Long time semiclassical Egorov theorem for ℏ -pseudodifferential systems

1. Introduction

Known as Bohr’s correspondence principle in physics and Egorov theorem in mathematical literature, the semiclassical approximation ensures the transition between quantum and classical evolutions of observables. The relation between the classical and quantum evolutions may be considered as one of the oldest problems of semiclassical analysis. The starting point was the famous Bohr’s correspondence principle proposed by Niels Bohr on 1923. According to this principle, the quantum evolution of an observable is closer and closer to its classical evolution as the Planck constant ℏ becomes negligible. For many years, the semiclassical approximation has been the object of several investigations following two approaches. The first one uses semiclassical wave packets (or coherent states) as initial data and aims to approximate the evolved wave packet by a linear combination of coherent states (see [7,13,14,24] and [8] for a complete description of coherent states). The second approach, that will be our principal subject in this paper, considers the Heisenberg evolution of suitable bounded observables and seeks to construct an approximation for the corresponding observables in terms of ℏ-pseudodifferential operators (see [4,16,25,28]). Let us start by recalling the general setting of this approach.

Let H ∈ C ∞ ( R 2 n ; R ) be a classical Hamiltonian and Q ∈ C ∞ ( R 2 n ; R ) be a real observable. We consider the system of Hamilton equations (1.1) d x d t = ∂ ξ H ( x , ξ ) , d ξ d t = − ∂ x H ( x , ξ ) . These equations generate a flow ϕ H t called the Hamiltonian flow associated to H defined by ϕ H t ( x = x ( 0 ) , ξ = ξ ( 0 ) ) = ( x ( t ) , ξ ( t ) ) and ϕ H t | t = 0 ( x , ξ ) = ( x , ξ ) . In the sequel, additional assumption will be required on H so that for all initial data ( x , ξ ) ∈ R 2 n , ϕ H t ( x , ξ ) is well defined for all t ∈ R .

The time evolution of Q under the flow ϕ H t given by q 0 ( t ) : = Q ∘ ϕ H t is described by the equation (1.2) d d t q 0 ( t ) = { H , Q } ∘ ϕ H t , q 0 ( t ) | t = 0 = Q , where { H , Q } denotes the Poisson bracket of H, Q defined by { H , Q } : = ∂ ξ H · ∂ x Q − ∂ x H · ∂ ξ Q . Let H w : = H w ( x , ℏ D x ) and Q w : = Q w ( x , ℏ D x ) be the self-adjoint operators in L 2 ( R n ) associated to H and Q, respectively. Here we use the ℏ-Weyl quantization (see formula (2.1)). Let e − i t ℏ H w be the unitary operator solution of the evolution equation i ℏ ∂ t u ( t ) = H w u ( t ) . The time evolution of the quantum observable Q w under e − i t ℏ H w given by Q ( t ) : = e i t ℏ H w Q w e − i t ℏ H w is described by the quantum analogous of (1.2) called the Heisenberg equation of motion (1.3) d d t Q ( t ) = i ℏ [ H w , Q ( t ) ] , Q ( t ) | t = 0 = Q w , where [ H w , Q ( t ) ] : = H w Q ( t ) − Q ( t ) H w denotes the commutator of H w , Q ( t ) .

Under suitable growth assumptions on H, the semiclassical Egorov theorem (see [4, Theorem 1.2] for the precise statement) states that for every fixed t, Q ( t ) is an ℏ-pseudodifferential operator with principal symbol q 0 ( t , x , ξ ) = Q ∘ ϕ H t ( x , ξ ) . More precisely, there exists a family of symbols ( q j ( t , · , · ) ) j ⩾ 1 with supp ( q j ( t , · , · ) ) ⊂ ϕ H − t ( K ) , where K is the union of the supports of the q j , such that for every finite time t ¯ > 0 and for all N ∈ N , the following estimate (1.4) ‖ Q ( t ) − ∑ j = 0 N ℏ j ( q j ( t ) ) w ‖ L ( L 2 ( R n ) ) ⩽ C N ℏ N + 1 , holds uniformly for | t | ⩽ t ¯ . The time-dependent family of operators ( ( q j ( t ) ) w ) j ⩾ 0 is called semiclassical approximation of Q ( t ) . The construction of such approximation is obtained in a formal way by solving the Heisenberg equation (1.3) in the space of formal power series of ℏ. If one assumes that Q ( t ) is an ℏ-pseudodifferential operator with Weyl symbol q ( t ) , where for now we look at q ( t ) as a formal power series of ℏ, q ( t ) ∼ ∑ j ⩾ 0 ℏ j q j ( t ) , by (1.3) it fulfils the equation (1.5) d d t q ( t ) = i ℏ [ H , q ( t ) ] # , q ( t ) | t = 0 = Q , where [ H , q ( t ) ] # : = H # q ( t ) − q ( t ) # H , with H # q ( t ) denotes the Moyal product of H, q ( t ) defined as the symbol of H w ∘ Q ( t ) (see Appendix A).

By expanding the symbol i ℏ [ H , q ( t ) ] # in powers of ℏ and then equating equal powers of ℏ in both sides, the Cauchy problem (1.5) implies similar Cauchy problems for the symbols q j ( t ) given by (1.6) ( C j ) d d t q j ( t ) = ( i ℏ [ H , q ( t ) ] # ) j , q 0 ( t ) | t = 0 = Q , q j ( t ) | t = 0 = 0 , ∀ j ⩾ 1 , where ( i ℏ [ H , q ( t ) ] # ) j is the jth term (i.e. the coefficient of ℏ j ) in the asymptotic expansion of i ℏ [ H , q ( t ) ] # .

For j = 0 , the principal symbol of [ H , q ( t ) ] # given by the commutator [ H , q 0 ( t ) ] vanishes. Therefore, the term of order ℏ 0 in the asymptotic expansion of i ℏ [ H , q ( t ) ] # coincides with the sub-principal symbol of i [ H , q ( t ) ] # which is equal to { H 0 , q 0 ( t ) } . Consequently, ( C 0 ) reads d d t q 0 ( t ) = { H , q 0 ( t ) } , q 0 ( t ) | t = 0 = Q , and we get q 0 ( t ) = Q ∘ ϕ H t . The symbols q j ( t ) for j ⩾ 1 are obtained in the same manner by following the above algorithm (see Remark 3.1 for the general formula of the symbols q j ( t ) ).

The semiclassical approximation given by the semiclassical Egorov theorem is limited to the evolutions in finite time intervals. Several works was devoted to the investigation of its validity for large times which may depend on ℏ (see [7] for the time evolution of coherent states, and [1,4] for the time evolution of observables). These investigations was based on a conjecture going back to the physicists Chirikov [6] and Zaslavsky [29] which claim that the semiclassical approximation remains valid in a large time interval of length T ( ℏ ) ≃ log ( ℏ − 1 ) known as the Ehrenfest time. The optimal time for which this claim was proved has been obtained by Bouzouina and Robert [4]. They showed that the L 2 -operator norm of the remainder term in the asymptotic expansion of Q ( t ) given by the left hand side of (1.4) is uniformly dominated, at any order, by an exponential term whose argument is linear in time. In particular, this allows them to recover the Ehrenfest time for the validity of the semiclassical approximation.

The purpose of this paper is to study the extension of the above results to the case of matrix-valued observables. Given a bounded quantum observable Q w ( x , ℏ D x ; ℏ ) associated to a ( m × m ) matrix-valued symbol Q ( x , ξ ; ℏ ) ∼ ∑ j ⩾ 0 ℏ j Q j ( x , ξ ) , we study the time evolution Q ( t ) : = e i t ℏ H w Q w e − i t ℏ H w generated by a semiclassical ( m × m ) hermitian-valued Hamiltonian H ( x , ξ ; ℏ ) . We establish a long time matrix-valued version for the semiclassical Egorov theorem by giving a semiclassical approximation for Q ( t ) valid in a large time interval of Ehrenfest type. To the best of our knowledge, there is no results concerning the large time behaviour of the semiclassical approximation in the matricial case.

Matrix-valued version for Egorov’s theorem has been discussed several times in the literature ([2,5,10,16,23]). Brummelhuis and Nourrigat [5] have studied the particular case of matrix-valued Hamiltonian with scalar principal symbol and proved an extension of the semiclassical Egorov theorem valid for evolutions in finite time intervals. This result has been extended to the general case by Bolte and Glaser [2] under an assumption on the gap between the eigenvalues of the principal symbol of the Hamiltonian (see assumption (A1) in the next section). However, their result is again only valid for finite time. Here we require the same assumption as in [2] and we are concerned with the large time behaviour of the approximation. Some ideas from [2] are still present here.

Let us explain the main difficulties arising from the matrix structure of the problem. By going back to the Cauchy problem ( C 0 ) satisfied by the principal symbol q 0 ( t ) , one immediately sees that in the case where H and Q are matrix-valued functions, the principal symbol of the Moyal commutator [ H , q ( t ) ] # which is equal to the matrix commutator [ H 0 , q 0 ( t ) ] is no longer zero. Here H 0 denotes the principal symbol of H. Then, at leading semiclassical order, we have an equation of the type (1.7) d d t q 0 ( t ) = i ℏ [ H 0 , q 0 ( t ) ] + O ( ℏ 0 ) , ℏ ↘ 0 . In order to get a solvable equation for q 0 ( t ) , the factor ℏ − 1 forces us to restrict ourself to a class of observables for which the commutativity between H 0 and q 0 ( t ) is preserved under the time evolution. For t = 0 , this is equivalent to a block-diagonal form of Q 0 with respect to the eigenprojectors of H 0 . Under a non-crossing assumption on the eigenvalues of H 0 , we use an idea due to Helffer and Sjöstrand [15] which consists in decomposing the Hilbert space L 2 ( R n ) ⊗ C m into almost invariant subspaces with respect to the time evolution generated by H w . By considering a class of observables that are block-diagonal with respect to the projections onto these almost invariant subspaces, we reduce the study of Q ( t ) to that of a family of block-diagonal Heisenberg observables for each of them we construct a formal asymptotic expansion in powers of ℏ by solving the corresponding symbolic Heisenberg problem. This reduction is modulo O ( ℏ ∞ ) in L ( L 2 ( R n ) ⊗ C m ) uniformly in time in large time intervals which cover the Ehrenfest time. Then, to justify this asymptotic expansion, we control the remainder term at any order by giving a uniform exponential estimate with linear argument in time. In particular, this estimate allows us to recover the Ehrenfest time for the validity of the semiclassical approximation.

Another difficulty related to the matrix structure of H and Q lies in the fact that the time evolution of the symbols of the constructed approximation will be governed not only by the Hamiltonian flows (generated by the eigenvalues of H 0 ), but also by a conjugation by a family of transport matrices that we have to control the behaviour of theirs derivatives uniformly in time.

The paper is organised as follows. In Section 2, after introducing the classes of symbols that we shall use through the paper, we state our main results. Section 3 will be devoted to the study of the particular case of Hamiltonian with scalar principal symbol (and matrix-valued sub-principal symbol). In Section 4, we generalize the results of Section 3 to the case of matrix-valued principal symbol without crossing eigenvalues. Appendix A contains a short background on some basic results of ℏ-pseudodifferential calculus in the context of operators with matrix-valued symbols. In Appendices B and C, we give the proofs of some technical results.

Some notations. Let M m ( C ) be the space of ( m × m ) complex-valued matrices endowed with the operator norm denoted by ‖ · ‖ . We denote I m the corresponding identity matrix.

In this paper, three types of commutators appear: for P, Q two matrix-valued functions in some suitable classes of symbols, [ P , Q ] : = P Q − Q P is the usual matrix commutator. We use the same notation for the standard operators commutator [ P w , Q w ] : = P w Q w − Q w P w . Finally, the symbol of [ P w , Q w ] will be denoted [ P , Q ] # : = P # Q − Q # P and called the Moyal commutator of P and Q.

Through the paper smooth means C ∞ . For A ∈ C ∞ ( R 2 n ) ⊗ M m ( C ) and α , β ∈ N n , we introduce the notation A ( β ) ( α ) ( x , ξ ) : = ∂ ξ β ∂ x α A ( x , ξ ) . Given a function f ℏ depending on the semiclassical parameter ℏ ∈ ( 0 , 1 ] , the asymptotic relation f ℏ = O ( ℏ ∞ ) means that f ℏ = O ( ℏ N ) , for all N ∈ N .

The identity operator on L 2 ( R n ) ⊗ C m will be denoted id L 2 ( R n ) ⊗ C m . For ζ = ( ζ 1 , … , ζ 2 n ) ∈ R 2 n , we use the standard notation ⟨ ζ ⟩ : = ( 1 + | ζ | 2 ) 1 2 = ( 1 + | ζ 1 | 2 + ⋯ + | ζ 2 n | 2 ) 1 2 . Finally, our convention for the Poisson bracket of matrix-valued functions A , B ∈ C ∞ ( R 2 n ) ⊗ M m ( C ) is { A , B } : = ∂ ξ A ∂ x B − ∂ x A ∂ ξ B . Notice that in general { A , B } ≠ − { B , A } .

2. Assumptions and main results

Let us begin by recalling some notions about semiclassical classes of symbols. We refer to [11, Chapter 7] and [30, Chapter 4] for more details. For the context of operators with matrix-valued symbols see [16, Chapter 1].

In this paper we use the standard ℏ-Weyl quantization defined for A ∈ S ( R 2 n ) ⊗ M m ( C ) (the space of Schwartz functions on R 2 n with values in M m ( C ) ) by the formula (2.1) A w ( x , ℏ D x ) u ( x ) : = 1 ( 2 π ℏ ) n ∫ ∫ R 2 n e i ℏ ⟨ x − y , ξ ⟩ A ( x + y 2 , ξ ) u ( y ) d y d ξ , u ∈ S ( R n ) ⊗ C m . For short, we shall sometimes simply write A w .

Let g : R 2 n → [ 1 , + ∞ [ be an order function, i.e. g satisfies: there exists C , N > 0 such that g ( v ) ⩽ C ⟨ v − w ⟩ N g ( w ) , ∀ v , w ∈ R 2 n . The typical example is g ( x , ξ ) = ⟨ ( x , ξ ) ⟩ a , a ⩾ 0 .