , C Ω ∣ z − z ∣ − ̵ h ≤ d ( ψ z , ψ z ) ≤ √∣ z − z ∣ + d ̵ h + C d ̵ h, where C d is defined in Lemma 5.3 Section 5 below. Applications to the classical limit
The estimates provided by the results of the two preceding sections do not require ∇ U to be Lipschitz continuous — in other words, the classical dynamics underlyingthe quantum dynamics generated by H ǫ , ǫ >
0, may fail to satisfy the assumptionsof the Cauchy-Lipschitz theorem.Let us recall that one way to look at the transition from quantum to classicaldynamics as ̵ h → R ̵ h on H (density operator) with integral kernel r ̵ h ( x, x ′ ) ,e.g. a pure state R ̵ h = ∣ ψ ̵ h ⟩⟨ ψ ̵ h ∣ , for any vector ψ ̵ h in H the so-called Wignertransform defined on phase-space by (with a slight abuse of notation again) W ̵ h [ R ̵ h ]( x, ξ ) ∶ = ( π ) d ∫ R d e − iξ ⋅ y r ̵ h ( x + ̵ hy, x − ̵ hy ) dy (5) W ̵ h [ ψ ̵ h ]( x, ξ ) ∶ = ( π ) d ∫ R d e − iξ ⋅ y ψ ̵ h ( x + ̵ hy ) ψ ǫ ( x − ̵ hy ) dy . (6)An easy computation shows that W ̵ h [ ψ ̵ h ] is linked to ψ ̵ h by the two followingmarginal properties ∫ R d W ̵ h [ ψ ̵ h ]( x, ξ ) dξ = ∣ ψ ̵ h ( x )∣ (7) ∫ R d W ̵ h [ ψ ̵ h ]( x, ξ ) dx = ∣ ̂ ψ ̵ h ( p )∣ , ̂ ψ ̵ h ( p ) ∶ = ∫ R d e − ip.x /̵ h ψ ̵ h ( x ) dx ( π ̵ h ) d / (8)It has been proved, see e.g. [LP93], that, under the tightness conditionslim R → +∞ sup ̵ h ∈( , ) ∫ R d ∖ B ( d ) R r ̵ h ( x, x ) dx = , (9) lim R → +∞ sup ̵ h ∈( , ) ( π ̵ h ) d ∫ R d ∖ B ( d ) R F r ̵ h ( p ̵ h , p ̵ h ) dp = , (10) F. GOLSE AND T. PAUL where B ( d ) R is the ball of radius R in R d and F is the Fourier transform on R d ,the family of Wigner functions W ̵ h [ R ̵ h ] converges weakly, after extraction of asubsequence of values of ̵ h to W ∈ P ( R d ) , the space of probability measures on R d . W is called the Wigner measure of the family R ̵ h .As it is quite standard, we will omit to mention the extraction of subsequences,together with the explicit dependence of states in the Planck constant, and we willjust write, when it doesn’t create any confusion,lim ̵ h → W ̵ h [ R ] = W . Note that when R = ∣ ψ ⟩⟨ ψ ∣ is a pure state, (9)-(10) readslim R → +∞ sup ̵ h ∈( , ) ∫ R d ∖ B ( d ) R ∣ ψ ( x )∣ dx = , (11) lim R → +∞ sup ̵ h ∈( , ) ∫ R d ∖ B ( d ) R ∣ ̂ ψ ( p )∣ dp = H ǫ , the expected underlying classical dy-namics is the one driven by the Liouville equation(13) ∂ t ρ = { ( p + λq ) + µV ( q ) + ǫU, ρ } , ρ t ∣ t = = ρ in where { ., . } is the Poisson bracket on the symplectic manifold T ∗ R d ∼ R d .When ǫ =
0, the Hamiltonian vector field of Hamiltonian p + λq + µV ( q ) is Lip-schitz continuous C ([ , T ] ; P ( R d )) . Moreover, it was proven, [LP93], that R (̵ h t ) is tight for any t ∈ R and W t ∶ = lim ̵ h → W ̵ h [ R ̵ h ( t )] = ρ t solving (13) with ρ in = W .When ǫ >
0, the Liouville equation (13) exits the Cauchy-Lipschitz category: theassociated Hamiltonian vector field might not have a unique characteristic out ofevery point of the phase-space. Nevertheless, as shown in [AFFGP10, Theorem6.1], (13) is still well posed in L ∞+ ([ , T ] ; L ( R d ) ∩ L ∞ ( R d )) , and it was provenin [FLP13] (after [AFFGP10], that the Wigner function W ̵ h [ R ̵ h ( t )] of the solutionof the von Neumann equation i ̵ h∂ t R ǫ ( t ) = [ H ǫ , R ǫ ( t )] , R ǫ ( ) = R inǫ tends weakly to the solution of (13), under certain conditions on R inǫ .Unfortunately, these conditions exclude definitively pure states, as, for example,one of them impose that ∥ R inǫ ∥ = O (̵ h d ) and, to our knowledge, nothing is knownconcerning the dynamics of the (possible) limit of W ̵ h [ ψ ǫ ( t )] as ̵ h → ψ ǫ ( t ) solves the Schr¨odinger equation (3).Our next result will show that such a limit remains √ ǫ -close to the pushfrowardof the Wigner measure of the initial condition by the flow of the unperturbedclassical Hamiltonian. Theorem 3.1.
Let R ǫ ( t ) be the solution of the Schr¨odinger equation (3) with Rǫ in satisfying ∆ ( R inǫ ) = O (√̵ h ) . Let R inǫ be tight, in the sense that it satisfies (9) - (10) , so that W ̵ h [ R inǫ ] → W in as ̵ h → , its Wigner measure.Then, for any t ∈ R , UANTUM PERTURBATIONS 7 (1) the family R ǫ ( t ) is tight, so that W ̵ h [ R ǫ ( t )] → W ( t ) ∈ P ( R d ) as ̵ h → (2) W ( t ) is √ ǫ -close to Φ t W in , where Φ t is the flow of Hamiltonian p + ( − λ ) q + µV ( q ) , in the sense that sup ∫ R d sup q ∈ R d ∣F p f ( q,z )∣ dz ≤ ∣ α ∣+∣ β ∣≤ [ d / ]+ ∥ ∂ αq ∂ βp f ∥ L ∞( R d ) ≤ ∣ ∫ R d f ( q, p ) ( W ( t ) − Φ t W in ) ( q, p ) dqdp ∣ ≤ − d √ C ( t )√ ǫ, where F p f ( q, z ) ∶ = ∫ R d e − ip.z f ( q, p ) dp and C ( t ) is as in Theorem 2.1 afterreplacing ( ψ inǫ , H ψ inǫ ) by ∫ R d ( p + q ) W ( dpdq ) .(3) in particular, W ( t ) is weakly √ ǫ -close to Φ t W in in the sense of distri-bution as, for all test functions ϕ ∈ S ( R d ) , ∣ ∫ R d ϕ ( q, p ) ( W ( t ) − Φ t W in ) ( dp, dq )∣ ≤ Cϕ ( t )√ ǫ with C ϕ ( t ) = max ( ∫ R d sup q ∈ R d ∣ F p f ( q, z )∣ dz, max ∣ α ∣ + ∣ β ∣ ≤ [ d / ] + ∥ ∂ αq ∂ βp f ∥ L ∞ ( R d ) ) − d √ C ( t ) . .Proof. (1) the propagation of tightness is proved as follows.Let χ ∈ C ∞ ( R d ) , 0 ≤ χ ≤ χ ( x ) = ∣ x ∣ < / χ ( x ) = ∣ x ∣ >
1, and define χ R ( x ) ∶ = χ ( x / R ) . Obviously ∫ R d ∖ B ( d ) R ∣ ψ ǫ ( t )( x )∣ dx ≤ ∫ R d χ R ( x )∣ ψ ǫ ( t )( x )∣ dx Moreover, for some C > ∥ ∇ χ R ∥ ∞ , ∥ ∆ χ R ∥ ∞ ≤ C / R , and − i ̵ h [ χ R , H ǫ ] = − i ̵ h [ χ R , − ̵ h ∆ ] = − i ̵ h ( ∆ χ R − i ∇ χ R . ∇ ) . Therefore ∂ t ∫ R d χ R ( x )∣ ψ ǫ ( t )( x )∣ dx = − i ̵ h ∫ R d ¯ ψ ǫ ( t )( x )(( ∆ χ R − i ∇ χ R . ∇ ) ψ ǫ ( t ))( x ) dx = ∫ R d ( − i ̵ h ∆ χ R ( x )∣ ψ ǫ ( t )∣ + ¯ ψ ǫ ( t )( x ) ∇ χ R ( x ) . ( − i ̵ h ∇ ψ ǫ ( t )( x )) dx, F. GOLSE AND T. PAUL so that ∂ t ∫ R d χ R ( x )∣ ψ ǫ ( t )( x )∣ dx ≤ ̵ h C R + CR ∥ − i ̵ h ∇ ψ ǫ ( t )∥ L ( R d ) = ̵ h C R + CR ( ψ ǫ ( t ) , H , ψ ǫ ( t )) L ( R d ) ≤ ̵ h C R + CR (( ψ ǫ ( t ) , H λ,µǫ ψ ǫ ( t )) L ( R d ) + µ ∥ V ∥ ∞ + ǫ ∥ vd ∥ ∞ ) = ̵ h C R + CR (( ψ inǫ , H λ,µǫ ψ inǫ ) L ( R d ) + µ ∥ V ∥ ∞ + ǫ ∥ vd ∥ ∞ ) and finally, for t ∈ [ , T ] ∫ R d χ R ( x )∣ ψ ǫ ( t )( x )∣ dx ≤ ∫ R d χ R ( x )∣ ψ inǫ ( x )∣ dx + (̵ h C R + CR (( ψ inǫ , H λ,µǫ ψ inǫ ) L ( R d ) + µ ∥ V ∥ ∞ + ǫ ∥ vd ∥ ∞ )) T. Therefore ψ ǫ ( t ) satisfies (12) as soon as ψ inǫ does.Finally, let us remark that ∫ R d ∖ B ( d ) R ∣ ̂ ψ ( p )∣ dp ≤ R ∫ R d p ∣ ̂ ψ ( p )∣ dp = R ( ψ ǫ ( t ) , H , ψ ǫ ( t )) L ( R d ) (14) and one concludes the same way.(2) one knows from [LP93] that the convergence of Wigner functions to Wignermeasure as ̵ h → f on R d satisfying ∫ R d sup q ∈ R d ∣ F p f ( q, z )∣ dz < ∞ . Since V ∈ C , one knows that, for such a test function,lim ̵ h → ∫ R d f ( x, ξ ) W ̵ h [ ψ ( t )]( x, ξ ) dxdξ = ∫ R d f ( x, ξ ) Φ t W ( x, ξ ) dxdξ. On the other site, we have the slight variant of Theorem 2.1, proven alsoin Section 5.
Proposition 3.2.
Let δ be defined by (17) below. Then δ ( W ̵ h [ ψ ( t )] , W ̵ h [ ψ ǫ ( t )]) ≤ − d √ C ( t ) ǫ + D ( t )̵ h, where C ( t ) , D ( t ) are the constants defined in Theorem 2.1. Proposition 3.2 tells us that, for any f satisfyingmax ∣ α ∣ + ∣ β ∣ ≤ [ d / ] + ∥ ∂ αq ∂ βp f ∥ L ∞ ( R d ) ≤ , ∣ ∫ R d f ( q, p )( W ̵ h [ ψ ǫ ( t )] − W ̵ h [ ψ ( t )])( dpdq )∣ ≤ − d √ C ( t ) ǫ + D ( t )̵ h Hence for any f satisfying ∫ R d sup q ∈ R d ∣ F p f ( q, z )∣ dz ≤ , max ∣ α ∣ + ∣ β ∣ ≤ [ d / ] + ∥ ∂ αq ∂ βp f ∥ L ∞ ( R d ) ≤ , UANTUM PERTURBATIONS 9 we have ∣ ∫ R d f ( q, p )( W ̵ h [ ψ ǫ ( t )] − Φ t W ( x, ξ )])( dpdq )∣ ≤ − d √ C ( t ) ǫ + D ( t )̵ h + ∣ ∫ R d f ( q, p )( W ̵ h [ ψ ( t )] − Φ t W ( x, ξ ))( dpdq )∣ and we conclude by taking the supremum on the functions f and the limit ̵ h → (cid:3) The case of mixed states
Consider the family of von Neumann equations, for ǫ ∈ [ , ] ,(15) i ̵ h∂ t R ǫ ( t ) = [ H ǫ , R ǫ ( t )] , R ǫ ( ) = R inǫ ∈ D ( H ) . , Let R inǫ = R in = R ∈ D ( H ) satisfy one of the five following hypothesis:(i) ∆ ( R in ) = O (√̵ h ) where the standard deviation ∆ ( R ) is defined in Lemma5.4 below. .(ii) √ R in satisfies, for some C > ∣ β ∣ ,..., ∣ β d ∣ ≤ ∣ d ∏ m = D β m ( x,ξ ) W ̵ h [√ R in ]( x, ξ )∣ ≤ C ( π ̵ h ) − d (( ξ + x ) + d ) + ǫ ∀ ( x, ξ ) ∈ R d . where W ̵ h [√ R in ] is the Wigner transform of R in (iii) √ R in satisfies, for C > j ∈ N d ,(a) ∣( H i , √ R in H j )∣ ≤ C ( π ̵ h ) d ∏ ≤ l ≤ d ∣̵ hj l + ∣ − − ǫ (∣ i l − j l ∣ + ) − − ǫ ,(b) sup O ∈ Ω ∣( H i , i ̵ h [ O, √ R in ] H j )∣ ≤ C ( π ̵ h ) d ∏ ≤ l ≤ d ∣̵ hj l + ∣ − − ǫ (∣ i l − j l ∣ + ) − − ǫ ,where Ω = { y j , ± ̵ h∂ y j on L ( R d , dy ) , j = , . . . , d } and the H j s arethe semiclassical Hermite functions..(iv) R in is a T¨oplitz operator.(v) there exist a T¨oplitz operator T F such that T − F √ R in , √ R in T − F and T − F √ R in i ̵ h [ O, √ R in T − F ] , O ∈ Ω are bounded on L ( R d ) . Theorem 4.1.
For every t ≥ , d ( R ( t ) , R ǫ ( t )) ≤ C ( t ) ǫ + D ( t )̵ h, where C ( t ) , D ( t ) are the same given by (19) , (20) , satisfy C ( t ) = e ∣ t ∣( − λ + µ Lip ( ∇ V )) − − λ + µ Lip ( ∇ V ) C ∥H R in ∥ , ∥ V ∥ ∞ , ∥ U ∥ ∞ , ∥ ∇ U ∥ ∞ < ∞ D ( t ) = e ∣ t ∣( − λ + µ Lip ( ∇ V )) D d < ∞ Corollary 4.2. d ( R ( t ) , R ǫ ( t )) ≤ E ( t ) ǫ , In Theorem 4.1 and Corollary 4.2, the constants C ( t ) , D ( t ) , E ( t ) are the same asin Theorem2.1 and Corollary 2.2 after replacing ( σ in , H ψ in ) by ∥ H R in ∥ in C ( t ) . Proof of Theorems 2.1 and 4.1, and Proposition 3.2
For all
R, S ∈ D ( H ) , we denote by C ( R, S ) the set of couplings of R and S , i.e. C ( R, S ) ∶ = { Q ∈ D ( H ⊗ H ) s.t. trace H ⊗ H (( A ⊗ I + I ⊗ B ) Q ) = trace H ( AR + BS )} . We recall the definition of the pseudo-distance
M K ̵ h (see Definition 2.2 in [GMP16]). Definition 5.1.
For each
R, S ∈ D ( H ) , M K ̵ h ( R, S ) ∶ = inf Q ∈ C( R,S ) √ trace H ⊗ H ( Q / CQ / ) , where C ∶ = d ∑ j = (( q j ⊗ I − I ⊗ q j ) − ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j ) ) . Theorem 2.1 and Proposition 3.2 are a consequence of the following inequality,which controls the continuous dependence of the solution to the von Neumannequation in terms of the initial data and on the potential.
Theorem 5.2.
Let R inǫ ∈ D ( H ) and R ǫ ( t ) be the solution of (15) , ǫ ∈ [ , ] . Then,for each t ∈ R , ǫ ∈ [ , ] , one has M K ̵ h ( R ( t ) , R ǫ ( t )) ≤ e ∣ t ∣ Λ ( ∇ V ) M K ̵ h ( R in , R inǫ ) + ǫ e ∣ t ∣ Λ ( ∇ V ) − ( ∇ V ) ∥ ∇ U ∥ L ∞ ( R d ) √ trace (( R in ) / H ( R in ) / ) + µ ∥ V ∥ L ∞ ( R d ) + ǫ e ∣ t ∣ Λ ( ∇ V ) − ( ∇ V ) ∥ ∇ U ∥ L ∞ ( R d ) √ trace (( R inǫ ) / H ( R inǫ ) / ) + ( µ ∥ V ∥ L ∞ ( R d ) + ǫ ∥ U ∥ L ∞ ( R d ) ) , where Λ ( ∇ V ) = − λ + µ Lip ( ∇ V ) . Note that when λ =
1, Λ ( ∇ V ) = µ Lip ( ∇ V ) and in the inequality above, thefunction z ↦ e z − z is extended by continuity at z = Proof.
In order to lighten the formulas we will use the following notations(16) V = µV, V = µV + ǫU, so that H = H λ, + V and H ǫ = H λ, + V . Let Q in ∈ C ( R in , R inǫ ) , and let Q be the solution of the von Neumann equation i ̵ h∂ t Q = [( H λ, ǫ + V ) ⊗ I + I ⊗ ( H λ, ǫ + V ) , Q ] , Q ∣ t = = Q in . Then Q ( t ) ∈ C ( R ( t ) , R ǫ ( t )) , for each t ≥ ddt trace H ⊗ H ( Q ( t ) / CQ ( t ) / ) = i ̵ h trace H ⊗ H ( Q ( t ) / [( H λ + V ) ⊗ I + I ⊗ ( H λ + V ) , C ] Q ( t ) / ) . UANTUM PERTURBATIONS 11
One finds that i ̵ h [ − ̵ h ( ∆ ⊗ I + I ⊗ ∆ ) , C ] = d ∑ j = ( q j ⊗ I − I ⊗ q j ) ∨ ( − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j )) , while i ̵ h [( V ⊗ I + I ⊗ V ) , C ] = − d ∑ j = ( ∂ q j V ⊗ I − I ⊗ ∂ q j V ) ∨ ( − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j )) , with the notation A ∨ B ∶ = AB + BA .
In particular i ̵ h [ (∣ q ∣ ⊗ I + I ⊗ ∣ q ∣ ) , C ] = − d ∑ j = ( q j ⊗ I − I ⊗ q j ) ∨ ( − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j )) . See [GMP16] on p. 190. Hence ddt trace H ⊗ H ( Q ( t ) / CQ ( t ) / ) − ( − λ ) trace H ⊗ H ⎛⎝ Q ( t ) / d ∑ j = ( q j ⊗ I − I ⊗ q j ) ∨ ( − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j )) Q ( t ) / ⎞⎠ = trace H ⊗ H ⎛⎝ Q ( t ) / d ∑ j = ( ∂ q j V ⊗ I − I ⊗ ∂ q j V ) ∨ ( − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j )) Q ( t ) / ⎞⎠ = trace H ⊗ H ⎛⎝ Q ( t ) / d ∑ j = ( ∂ q j V ⊗ I − I ⊗ ∂ q j V ) ∨ ( − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j )) Q ( t ) / ⎞⎠ + trace H ⊗ H ⎛⎝ Q ( t ) / d ∑ j = ( I ⊗ ∂ q j ( V − V ) ∨ ( − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j )) Q ( t ) / ⎞⎠ = ∶ τ + τ . At this point, we recall the elementary operator inequality A ∗ B + B ∗ A ≤ A ∗ A + B ∗ B .
Therefore, d ∑ j = ( q j ⊗ I − I ⊗ q j ) ∨ ( − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j )) ≤ C and, for each ℓ > Lip ( ∇ V ) / , one has d ∑ j = ( ∂ q j V ⊗ I − I ⊗ ∂ q j V ) ∨ ( − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j )) = d ∑ j = ℓ ( ∂ q j V ⊗ I − I ⊗ ∂ q j V ) ∨ ℓ ( − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j )) ≤ d ∑ j = ( Lip ( ∇ V ) ℓ ( q j ⊗ I − I ⊗ q j ) + ℓ ( − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j )) ) . Letting ℓ → Lip ( ∇ V ) / shows that d ∑ j = ( ∂ q j V ⊗ I − I ⊗ ∂ q j V ) ∨ ( − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j )) ≤ Lip ( ∇ V ) C , so that τ ≤ Lip ( ∇ V ) trace H ⊗ H ( Q ( t ) / CQ ( t ) / ) . For the term τ , we simply use the Cauchy-Schwarz inequality: τ = trace H ⊗ H ⎛⎝ Q ( t ) / d ∑ j = ( I ⊗ ∂ q j ( V − V ) ∨ ( − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j )) Q ( t ) / ⎞⎠ ≤ d ∑ j = ∥ Q ( t ) / ( I ⊗ ∂ q j ( V − V ))∥ ∣∣ − i ̵ h ( ∂ q j ⊗ I − I ⊗ ∂ q j ) Q ( t ) / ∣∣ ≤ d ∑ j = ∥ Q ( t ) / ( I ⊗ ∂ q j ( V − V ))∥ ∣∣( − i ̵ h∂ q j ⊗ I ) Q ( t ) / ∣∣ + d ∑ j = ∥ Q ( t ) / ( I ⊗ ∂ q j ( V − V ))∥ + ∣∣( I ⊗ ( − i ̵ h∂ q j )) Q ( t ) / ∣∣ = τ + τ . Now τ ≤ ⎛⎝ d ∑ j = ∥ Q ( t ) / ( I ⊗ ∂ q j ( V − V ))∥ ⎞⎠ / ⎛⎝ d ∑ j = ∥( − i ̵ h∂ q j ⊗ I ) Q ( t ) / ∥ ⎞⎠ / = ⎛⎝ d ∑ j = trace ( Q ( t ) / ( I ⊗ ∂ q j ( V − V )) Q ( t ) / )⎞⎠ / × ( trace ( Q ( t ) / ( − ̵ h ∆ ⊗ I ) Q ( t ) / )) / so that τ ≤ ∥ ∇ ( V − V )∥ L ∞ ( R d ) trace ( R ( t ) / H λ R ( t ) / ) / , and likewise τ ≤ ∥ ∇ ( V − V )∥ L ∞ ( R d ) trace ( R ǫ ( t ) / H λ R ǫ ( t ) / ) / . Summarizing, we have proved that ddt trace H ⊗ H ( Q ( t ) / CQ ( t ) / ) ≤ trace H ⊗ H ( Q ( t ) / CQ ( t ) / ) + Lip ( ∇ V ) trace H ⊗ H ( Q ( t ) / CQ ( t ) / ) + ∥ ∇ ( V − V )∥ L ∞ ( R d ) ( trace ( R ( t ) / H λ R ( t ) / ) / + trace ( R ǫ ( t ) / H λ R ǫ ( t ) / ) / ) . On the other hand, since i ̵ h∂ t R j ( t ) = [ H λ + V j , R j ( t )] , one hastrace ( R j ( t ) / ( H λ + V j ) R j ( t ) / ) = trace (( R inj ) / ( H λ + V j )( R inj ) / ) so thattrace ( R j ( t ) / H λ R j ( t ) / ) ≤ trace (( R inj ) / H λ ( R inj ) / ) + ∥ V j ∥ L ∞ ( R d ) . UANTUM PERTURBATIONS 13
Hence ddt trace H ⊗ H ( Q ( t ) / CQ ( t ) / ) ≤ ( − λ + Lip ( ∇ V )) trace H ⊗ H ( Q ( t ) / CQ ( t ) / ) + ∥ ∇ ( V − V )∥ L ∞ ( R d ) √ trace (( R in ) / H λ ( R in ) / ) + ∥ V ∥ L ∞ ( R d ) + ∥ ∇ ( V − V )∥ L ∞ ( R d ) √ trace (( R inǫ ) / H λ ( R inǫ ) / ) + ∥ V ∥ L ∞ ( R d ) . By Gronwall’s inequality, choosing Q in to be an optimal coupling of R in and R inǫ ,one finds that, denoting Λ ( ∇ V ) = − λ + Lip ( ∇ V ) , M K ̵ h ( R ( t ) , R ǫ ( t )) ≤ trace H ⊗ H ( Q ( t ) / CQ ( t ) / ) ≤ e t Λ ( ∇ V ) M K ̵ h ( R in , R inǫ ) + e t Λ ( ∇ V ) − ( ∇ V ) ∥ ∇ ( V − V )∥ L ∞ ( R d ) √ trace (( R in ) / H λ ( R in ) / ) + ∥ V ∥ L ∞ ( R d ) + e t Λ ( ∇ V ) − ( ∇ V ) ∥ ∇ ( V − V )∥ L ∞ ( R d ) √ trace (( R inǫ ) / H λ H λ ( R inǫ ) / ) + ∥ V ∥ L ∞ ( R d ) , which is the desired inequality by coming back to V, U through (16) and using ∥ µV + ǫU ∥ ≤ µ ∥ V ∥ + ǫ ∥ U ∥ . (cid:3) Proof of Theorem 2.1 and Proposition 3.2.
Observe that the estimate above is uni-form in ̵ h — more precisely, the moduli of continuity in the initial data and in thepotential are independent of ̵ h . Of course, the pseudo-distance M K ̵ h itself is notindependent of ̵ h . For R, S ∈ D ( H ) let us define(17) δ ( W ̵ h [ R ] , W ̵ h [ S ]) ∶ = sup max ∣ α ∣+∣ β ∣≤ [ d / ]+ ∥ ∂ αq ∂ βp f ∥ L ∞( R d ) ≤ ∣ ∫ R d f ( q, p )( W ̵ h [ R ] , W ̵ h [ S ]) ∗ q, p ) dqdp ∣ Lemma 5.3.
For any
R, S ∈ D ( H ) , d ( R, S ) ≤ d δ ( W ̵ h [ R ] , W ̵ h [ S ]) ≤ d ( M K ̵ h ( R, S ) + C d ̵ h ) with C d = ( + γ d √ π ) d, where γ d ≤ d / ( e − π − ) d e ( d d ) / is the constant appearing in the Calderon-Vaillancourt theorem (see Appendix C in [GP20a] ).Proof. The proof consists in applying Theorem A.7 in [GJP20] and Theorem 2.3(2) in [GMP16]. (cid:3)
Lemma 5.3 shows that the inequality above implies that ( − d d ( R ( t ) , R ǫ ( t )) − C d ̵ h ) ( δ ( W ̵ h [ R ( t )] , W ̵ h [ R ǫ ( t )]) − C d ̵ h ) ≤ e t Λ ( ǫ ∇ V ) M K ̵ h ( R in , R inǫ ) + d ̵ h + ǫ e t Λ ( ǫ ∇ V ) − ( ∇ V ) ∥ ∇ ( V − V )∥ L ∞ ( R d ) √ trace (( R in ) / H λ ( R in ) / ) + ǫ ∥ V ∥ L ∞ ( R d ) + ǫ e t Λ ( ǫ ∇ V ) − ( ∇ V ) ∥ ∇ ( V − V )∥ L ∞ ( R d ) √ trace (( R inǫ ) / H λ ( R inǫ ) / ) + ǫ ∥ V ∥ L ∞ ( R d ) ∶ = e t Λ ( ∇ V ) M K ̵ h ( R in , R inǫ ) + d ̵ h + γ ( t ) ǫ. Therefore Theorem 2.1 and Proposition 3.2 are proven as soon as(18)
M K ̵ h ( R in , R in ) ≤ D ′ ̵ h since then d ( R ( t ) , R ǫ ( t )) ≤ d δ ( W ̵ h [ R ( t )] , W ̵ h [ R ǫ ( t )]) ≤ √ d ( e ∣ t ∣ µ Lip V D ′ + d )̵ h + d γ ( t ) ǫ + d C d ̵ h ≤ √ C ( t ) ǫ + D ( t )̵ h with C ( t ) = d γ ( t ) (19) D ( t ) = e ∣ t ∣ µ Lip V d ( D ′ + d, C d ) (20) Lemma 5.4.
For R ∈ D ( H ) let ∆ ( R ) ∶ = √ trace ( R (( x − trace ( R xR )) + ( − i ∇ x − trace ( R ( − i ∇ x ) R )) ) R ) . Note that when R is a pure state R ∣ ψ ⟩⟨ ψ ∣ , ∆ ( R ) is, modulo a slight abuse of nota-tion, the same as in the definition (1) .Then M K ̵ h ( R, R ) ≤ √ ( R ) . Moreover, for any ψ ∈ H , M K ̵ h (∣ ψ ⟩⟨ ψ ∣ , ∣ ψ ⟩⟨ ψ ∣) = √ (∣ ψ ) . Proof.
The proof consists in remarking that R ⊗ R is indeed a coupling between R and itself. Therefore M K ̵ h ( R ) ≤ trace (( R ⊗ R ) C ( R ⊗ R )) = trace (( R ⊗ R )( x ⊗ I + I ⊗ x − x ⊗ x )( R ⊗ R )) + same with x ↔ − i ̵ h ∇ x = trace ( R x R + R x R − R xR .R xR ) + same with x ↔ − i ̵ h ∇ x = ( R x R − ( R xR ) ) + same with x ↔ − i ̵ h ∇ x = ( R ) . The equality is proven the same way, after Lemma 2.1 (ii) in [GP20b] which stipu-lates that the only coupling between ∣ ψ ⟩⟨ ψ ∣ and itself is ∣ ψ ⟩⟨ ψ ∣ ⊗ ∣ ψ ⟩⟨ ψ ∣ . (cid:3) This proves Theorem 2.1, and Theorem 4.1 when R in satisfies hypothesis (i).If the initial data R in is e T¨oplitz operator, specifically if R in = OP T ̵ h [( π ̵ h ) d µ in ] , with the notation of [GMP16] one can go further and apply Theorem 2.3 (1) in[GMP16]: M K ̵ h ( R in , R in ) ≤ d ̵ h, UANTUM PERTURBATIONS 15 so that (18) is again satisfied and Theorem 4.1 is proven when R in satisfies thehypothesis (iv).The proof in the case of the hypothesis (ii), (iii) and (v) follows directly the firstinequality of Theorem 8.1 in [GP20b] with R = S = R in , together with item ( I ) (through the Corollary of Theorem 3.1 in [GP20a]) and item ( II ) of Theorem 4.1in [GP20a], with µ (̵ h ) = µ ′ (̵ h ) = C, ν (̵ h ) = ν ′ (̵ h ) = C √̵ h and τ (̵ h ) = ̵ h , respectively.Indeed, [GP20b, Theorem 8.1, (iii) first inequality] stipulates that, for all densitymatrix R , M K ̵ h ( R in , R in ) ≤ E ̵ h (̃ W ̵ h [ R in ] , R in ) where E ̵ h is a semiquantum pseudometric whose knowledge of the definition [GP17,Definition 2.2] is not strictly necessary for our purpose here since Theorem 4.1 in[GP20a]) shows that, when µ (̵ h ) = µ ′ (̵ h ) = C, ν (̵ h ) = ν ′ (̵ h ) = C √̵ h and τ (̵ h ) = ̵ h , E ̵ h (̃ W ̵ h [ R in ] , R in ) = O (√̵ h ) . Hence
M K ̵ h ( R in , R in ) ≤ D ′ ̵ h for some constant D ′ explicitely recoverable from [GP20a, Theorem 4.1].This completes the proof of Theorems 2.1 and 4.1. (cid:3) Proof of Corollaries 2.2 and 4.2
Let us first derive the easy standard following estimate.
Proposition 6.1.
For every t ∈ R , ∥ R ( t ) − R ǫ ( t )∥ ≤ t ǫ ̵ h ∥ U ∥ ∞ Proof.
The solution of (15) is explicitly given by R ǫ ( t ) = e − i t H ǫ ̵ h R in e i t H ǫ ̵ h Therefore, one easily shows that R ( t ) − R ǫ ( t ) = i ̵ h ∫ t e − i ( t − s )H ǫ ̵ h [ ǫU, R ( t )] e i ( t − s )H ǫ ̵ h . and the result follows by ∥[ ǫU, R ( t )]∥ ≤ ǫ ∥ U ∥∥ R ( t )∥ . (cid:3) Corollaries 2.2, 4.2 will follow by interpolation between Theorem 5.2 and Propo-sition 6.1, through the following inequality.
Lemma 6.2.
For any
R, S ∈ D ( H ) , d ( R, S ) ≤ d ∥ R − S ∥ Proof.
This is Theorem A7 in [GJP20], item ( ) (cid:3) Therefore, by Theorem 2.1 and Lemma 6.2 we get, for each ̵ h, ǫ ∈ [ , ] , d ( R ( t ) , R ( t )) ≤ min (√ C ( t ) ǫ + D ( t )̵ h, ∣ t ∣ ǫ ̵ h ∥ U ∥ ∞ ) . Obviously, min (√ C ( t ) ǫ + D ( t )̵ h, ∣ t ∣ ǫ ̵ h ∥ U ∥ ∞ ) ≤ min (√ C ( t ) + D ( t ) , ∣ t ∣∥ U ∥ ∞ ) ǫ for ǫ, ̵ h ≤
1, since, when ̵ h ≤ ǫ , ̵ h, ǫ ≤ ǫ , and, when ̵ h ≥ ǫ , ǫ ̵ h ≤ ǫ . The Corollary isproved. Proof of Proposition 2.4
The upper bound is given simply by Lemma 5.3 and the following inequality,proved in [GP18, Section 4]
M K ̵ h (∣ ψ z ⟩⟨ ψ z ∣ , ∣ ψ z ⟩⟨ ψ z ∣) ≤ ∣ z − z ∣ + d ̵ h. For the lower bound, we will pick up a test operator in the form of a T¨oplitzoperator with symbol f ≥ F ∶ = OP T ̵ h [( π ̵ h ) d f ] , One easily verifies that trace F = ∫ R d f ( q, p ) dqdp .Moreover, see [GJP20, Appendix B], i ̵ h [ F, − i ̵ h∂ x j ] = OP T ̵ h [( π ̵ h ) d ∂ p j f ] , j = , . . . , d i ̵ h [ F, x j ] = OP T ̵ h [( π ̵ h ) d ( − ∂ q j f )] , j = , . . . , d ∥ F ∥ ≤ ∥ f ∥ L ( R d ) Therefore, it is easy to construct functions f such that F ∶ = OP T ̵ h [( π ̵ h ) d f ] satisfiesthe constraints of the maximization problem in the definition of d .Moreover, denoting z = ( q, p ) ∈ R d , d (∣ ψ z ⟩⟨ ψ z ∣ , ∣ ψ z ⟩⟨ ψ z ∣) ≥ ∣ trace ( F (∣ ψ z ⟩⟨ ψ z ∣ − ∣ ψ z ⟩⟨ ψ z ∣))∣ = ∣⟨ ψ z ∣ F ∣ ψ z ⟩ − ⟨ ψ z ∣ F ∣ ψ z ⟩∣ = ∣ ∫ R d f ( z )(∣⟨ ψ z ∣ ψ z ⟩∣ − ∣⟨ ψ z ∣ ψ z ⟩∣ ) dqdp ( π ̵ h ) d ∣ = ∣ ∫ R d f ( z )( e − ∣ z − z ∣ ̵ h − e − ∣ z − z ∣ ̵ h ) dqdp ( π ̵ h ) d ∣ ≥ ∣ f ( z ) − f ( z )∣ − e ̵ h max ∥ α ∣ , ∣ β ∣ ≤ ∫ R d ∣ ∂ αq ∂ βp f ( q, p )∣ dqdp. Let us suppose now that f ∈ S ( R d ) and f is convex in a convex domain con-taining z , z . Then, one can certainly rescale, translate and rotate f such that ● f ( z ) − f ( z ) ≥ ∇ f ( z ) . ( z − z ) convexity ● ∣ f ( z ) − f ( z )∣ ≥ C ∣ z − z ∣ , C > ● max ∣ α ∣ , ∣ β ∣ ≤ [ d ] + ∥ D α − i ̵ h ∇ D βx F ∥ ≤ max ∥ α ∣ , ∣ β ∣ ≤ ∥ ∂ αq ∂ βp f ∥ L ( R d ) ≤ C ∣ z − z ∣ − ̵ h ≤ d (∣ ψ z ⟩⟨ ψ z ∣ , ∣ ψ z ⟩⟨ ψ z ∣) . References [AFFGP10] L.Ambrosio, A. Figalli, G. Friesecke, J. Giannoulis, T. Paul :
Semiclassical limitof quantum dynamics with rough potentials and well posedness of transport equations withmeasure initial data.
Comm. Pure Appl. Math. (2011), 1199–1242.[B28] G. D. Birkhoff, “Dynamical systems”, American Mathematical Society Colloquium Publi-cations, Vol. IX American Mathematical Society, Providence, R.I. (1966).[B25] M. Born, “Vorlesungen ¨uber Atommechanik”, Springer, Berlin, (1925). English translation:“The mechanics of the atom”, Ungar, New-York, (1927).[D91] M. Degli Esposti, S. Graffi, J. Herczynski, Quantization of the classical Lie algorithm inthe Bargmann representation , Annals of Physics, (1991), 364–392.
UANTUM PERTURBATIONS 17 [GJP20] F. Golse, S. Jin, T. Paul:
On the convergence of the time splitting methods for quan-tum dynamics , Foundations of Computational Mathematics (2020), doi:10.1007/s10208-020-09470-z.[GMP16] F. Golse, C. Mouhot, T. Paul:
On the Mean Field and Classical Limits of QuantumMechanics , Commun. Math. Phys. (2016), 165–205.[GP17] F. Golse, T. Paul:
The Schr¨odinger Equation in the Mean-Field and SemiclassicalRegime , Arch. Rational Mech. Anal. (2017), 57–94.[GP18] F. Golse, T. Paul:
Wave Packets and the Quadratic Monge-Kantorovich Distance inQuantum Mechanics , Comptes Rendus Mathematique (2018), 177–197.[GP20a] F. Golse, T. Paul:
Semiclassical evolution with low regularity , preprint hal-02619489 and arXiv:2011.14884 , to appear in J. Math. Pures et Appl..[GP20b] F. Golse, T. Paul:
Quantum and Semiquantum Pseudometrics and applications ,preprint.[G87] S. Graffi, T. Paul,
Schr¨odinger equation and canonical perturbation theory , Comm. Math.Phys., (1987), 25–40.[H25] W. Heisenberg,
Matrix mechanik , Zeitscrift f¨ur Physik, (1925), 879–893.[LP93] P.L.Lions, T.Paul: Sur les mesures de Wigner.
Rev. Mat. Iberoamericana, (1993), 553–618.[FLP13] A. Figalli, M. Ligab`o, T. Paul, Semiclassical limit for mixed states with singular andrough potentials , Indiana University Mathematics Journal, (2013), 193–222.[NPST18] J-C. Novelli, T. Paul, D. Sauzin, J-Y. Thibon: Rayleigh-Schr¨odinger series andBirkhoff decomposition , Letters in Mathematical Physics (2018),1583–1600.[P16] T. Paul, D. Sauzin,
Normalization in Lie algebras via mould calculus and applications ,Regular and Chaotic Dynamics (6) (2017), 616–649, special issue in memory of VladimirArnold.[P162] T. Paul, D. Sauzin, Normalization in Banach scale Lie algebras via mould calculus andapplications , Discrete and Continuous Dynamical Systems A (2017), 4461–4487.[P1892] H. Poincar´e, “Les m´ethodes nouvelles de la m´ecanique c´eleste”, Volume 2, Gauthier-Villars, Paris, (1892), Blanchard, Paris, (1987).(F.G.) CMLS, ´Ecole polytechnique, CNRS, Universit´e Paris-Saclay , 91128 PalaiseauCedex, France
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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