An Egorov Theorem for avoided crossings of eigenvalue surfaces
AAN EGOROV THEOREM FOR AVOIDED CROSSINGS OFEIGENVALUE SURFACES
CLOTILDE FERMANIAN KAMMERER AND CAROLINE LASSER
Abstract.
We study nuclear propagation through avoided crossings of elec-tron energy levels. We construct a surface hopping semigroup, which givesan Egorov-type description of the dynamics. The underlying time-dependentSchr¨odinger equation has a two-by-two matrix-valued potential, whose eigen-value surfaces have an avoided crossing. Using microlocal normal forms remi-niscent of the Landau-Zener problem, we prove convergence to the true solutionin the semi-classical limit. Introduction
We consider a system of Schr¨odinger equations(1) (cid:26) iε∂ t ψ εt = − ε ∆ q ψ εt + V ( q ) ψ εt , ( t, q ) ∈ R × R d ψ εt =0 = ψ ε , where V is a smooth function on R d , whose values are 2 × V ( q ) = α ( q )Id + (cid:18) β ( q ) γ ( q ) γ ( q ) − β ( q ) (cid:19) , q ∈ R d . The smooth functions α, β, γ ∈ C ∞ ( R d , R ) are of subquadratric growth such thatthe Schr¨odinger operator in equation (1) is essentially self-adjoint, and there existsa unique solution for all times t ∈ R . We aim at the semi-classical limit ε → λ + ( q ) ≥ λ − ( q ) of the potential matrix V ( q )get close to each other and non-adiabatic transitions occur to leading order in ε .Our method of proof allows for rather general initial data ( ψ ε ) ε> that are uniformlybounded in L ( R d , C ).Schr¨odinger systems with matrix-valued potentials can be rigorously derived inthe context of the Born–Oppenheimer approximation of molecular quantum dy-namics, and we refer to [ST] and [MS] where this theory is carefully carried out.Born–Oppenheimer theory also applies for the present two-level system, providedthat the eigenvalues λ + and λ − are uniformly separated, that is, if there exists asmall gap parameter δ >
0, independent of the semi-classical parameter ε > g ( q ) = λ + ( q ) − λ − ( q ) Date : September 17, 2018.
Key words and phrases.
Time-dependent Schr¨odinger system, eigenvalue crossing, avoidedcrossings, Egorov theorem, microlocal normal form, surface hopping.C. Lasser acknowledges support by the German Research Foundation (DFG), CollaborativeResearch Center SFB-TR 109. a r X i v : . [ m a t h - ph ] M a r atisfies g ( q ) ≥ δ for all q ∈ R d . In this situation, the eigenspaces are adiabaticallydecoupled in the following sense: If Π ± ( q ) denote the eigenprojectors onto theeigenspaces of V ( q ), then for initial data with ψ ε = Π + ψ ε , one obtains only asmall non-adiabatic contribution (cid:107) Π − ψ ε ( t ) (cid:107) L = O ( ε ) at time t , and the analogousstatement holds true if ψ ε = Π − ψ ε . If the gap-condition is violated because thegap becomes small (with respect to ε ) or vanishes, then adiabatic decoupling nolonger holds. For concrete molecular systems, the semi-classical parameter ε andthe gap parameter δ are given numbers, and an asymptotic analysis just takinginto account the smallness of the semi-classical parameter ε does not provide thenecessary information.Especially for Schr¨odinger systems, non-adiabatic transitions have been of inter-est over decades, since they occur in many applications ranging from atmosphericchemistry to photochemistry, see the recent perspective article [Tu2]. Typically,non-adiabatic phenomena are attributed to avoided or conical crossings of eigen-values. Conical crossings occur, when the eigenvalue gap vanishes and the eigen-projectors have a conical singularity at these points. Generic conical crossings havebeen classified by symmetry, see [Hag1], and have been analysed by semiclassicalwavepackets [Hag1] as well as by pseudodifferential operators [CdV1, FG02, LT],see also the kinetic model for graphene [FM] that encorporates a conical crossing.Here we aim at the analysis of avoided crossings. For them, the following definitionhas been proposed [Hag2]: Definition 1.1.
Suppose V ( q, δ ) is a family of real symmetric × matrices de-pending smoothly on q ∈ Ω and δ ∈ I , V ∈ C ∞ (Ω × I, R × ) , where Ω is an opensubset of R d and I ⊂ R some interval containing . Suppose that V ( q, δ ) has twoeigenvalues λ + ( q, δ ) and λ − ( q, δ ) that depend continuously on q and δ . Assumethat { q ∈ Ω , λ + ( q,
0) = λ − ( q, } is a non-empty submanifold of Ω , such that λ + ( q, δ ) (cid:54) = λ − ( q, δ ) for all q ∈ Ω and δ (cid:54) = 0 . Then we say that V ( q, δ ) has an avoided crossing of eigenvalues. Avoided crossings have a similar symmetry classification [Hag2] as the conicalintersections. In [HJ1, HJ2, Rou], the perturbation parameter δ of Definition 1.1has been linked with the semiclassical parameter ε , and a leading order analysis of iε∂ t ψ εt = − ε ∆ q ψ εt + V ( q, √ ε ) ψ εt , ψ εt =0 = ψ ε has been carried out for families of initial data ( ψ ε ) ε> which are semiclassicalwavepackets. For Schr¨odinger systems in one space dimension, avoided crossingshave also been considered without assuming that the perturbation δ and the semi-classical parameter ε are coupled. In this situation, the non-adiabatic contributionsare exponentially small with respect to ε . In [HJ3], it is assumed that the potential V ( q ) of the Schr¨odinger system (1) belongs to a family V ( q, δ ) with an avoidedcrossing such that V ( q ) = V ( q, δ ) for all q ∈ R and some fixed δ >
0. Then, thescattering wave function of semiclassical wavepackets is determined together withits non-adiabatic contributions. In [BGT, BG], superadiabatic representations ofone-dimensional avoided crossings have been developed together with an explicitheuristic formula for the outgoing nonadiabatic component. Our results interpolatebetween the existing ones in the following sense. On the one hand, we allow forgeneral families ( ψ ε ) ε> of initial data in L ( R d , C ) without restricting to coherent tates or a single space dimension. Also, we will not explicitly link the semiclassicalparameter ε and the gap parameter δ . On the other hand, we will only reach forthe leading order behaviour with respect to ε .1.1. Wigner transforms.
It is impossible to directly study the densities n ε ± ( q, t ) = | Π ± ( q ) ψ ε ( q, t ) | C or the dynamics of the so-called level populations t (cid:55)→ (cid:90) R d n ε ± ( t, q ) dq for general initial data. Thus, we focus on providing an asymptotic description forthe time evolution of the Wigner transform of ψ ε ( q, t ) in a suitable ε -dependentscaling, W ε ( ψ εt )( q, p ) = (2 π ) − d (cid:90) R d ψ ε (cid:0) q − ε v, t (cid:1) ⊗ ψ ε (cid:0) q + ε v, t (cid:1) e i v · p d v with ( q, p ) ∈ R d . The Wigner transform plays the role of a generalized probabilitydensity on phase space. For square integrable wave functions ψ ∈ L ( R d , C ), theWigner function W ε ( ψ ) is a square integrable function on phase space with valuesin the space of Hermitian 2 × n ε ± ( q, t ) = tr (cid:90) R d Π ± ( q ) W ε ( ψ εt )( q, p ) d p. Besides, the action of the Wigner function against compactly supported smoothtest functions a ∈ C ∞ c ( R d , C × ) is simply expressed in terms of the semi-classicalpseudodifferential operator of symbol a , which is defined byop ε ( a ) ψ ( q ) = (2 πε ) − d (cid:90) R d a (cid:0) ( q + q (cid:48) ) , p (cid:1) e iε p · ( q − q (cid:48) ) ψ ( q (cid:48) ) d q (cid:48) d q for ψ ∈ L ( R d , C ). Indeed, we have (cid:90) R d tr ( W ε ( ψ )( q, p ) a ( q, p )) d q d p = (op ε ( a ) ψ , ψ ) L ( R d , C ) . The Wigner transform is perfectly suited for the analysis of quadratic functions ofthe wave function, which do not require all the phase information.Our aim is the study of the diagonal parts of the Wigner transformΠ ± ( q ) W ε ( ψ εt )Π ± ( q ) = w ± ε ( t )Π ± ( q ) ,w ε ± ( t ) = tr (cid:0) Π ± ( q ) W ε ( ψ εt )Π ± ( q ) (cid:1) , and to describe the evolution of the coefficients w ε ± ( t ) in terms of w ε + (0) and w ε − (0)as ε →
0. The oscillatory dynamics of the off-diagonal part of the Wigner functionimplies that it can be neglected far from the crossing set (see Remark A.3 in theAppendix). However, these effects could restrict our results, see the comments afterour main Theorem 2.3 and the corresponding numerical experiment in § .2. Egorov’s theorem.
We consider the classical flowΦ t ± : R d → R d , Φ t ± ( q , p ) = (cid:0) q ± ( t ) , p ± ( t ) (cid:1) associated with the Hamiltonian curves of Λ ± ( q, p ) = | p | + λ ± ( q ). These curvesare solutions to the Hamiltonian systems(3) (cid:26) ˙ q ± ( t ) = p ± ( t ) , ˙ p ± ( t ) = −∇ λ ± ( q ± ( t )) ,q ± (0) = q , p ± (0) = p which can be solved for all t ∈ R , since the maps q (cid:55)→ λ ± ( q ) are smooth foreigenvalues, which do not intersect each other.If the eigenvalues are uniformly separated from each other, then the classicalflows Φ t ± are enough for an approximate description of the dynamics up to an errorof order ε . Indeed, the action of the diagonal part of the Wigner transform onscalar test functions a ∈ C ∞ c ( R d +1 , C ) obeys(4) (cid:90) R d +1 (cid:0) w ε ± ( t ) − w ε ± (0) ◦ Φ − t ± (cid:1) ( q, p ) a ( t, q, p ) d( t, q, p ) = O ( ε ) . Such dynamical descriptions in the spirit of Egorov’s theorem are well established,see for example [GMMP].If the gap g ( q ) = λ + ( q ) − λ − ( q ) is not uniformly bounded from below and small,but not too small, this description is still valid. More precisely, one proves in [FL08](see also the proof in Appendix A) that as long as the trajectories of Φ t ± whichreach the support of the observable a remain in a zone where g ( q ) > R √ ε = ε / for R = R ( ε ) = ε − / , then (cid:90) R d +1 (cid:0) w ε ± ( t ) − w ε ± (0) ◦ Φ − t ± (cid:1) ( q, p ) a ( t, q, p ) d( t, q, p ) = O ( ε / ) , where the error estimate just depends on derivatives of the potential matrix V andthe symbol a , while it is independent of the gap parameter δ .However, on regions with smaller eigenvalue gap the approximation of the diag-onal Wigner components w ε ± ( t ) by mere classical transport is no longer valid, andthere are non-adiabatic transitions between the levels. The components propagateduntil the crossing region on one level may pass (partially or utterly) the other level.1.3. Surface hopping.
For a particular isotropic conical crossing [LT] and laterfor general conical crossings [FL08], it has been proved that the diagonal parts ofthe Wigner transform can effectively be described by the following random walkconstruction: We consider a classical trajectory Φ t + ( q, p ) with associated weight w ε + ( q, p, t ). If the gap function t (cid:55)→ g ( q + ( t ))attains a local minimum at time t ∗ for the phase space point ( q ∗ , p ∗ ), such that g ( q ∗ ) ≤ R √ ε , then one opens a new trajectory Φ t − t ∗ − ( q ∗ , p ∗ out ) with p ∗ out = p ∗ + ω ∗ , ω ∗ = g ( q ∗ ) p ∗ | p ∗ | . The two trajectories Φ t − t ∗ + ( q ∗ , p ∗ ) and Φ t − t ∗ − ( q ∗ , p ∗ out ) are equipped with the weights w + ε ( q ∗ , p ∗ , ( t ∗ ) + ) = (1 − T ε ( q ∗ , p ∗ )) w + ε ( q ∗ , p ∗ , ( t ∗ ) − ) ,w − ε ( q ∗ , p ∗ out , ( t ∗ ) + ) = T ε ( q ∗ , p ∗ ) w + ε ( q ∗ , p ∗ , ( t ∗ ) − ) , espectively. The transition probability is given by the Landau–Zener formula(5) T ε ( q ∗ , p ∗ ) = exp (cid:18) − π ε g ( q ∗ ) | det( p ∗ · ∇ q V ( q ∗ )) | / (cid:19) , where V ( q ∗ ) denotes the trace-free part of the potential matrix V ( q ∗ ). The anal-ogous construction applies to the classical trajectories entering the region of smallgap on the other eigenvalue surface.This combination of classical transport and Landau–Zener transitions yields aneasy algorithm for the numerical simulation of non-adiabatic quantum dynamics,see [FL12] and its applications to a three-dimensional model of the pyrazine mole-cule [LS] and the twelve-dimensional ammonia cation [BDLT]. Its striking proper-ties are, that only classical trajectories, local gap minima along classical trajectoriesand the Landau–Zener formula (5) are required. Many other surface hopping algo-rithms exist in the chemical literature starting with the pioneering work of Tully andPreston [TP], and it is worth mentioning that they are equally applied for systemswith avoided or conical eigenvalue crossings. In high dimensions, surface hoppingalgorithms are computationally much less demanding than the discretization of thefull wave function and thus often a popular choice. Despite the intense researchactivity in chemical physics on these algorithms, there are very few mathematicalresults on their justification.1.4. Aim and organisation of the paper.
We are interested in extending theLandau–Zener random walk through conical crossings [LT, FL08] to the case ofavoided crossings, thus obtaining a unified treatment for conical intersections andavoided crossings, regardless of the respective sizes of the gap and the semi-classicalparameter. As far as we know, this unified treatment of both crossing types andits rigorous mathematical analysis is new, see also [FL12] for comments on thesubject. Following [Hag2], we assume that the potential matrix V presents anavoided crossing in the sense of the definition below. Definition 1.2.
Let the potential V ( q ) = α ( q )Id + (cid:18) β ( q ) γ ( q ) γ ( q ) − β ( q ) (cid:19) , q ∈ R d , be a smooth function on R d , whose values are × real symmetric matrices. Denoteby λ ± ( q ) and g ( q ) = λ + ( q ) − λ − ( q ) the eigenvalues and the gap function of V ( q ) and by V ( q ) its trace-free part, that is, (6) V ( q ) = (cid:18) β ( q ) γ ( q ) γ ( q ) − β ( q ) (cid:19) , q ∈ R d . We say that V has an avoided gap in an open subset Ω ⊂ R d if it satisfies thefollowing conditions : (1) There exists δ > , which is the minimum of g in Ω , (2) Let S = { q ∈ R d , g ( q ) = δ } . The set S ∩ Ω is a hypersurface. (3) There exists a system of local coordinates ( y , y (cid:48) ) with S ∩ Ω = { y = 0 }∩ Ω and V ( y ) = y V ( y ) + V ( y (cid:48) ) here V ( y ) is an invertible matrix, while V ( y ) and V ( y ) are linearly in-dependent for all y ∈ Ω . An illustrative easy example for an avoided crossing in the sense of Definition 1.2is provided by(7) V ( q ) = (cid:18) q δ δ − q (cid:19) , q ∈ R d , δ > . Here, the eigenvalues λ ± ( q ) = ± (cid:113) q + δ have a global minimal gap of size δ at the hyperplane S = { q ∈ R d , q = 0 } , andthe potential matrix can be written as V ( q ) = q V + V with V = (cid:18) − (cid:19) , V = (cid:18) δ δ (cid:19) . On the contrary, the matrix V ( q ) = (cid:113) q + δ (cid:18) − (cid:19) not satisfy the assumptions of Definition 1.2. Remark . We note that if one can write the minimal gap set in local coordinatesas S ∩ Ω = { y = 0 } and V ( y ) = y V ( y ) + V ( y (cid:48) ) with V ( y ) an invertible matrix,then V ( y ) and V ( y (cid:48) ) are necessarily linearly independent, in the sense, that thereexists no smooth function f : Ω → R with V ( y ) = f ( y ) V ( y (cid:48) ) for all y ∈ Ω. Remark . The avoided crossing of Definition 1.2 is associated with a minimal gapmanifold S ⊂ R d of codimension one. In the symmetry classification of avoidedcrossings given in [Hag2], higher codimensions also occur. We expect that ouranalysis of the codimension one case can be generalized, see also Remark 2.4.If potential V has an avoided gap with minimal gap size δ > V ( q, δ ) with an avoided crossingof eigenvalues in the sense of Definition 1.1 with two crucial properties. First, wehave V ( q ) = V ( q, δ ) , q ∈ Ω , and second the mapping ( q, δ ) (cid:55)→ V ( q, δ ) has a conical intersection of its eigenvaluesfor ( q, δ ) ∈ S × { δ = 0 } , see Theorem 3.1 below. Considering momenta p ∈ R d which are transverse to the hypersurface S ensures that the crossing is generic in T ∗ ( R dq × R δ ) in the sense of [CdV1] and [FG03]. This link between avoided andconical crossings – which is already indicated in Colin de Verdi`ere’s paper [CdV1, § δ as a controlled parameter.We start in Section 2 by providing the precise mathematical statement of ourresult. In Section 3 we prove the relation between avoided and conical cross-ings. In Section 4 we perform an elementary reduction to a Landau–Zener modelparametrized by the gap parameter δ . Both these new results are crucial for theproof of the surface hopping approximation in Section 5. We then describe the as-sociated surface hopping algorithm in Section 6 and present numerical experiments or Tully’s well-known avoided crossing models [Tu1]. The Appendix A presentsthe proof of classical transport in the zone of large gap.2. An Theorem
We now give a precise statement for the Egorov type description of the propa-gation of the diagonal part of the Wigner transform for systems with an avoidedeigenvalue crossing in the sense of Definition 1.2. We consider the classical trajec-tories ( q ± ( t ) , p ± ( t )) of the Hamiltonian systems (3) and monitor the phase spacepoints, where the classical trajectories attain a local minimal gap between the twoeigenvalues. At such points we have(8) dd t (cid:0) g ( q ± ( t )) (cid:1) = p ± ( t ) · ∇ q g ( q ± ( t )) = 0 . Therefore, one performs an effective non-adiabatic transfer of weight, whenever atrajectory passes the setΣ ε = (cid:8) ( q, p ) ∈ R d | g ( q ) ≤ R √ ε, p · ∇ g ( q ) = 0 (cid:9) where R = R ( ε ) = ε − / (cid:29)
1. This choice of R is motivated from the analysisin [FL08] and ensures that the largest occuring error terms R √ ε and R − ε − / are of the same order η ε = ε / .2.1. The random trajectories.
We attach the labels j = − j = +1 to thephase space R d and consider the random trajectories T ( q,p,j ) ε : [0 , + ∞ ) → R d × {− , +1 } , with T ( q,p,j ) ε ( t ) = (cid:0) Φ tj ( q, p ) , j (cid:1) as long as Φ tj ( q, p ) (cid:54)∈ Σ ε . Whenever the deterministic flow Φ tj ( q, p ) hits the set Σ ε at a point ( q ∗ , p ∗ ), a jump( q ∗ , p ∗ , j ) → ( q ∗ , p ∗ + j ω ∗ , − j )occurs with the transition probability T ε ( q ∗ , p ∗ ) defined in (5). The drift(9) ω ∗ = ω ∗ ( q ∗ , p ∗ ) = g ( q ∗ ) | p ∗ | p ∗ , is applied to preserve the energy of the trajectoriesΛ ± ( q, p ) = | p | + λ ± ( q ) = | p | + α ( q ) ± g ( q )up to order R ε . Indeed, let us suppose that the incoming trajectory is on the plusmode. Then, one chooses the momentum p ∗ out = p ∗ + ω ∗ of the trajectory generatedon the minus mode such that its energy Λ − ( q ∗ , p ∗ out ) satisfiesΛ − ( q ∗ , p ∗ out ) = Λ + ( q ∗ , p ∗ ) + O ( R ε ) . Since Λ − ( q ∗ , p ∗ out ) = | p ∗ + ω ∗ | + α ( q ∗ ) − g ( q ∗ ), it is enough to choose ω ∗ suchthat ω ∗ · p ∗ = g ( q ∗ ) , whence (9). Remark . Let us comment about various aspects of the drift. The drift p ∗ ± ω ∗ will be crucial later on for localizing the solution at a distance of size R √ ε tothe energy surfaces { τ + Λ ± ( q, p ) = 0 } . The drift is performed in the momentumcoordinates, since the difference of the two Hamiltonian vector fields H λ ± ( q, p ) = ( p, −∇ λ ± ( q )) anishes identically in the position coordinates. Moreover, after a change of space-time coordinates, see Section 5.2.6, the drift is exact when performed in this direc-tion. A similar drift has been used in [HJ1] and [HJ2] for analysing wave packetpropagation through avoided crossings.The transversality condition that will be stated in assumption (A0) of our mainTheorem 2.3 excludes trajectories with small momenta at the jump manifold.Therefore, ω ∗ is of the order of the gap size and thus bounded by R √ ε . Out-side the jump manifold Σ ε the gap and thus the drift are large. However, theLandau–Zener transition rates are exponentially small there. Consequently, in thisregime the drift would be harmless (if performed).2.2. The semigroup.
Within a bounded time-interval [0 , T ], each path( q, p, j ) → T ( q,p,j ) ε ( t )only has a finite number of jumps, remains in bounded regions of R d × {− , +1 } ,and is smooth away from the jump manifold Σ ε × {− , +1 } . Hence, the randomtrajectories define a Markov process (cid:110) X ( q,p,j ) | ( q, p, j ) ∈ R d × {− , +1 } (cid:111) . The associated transition function P ( p, q, j ; t, Γ) describes the probability of beingat time t in the set Γ ⊂ R d × {− , +1 } having started in ( q, p, j ). Its action on theset B = (cid:8) f : R d × {− , +1 } → C | f is measurable, bounded (cid:9) defines a semigroup ( L tε ) t ≥ by L tε f ( q, p, j ) = (cid:90) R d ×{− , +1 } f ( x, ξ, k ) P ( q, p, j ; t, d( x, ξ, k )) . Remark . We associate with f ∈ B two functions f ± : R d → C via(10) f ± ( q, p ) = f ( q, p, ± . Reversely, relation (10) implies that two bounded measurable functions f ± on R d define a function f ∈ B . We shall use this identification all over the paper.We now define the action of the semigroup on Wigner functions by duality. Moreprecisely, let ψ ∈ L ( R d , C ) and W ε ( ψ ) be its Wigner transform. Denote by w ε ± ( ψ )( q, p ) = tr (cid:0) Π ± ( q ) W ε ( ψ )( q, p ) (cid:1) the diagonal components of W ε ( ψ ) and define w ε ( ψ ) ∈ B according to relation (10).For a ∈ B such that a + and a − have compact support, we set( w ε ( ψ ) , a ) = (cid:90) R d w ε + ( ψ )( q, p ) a + ( q, p ) d( q, p ) + (cid:90) R d w ε − ( ψ )( q, p ) a − ( q, p ) d( q, p )and define L tε w ε ( ψ ) ∈ B by (cid:0) L tε w ε ( ψ ) , a (cid:1) = (cid:0) w ε ( ψ ) , L tε a (cid:1) . The result.
Let V be a potential matrix presenting an avoided crossing ofeigenvalues in the sense of Definition 1.2, the notations of which we shall use inthe following. The semi-group ( L tε ) t> approximates the non-adiabatic dynamicsgenerated by this avoided crossing in Ω ⊂ R d , if we assume the following: .3.1. Initial data (A0).
The initial data ( ψ ε ) ε> is a bounded family in L (cid:0) R d , C (cid:1) associated either with RanΠ + or RanΠ − , meaning that either (cid:12)(cid:12)(cid:12)(cid:12) Π − ψ ε (cid:12)(cid:12)(cid:12)(cid:12) L ( R d , C ) = O ( ε β ) , β ≥ / , or the analogous condition on Π + ψ ε holds. We suppose that the initial data arelocalized away from S ∩ Ω, that is, there is some C > (cid:90) { d( q,S )
The observable a ∈ B satisfies a ± ∈ C ∞ c (cid:0) R d , C (cid:1) and hasits support at a distance larger than ε β from S , i. e.d(supp ( q,p ) ( a ± ) , S ) (cid:29) ε β , β ≥ / . Time-interval (A2).
Let
T >
0. Within the time-interval [0 , T ], the classicaltrajectories issued from the support of W ε ( ψ ε ) reach their minimal gap points onlyonce.These assumptions on the initial data, the observables, and the time intervalallow us to effectively describe the dynamics through an avoided crossing by surfacehopping. Theorem 2.3.
Let ε > and ψ ε be the solution of the Schr¨odinger equation iε∂ t ψ εt = − ε ∆ q ψ εt + V ( q ) ψ εt , ψ εt =0 = ψ ε , where the potential V has an avoided crossing in the sense of Definition 1.2 witha gap parameter δ ∈ ]0 , . Assuming (A0), (A1) and (A2), we have for all testfunctions χ ∈ C ∞ c ([0 , T ]) a constant C > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T χ ( t ) (cid:0) w ε ( ψ εt ) − L tε w ε ( ψ ε ) , a (cid:1) d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ε / , where the constant C depends on a finite number of upper bounds of derivatives ofthe smooth functions α, β, γ defining the potential V and a, χ and of lower boundsof the determinant of the matrix V . The semigroup ( L tε ) t ≥ crucially depends on the jump manifold Σ ε , that com-prises those points in phase space with g ( q ) ≤ R √ ε , R = R ( ε ) = ε − / , wherethe classical trajectories ( q ± ( t ) , p ± ( t )) attain locally minimal surface gaps. If theminimal gap size δ > R √ ε , then the jumpmanifold is the empty set, so that Theorem 2.3 reduces to a leading order descriptionof expectation values for block-diagonal observables by mere classical transport, seeAppendix A. As a consequence, δ ≤ R √ ε is the only interesting regime, and thekey issue is to prove the hopping formula locally, close to any point of S ∩ Ω. Thisis done by using the possibility to parametrically link the avoided crossing with aconical intersection. This construction is carried out in Section 3, where we prove hat close to any point of Ω, the potential V is embeddable in a parametrized familyof potentials. We then prove a result analogous to Theorem 2.3 for the family ofsolutions to the Schr¨odinger equation associated with the parametrized potentials(see Theorem 3.4 below). The different steps of the proof of the surface hoppingapproximation are then developed in Section 5, via a reduction to a Landau–Zenermodel performed in Section 4.An interesting feature of Theorem 2.3 is that it justifies using a surface algorithmwithout assessing the size of the gap with respect to ε . This is of major interest forapplications, since for “real” molecular quantum systems the explicit comparisonof δ and ε might be difficult. The algorithm takes into account the three mainregimes:(1) δ (cid:29) √ ε propagation along the eigenvalue surfaces(2) δ ∼ √ ε partial transition between eigenspaces(3) δ (cid:28) √ ε total transition between the eigenspacesIn the context of semi-classical wave packet propagation, [HJ1] and [HJ2] haveanalysed the second regime, while [Rou] has considered the first and third one.In particular, if there are several sizes of minimal gaps in different open subsets,Theorem 2.3 proves that the algorithm can be used and the transition processwill be automatically adapted to the gap size. The Born–Oppenheimer result (4)uses plain classical transport and has an error constant that tends to infinity forshrinking minimal gap size δ . In constrast, the error bound of Theorem 2.3 onlydepends on bounds of the potential V that can be controlled with respect to δ .(The lower bound on the determinant of V is not related to the gap size.)We note that the transversality condition of assumption (A0) is crucial for themicrolocal normal form we use for effectively describing the nonadiabatic transi-tions. For the simple example (7), it means that the set { ( q, p ) ∈ R d , q = p = 0 } is negligible for the trajectories of˙ q = p, ˙ p = ∓ q (cid:112) q + δ (1 , , . . . , W ε ( ψ ε ). The firstcondition of assumption (A0) can be relaxed to initial data associated with bothRan Π + and Ran Π − , provided that the trajectories for both modes do not arrivesimultaneously at the same phase space point of the jump manifold Σ ε . However, asillustrated by the numerical experiments for the dual avoided crossing in Section 6.3,simultaneous arrival at the jump manifold is the situation where the off-diagonalcomponents of the Wigner transform become relevant such that the present surfacehopping approximation breaks down. Remark . The avoided crossing of Definition 1.2 has a minimal gap manifold S ⊂ R d of codimension one. The results of [FG03, Fe06, FL08] on eigenvaluecrossings of codimension three and five allow to extend Theorem 2.3 to avoidedcrossings with minimal gap manifolds of higher codimension as well.3. Reduction to a conical intersection
We now introduce a family of potentials ( V ( q, δ )) δ ∈ I locally extending the trace-free part V ( q ) of our original potential. We verify that V ( q, δ ) viewed as a functionon R d × I has a generic codimension two crossing for q ∈ S and δ = 0 in the ense of [CdV1] and [FG03], respectively. Then, we describe the parametrizedSchr¨odinger system that we shall consider afterwards.3.1. Parametrization of the gap.
We start by constructing the family of trace-free potentials ( V ( q, δ )) δ ∈ I that locally extends the original potential V ( q ) byadding the gap size as an additional coordinate. Theorem 3.1.
Let V have an avoided crossing in Ω ⊂ R d , in the sense of Defi-nition 1.2, with minimal gap hypersurface S . Then there exists an open interval I ⊆ R with , δ ∈ I , an open subset (cid:101) Ω ⊆ Ω and two functions β ( q, δ ) and γ ( q, δ ) smooth on (cid:101) Ω × I and affine in δ , such that the matrix V ( q, δ ) = (cid:18) β ( q, δ ) γ ( q, δ ) γ ( q, δ ) − β ( q, δ ) (cid:19) satisfies the following properties: (1) We have V ( q, δ ) = V ( q ) for all q ∈ (cid:101) Ω . (2) The eigenvalue gap g ( q, δ ) = 2 (cid:112) β ( q, δ ) + γ ( q, δ ) of V ( q, δ ) is of minimal size | δ | on S , that is, g ( q, δ ) ≥ | δ | for all q ∈ (cid:101) Ω , g ( q, δ ) = | δ | if and only if q ∈ S . (3) There exists a smooth orthogonal matrix R ( q ) and two smooth functions (cid:101) β ( q, δ ) and (cid:101) γ ( q ) , where (cid:101) β is affine in δ , such that R ( q ) V ( q, δ ) R ( q ) ∗ = (cid:32) (cid:101) β ( q, δ ) δ (cid:101) γ ( q ) δ (cid:101) γ ( q ) − (cid:101) β ( q, δ ) (cid:33) and (cid:101) γ ( q ) (cid:54) = 0 for all q ∈ (cid:101) Ω . All derivatives of (cid:101) β are bounded. (4) If y = ( y , y (cid:48) ) are local coordinates such that S ∩ (cid:101) Ω = { y = 0 } ∩ (cid:101) Ω , then ∀ k ∈ N ∃ c k > y ∈ (cid:101) Ω | ∂ ky (cid:101) γ ( y ) | < c k and ∀ α ∈ N d − ∃ c α > y ∈ (cid:101) Ω | ∂ αy (cid:48) (cid:101) γ ( y ) | < c α | y | δ , while all other derivatives of the function (cid:101) γ are of the order /δ . The deriv-ative bounds involve a lower bound on the determinant of the matrix V ( y ) of the decomposition V ( y ) = y V ( y ) + V ( y (cid:48) ) .Proof. We work close to some q ∈ S in local coordinates y = ( y , y (cid:48) ) such that S ∩ Ω = { y = 0 } ∩ Ω and V ( y ) = y V ( y ) + V ( y (cid:48) ) with V ( y ) invertiblefor all y ∈ Ω. Setting y = 0, one obtains that V and consequently V are trace-freeon Ω. We denote V ( y ) = (cid:18) a ( y ) b ( y ) b ( y ) − a ( y ) (cid:19) , V ( y (cid:48) ) = (cid:18) a ( y (cid:48) ) b ( y (cid:48) ) b ( y (cid:48) ) − a ( y (cid:48) ) (cid:19) nd write the gap as g ( y ) = 4 y (cid:0) a ( y ) + b ( y ) (cid:1) + 8 y ( a ( y ) a ( y (cid:48) ) + b ( y ) b ( y (cid:48) ))+ 4 (cid:0) a ( y (cid:48) ) + b ( y (cid:48) ) (cid:1) for all y ∈ Ω. From the relation g ( y ) = δ for all y ∈ S we then deduce δ = 4( a ( y (cid:48) ) + b ( y (cid:48) ) ) for all y ∈ Ω . We define for δ ∈ R V ( y, δ ) = y V ( y ) + δδ V ( y (cid:48) )such that for all y ∈ Ω V ( y, δ ) = V ( y ) , g ( y, δ ) = g ( y )and g ( y, δ ) = 4 y (cid:0) a ( y ) + b ( y ) (cid:1) + 8 y δδ ( a ( y ) a ( y (cid:48) ) + b ( y ) b ( y (cid:48) )) + δ . With respect to the original coordinates, this construction means V ( q, δ ) = (cid:18) β ( q, δ ) γ ( q, δ ) γ ( q, δ ) − β ( q, δ ) (cid:19) with β ( q, δ ) = y a ( y ) + δδ a ( y (cid:48) ) , γ ( q, δ ) = y b ( y ) + δδ b ( y (cid:48) ) . Let us prove now that the gap g ( y, δ ) is minimal on S for δ in some openinterval I which contains [0 , δ ]. Since the gap g ( y ) is minimal on S ∩ Ω, we have ∇ y ( g ( y ) ) = 0 for all y ∈ S ∩ Ω. Consequently, a (0 , y (cid:48) ) a ( y (cid:48) ) + b (0 , y (cid:48) ) b ( y (cid:48) ) = 0 , y = (0 , y (cid:48) ) ∈ S ∩ Ω . Therefore, there exist an open subset (cid:101) Ω ⊆ Ω and a continuous function Γ : (cid:101) Ω → R such that a ( y ) a ( y (cid:48) ) + b ( y ) b ( y (cid:48) ) = y Γ( y )and g ( y, δ ) = 4 y (cid:18) a ( y ) + b ( y ) + 2 δδ Γ( y ) (cid:19) + δ . Now it remains to find an open interval I such that(12) a ( y ) + b ( y ) + 2 δδ Γ( y ) > y, δ ) ∈ (cid:101) Ω × I with y (cid:54) = 0. We observe that g ( y, δ ) = g ( y ) > δ for y (cid:54) = 0implies 4 y (cid:0) a ( y ) + b ( y ) + 2Γ( y ) (cid:1) > y (cid:54) = 0 , while the invertibility of V ( y ), y ∈ (cid:101) Ω, implies a ( y ) + b ( y ) > , y ∈ (cid:101) Ω . Therefore, the affine function δ (cid:55)→ a ( y ) + b ( y ) + 2 δδ Γ( y ) akes nonnegative values in δ = 0 and δ = δ and thus for any δ ∈ ]0 , δ [, whichyields (12). We note that the we can choose the interval I small enough such thatthe quotient δ/δ remains bounded for all δ ∈ I . Consequently, the functions β ( · , δ )and γ ( · , δ ) have smooth bounded derivatives.For the rotation of V ( q, δ ) we set A ( q ) = − b ( y ( q )) (cid:112) b ( y ( q )) + a ( y ( q )) , B ( q ) = a ( y ( q )) (cid:112) b ( y ( q )) + a ( y ( q )) and note that A and B are smooth functions with bounded derivatives, where thebound involves a lower bound on the determinant of V . We also define the smoothrotation matrix R ( q ) of angle θ ( q ) such thatcos (2 θ ( q )) = B ( q ) , sin (2 θ ( q )) = A ( q ) . Then, we have R ( q ) V ( q, δ ) R ( q ) ∗ = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) (cid:18) β γγ − β (cid:19) (cid:18) cos θ sin θ − sin θ cos θ (cid:19) = (cid:18) β cos(2 θ ) − γ sin(2 θ ) β sin(2 θ ) + γ cos(2 θ ) β sin(2 θ ) + γ cos(2 θ ) − β cos(2 θ ) + γ sin(2 θ ) (cid:19) = (cid:32) (cid:101) β ( q, δ ) δ (cid:101) γ ( q ) δ (cid:101) γ ( q ) − (cid:101) β ( q, δ ) (cid:33) , where we define the δ -affine function (cid:101) β ( q, δ ) = B ( q ) β ( q, δ ) − A ( q ) γ ( q, δ )= y ( q ) (cid:112) a ( y ( q )) + b ( y ( q )) + δδ a ( y (cid:48) ( q )) a ( y ( q )) + b ( y ( q )) b ( y (cid:48) ( q )) (cid:112) b ( y ( q )) + a ( y ( q )) , whose derivatives are bounded functions, since δ/δ is uniformly bounded. Thesmooth function (cid:101) γ ( q ) is defined by (cid:101) γ ( q ) = 1 δ ( A ( q ) β ( q, δ ) + B ( q ) γ ( q, δ ))= − b ( y ( q )) a ( y (cid:48) ( q )) + a ( y ( q )) b ( y (cid:48) ( q )) δ (cid:112) b ( y ( q )) + a ( y ( q )) . Observing that a and b only depend on y (cid:48) and that a ( y (cid:48) ) + b ( y (cid:48) ) = δ ≤ δ , y = ( y , y (cid:48) ) ∈ (cid:101) Ω , we deduce that ∂ ky (cid:101) γ is a smooth bounded function for any k ∈ N . Using that a a + b b = 0 on S , we observe (cid:101) β ( q, δ ) = 0 , q ∈ S ∩ (cid:101) Ω , and δ = β ( · , δ ) + γ ( · , δ ) = (cid:101) β ( · , δ ) + δ (cid:101) γ , and we deduce that (cid:101) γ = 1 on S .Since (cid:101) γ is non-vanishing, due to the linear independence of V ( y ) and V ( y (cid:48) ), wethen conclude that ∂ αy (cid:48) (cid:101) γ (0 , y (cid:48) ) = 0 for all α ∈ N d − . A Taylor expansion togetherwith the rough estimate, that derivatives of (cid:101) γ are of the order 1 /δ , yields theclaimed bound on ∂ αy (cid:48) (cid:101) γ ( y ) for y ∈ (cid:101) Ω. (cid:3) .2. The geometry of the crossing.
We add half the trace to the δ -parametrizedtrace-free family of Theorem 3.1 and consider(13) V ( q, δ ) = α ( q )Id + V ( q, δ ) , ( q, δ ) ∈ (cid:101) Ω × I, with α ( q ) = tr V ( q ), such that the original potential can be written as V ( q ) = V ( q, δ ) , q ∈ (cid:101) Ω . We now verify that the symbol of the corresponding time-dependent Schr¨odingeroperator, the matrix-valued function P ( q, p, τ, δ ) := (cid:16) τ + | p | (cid:17) Id + V ( q, δ ) , has a generic codimension two crossing on S × { δ = 0 } in the sense of [CdV1, § § g ( q, δ ) = 0 if and only if ( q, δ ) ∈ ( S ∩ (cid:101) Ω) × { δ = 0 } , according to point (2) of Theorem 3.1.(2) The Poisson bracket (cid:110) τ + | p | + α ( q ) , V ( q, δ ) (cid:111) = p · ∇ q V ( q, δ )is invertible for ( q, δ ) ∈ ( S ∩ (cid:101) Ω) ×{ δ = 0 } and those momenta p ∈ R d whichare transverse to S at q . This is implied by the following Lemma 3.2. Lemma 3.2.
Let V have an avoided crossing in the sense of Definition 1.2 withminimal gap hypersurface S . Let q ∈ S and p ∈ R d transverse to S at q . Then,there exists a neighborhood Ω ⊂ R d of ( q , p ) , independent of δ such that for all δ ∈ I and ( q, p ) ∈ Ω , p · ∇ q V ( q, δ ) is invertible and p · ∇ q (cid:101) β ( q, δ ) < , where ( V ( q, δ )) δ ∈ I and (cid:101) β ( q, δ ) are defined as in Theorem 3.1.Proof. Let (cid:101)
Ω be the open set of Theorem 3.1. We again work close to some point q ∈ S ∩ (cid:101) Ω in local coordinates y = ( y , y (cid:48) ) such that S ∩ (cid:101) Ω = { y = 0 } ∩ (cid:101) Ω and (cid:101) β ( q, δ ) = y (cid:112) a ( y ) + b ( y ) (cid:18) a ( y ) + b ( y ) + δδ Γ( y ) (cid:19) , where by (12), a ( y ) + b ( y ) + 2 δδ Γ( y ) ≥ , y ∈ (cid:101) Ω , δ ∈ I. Then, for all ( q, p ), p · ∇ q (cid:101) β ( q, δ ) = p · ∇ q y (cid:112) a ( y ) + b ( y ) (cid:18) a ( y ) + b ( y ) + δδ Γ( y ) (cid:19) + y p · γ ( y, δ )where the function p · γ ( y, δ ) is bounded for δ ∈ I and ( p, q ) in any bounded set.Since y = 0 is an equation of the hypersurface S in (cid:101) Ω and p is transverse to S at q , we have p · ∇ q y ( q ) (cid:54) = 0. Therefore, if necessary, we turn y into − y , sothat p · ∇ q (cid:101) β ( q , δ ) < , or all δ ∈ I . Besides, by setting a bound on y and p , we can find a δ -independentneighborhood Ω ⊂ (cid:101) Ω of ( q , p ) such that ∀ δ ∈ I ∀ ( q, p ) ∈ Ω : p · ∇ q (cid:101) β ( q, δ ) < . The invertibility of p · ∇ q V ( q, δ ) in I × Ω is then implied by − det( p · ∇ q V ( q, δ )) = ( p · ∇ q (cid:101) β ( q, δ )) + δ ( p · ∇ q (cid:101) γ ( q )) . (cid:3) We now work in the set Ω and denote by λ ± ( q, δ ) = α ( q ) ± (cid:113) (cid:101) β ( q, δ ) + δ (cid:101) γ ( q ) the eigenvalues of the matrix V ( q, δ ) and still denote byΦ t ± : R d → R d , Φ t ± ( q , p ) = (cid:0) q ± δ ( t ) , p ± δ ( t ) (cid:1) the flow associated with the δ -dependent Hamiltonian system˙ q ± δ ( t ) = p ± δ ( t ) , ˙ p ± δ ( t ) = −∇ q λ ± ( q ± δ ( t ) , δ ) , that becomes singular on the hypersurface S if δ = 0. However, by the analysisof [FG02, §
3] and [FG03, § H Λ ± the Hamiltonian vector fields ofΛ ± ( q, p, δ ) = | p | + λ ± ( q, δ )and consider ( q, p ) ∈ S × R d with p · ∇ V ( q,
0) invertible and δ = 0. Then,lim t → − H Λ + (Φ t + ( q, p )) = lim t → + H Λ − (Φ t − ( q, p ))(14) = p · ∇ q − ∇ q α ( q ) · ∇ p − ∇ q (cid:101) β ( q, · ∇ p =: H, lim t → + H Λ + (Φ t + ( q, p )) = lim t → − H Λ − (Φ t − ( q, p ))(15) = p · ∇ q − ∇ q α ( q ) · ∇ p + ∇ q (cid:101) β ( q, · ∇ p =: H (cid:48) . Moreover, the standard symplectic product of H and H (cid:48) has a sign, since ω ( H, H (cid:48) ) = 2 p · ∇ q (cid:101) β ( q, < S × { δ = 0 } will be a crucial element of our analysis. Remark . For δ = 0, on ingoing trajectories, that is, on trajectories entering theconical crossing, we havedd t g ( q ± δ =0 ( t ) ,
0) = p ± δ =0 ( t ) · ∇ q g ( q ± δ =0 ( t ) , ≤ , which implies that they are included in the set (cid:110)(cid:16) p · ∇ q (cid:101) β ( q, (cid:17) (cid:101) β ( q, ≤ (cid:111) ⊂ (cid:110) (cid:101) β ( q, ≥ (cid:111) . Similarly, outgoing trajectories are included in (cid:110) (cid:101) β ( q, ≤ (cid:111) . .3. The parametrized Schr¨odinger system.
We analyse the time-dependentSchr¨odinger systems (cid:26) iε∂ t ψ εt = − ε ∆ q ψ εt + V ( q, δ ) ψ εt , ( t, q ) ∈ R × R d ,ψ εt =0 = ψ ε , defined by the family of potential matrices V ( q, δ ) = tr V ( q ) + V ( q, δ ) with δ ∈ I of Theorem 3.1 and equation (13).Literally as in Section 2, we construct a surface hopping semigroup ( L tε ) t ≥ forall δ ∈ I , and thus obtain an effective dynamical description comprising both theoriginal avoided crossing at δ = δ and the conical intersection at δ = 0. On theone hand we use classical transport along the flows Φ t ± : R d → R d of˙ q ± δ ( t ) = p ± δ ( t ) , ˙ p ± δ ( t ) = −∇ λ ± ( q ± δ ( t ) , δ ) . On the other hand we monitor the gap function along the classical trajectories anddetect local minima by checking whetherdd t g ( q ± δ ( t ) , δ ) = p ± δ ( t ) · ∇ q g ( q ± δ ( t ) , δ ) = 0 . The corresponding jump manifold readsΣ ε = (cid:8) ( q, p ) ∈ R d | g ( q, δ ) ≤ R √ ε, p · ∇ q g ( q, δ ) = 0 (cid:9) . The non-adiabatic transition probability for ( q ∗ , p ∗ ) ∈ Σ ε is given by(16) T ε ( q ∗ , p ∗ , δ ) = exp (cid:18) − π ε g ( q ∗ , δ ) | det( p ∗ · ∇ q V ( q ∗ , δ )) | / (cid:19) . Finally, the diagonal parts of the Wigner transform W ε ( ψ ) of a wave function ψ ∈ L ( R d , C ) are defined with respect to the eigenprojectors Π ± ( q, δ ), that is, by w ε ± ( ψ )( q, p, δ ) = tr (cid:0) Π ± ( q, δ ) W ε ( ψ )( q, p ) (cid:1) . The semigroup ( L tε ) t ≥ then acts on the function w ε ( ψ ) ∈ B constructed from thediagonal components w ε ± ( ψ ) according to relation (10).The assumptions ( A A A
2) of Theorem 2.3 refer to the potential V ( q ) = V ( q, δ ), and we denote by ( A δ , ( A δ , and ( A δ the corresponding assumptionswith respect to V ( q, δ ). Our aim is to prove the following result: Theorem 3.4.
Let V be a potential matrix with an avoided crossing in the senseof Definition 1.2 and V ( · , δ ) , δ ∈ I , the corresponding parametrized family of The-orem 3.1. We consider the time-dependent Schr¨odinger equation iε∂ t ψ εt = − ε ∆ q ψ εt + V ( q, δ ) ψ εt , ψ εt =0 = ψ ε , and assume that ( A δ , ( A δ and ( A δ hold for all δ ∈ I . Then, for all cut-offfunctions χ ∈ C ∞ c ([0 , T ]) , there exists a constant C > such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T χ ( t ) (cid:0) w ε ( ψ εt ) − L tε w ε ( ψ ε ) , a (cid:1) d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ε / . The constant C depends on a finite number of upper bounds of derivatives of thesmooth functions α, β, γ defining the potential V and a, χ and of lower bounds ofthe determinant of the matrix V . The particular choice δ = δ then implies Theorem 2.3. emark . By the construction of Theorem 3.1, the gap function g ( q, δ ) has | δ | as its minimal value. Hence, if | δ | > R √ ε , then the jump manifold Σ ε is empty.In this situation the semigroup ( L tε ) ε> reduces to mere classical transport, that isproven in Appendix A. Remark . If the off-diagonal function (cid:101) γ had derivatives uniformly bounded withrespect to the gap parameter δ , then Theorem 3.4 would hold with an error ofthe order ε / . We will indicate in Remark 4.5, 4.7 and 5.2 how the analysis wouldsimplify, if uniform estimates were available.4. Reduction to a Landau–Zener model
We now focus on points in the minimal gap hypersurface S and construct asymplectic change of space-time phase space coordinates that allows an elementarymicrolocal normal form reduction to a Landau–Zener model in § § δ is not treated as anothercoordinate but as a controlled parameter.4.1. The new symplectic coordinates.
Following the ideas of [FG02, § S = (cid:8) ( q, t, p, τ ) ∈ R d +2 , q ∈ S , τ + | p | + α ( q ) = 0 (cid:9) , that restricts both the energy shells E + and E − to the conical crossing situationfor ( q, δ ) ∈ S × { δ = 0 } . Proposition 4.1.
Consider ρ = ( q , t , p , τ ) ∈ R d +2 with q ∈ S and p ∈ R d transverse to S at q . There exists a neighborhood Ω ⊂ R d +2 of the point ρ andfor all δ ∈ I a positive function λ ( · , δ ) : Ω → R such that σ ( ρ, δ ) = − λ ( ρ, δ ) (cid:0) τ + | p | + α ( q ) (cid:1) and s ( ρ, δ ) = λ ( ρ, δ ) (cid:101) β ( q, δ ) satisfy { σ ( · , δ ) , s ( · , δ ) } = 1 on Ω , and (17) λ ( ρ, δ ) = − ( p · ∇ q (cid:101) β ( q, δ )) − , ρ ∈ S ∩ Ω . Moreover, all derivatives of λ ( · , δ ) are uniformly bounded with respect to δ ∈ I .Proof. By Lemma 3.2, we have for q ∈ S and p ∈ R d transverse to S at q (cid:110) τ + | p | + α ( q ) , (cid:101) β ( q, δ ) (cid:111) = p · ∇ (cid:101) β ( q, δ ) < . By [Ho, Lemma 21.3.4], we can find Ω and a positive function λ ( · , δ ) such that thefunctions σ ( · , δ ) and s ( · , δ ) satisfy for all δ ∈ I { σ ( · , δ ) , s ( · , δ ) } = 1 on Ω . Indeed, the proof of [Ho, Lemma 21.3.4] relies on solving differential equationsin the variable ( q, p ), which requires to restrict the set Ω . When this is donewith coefficients depending smoothly on δ , for δ in the bounded interval I , therestriction can be taken uniformly in Ω . Therefore, the set Ω does not dependon δ . It remains to compute for ρ ∈ S ∩ Ω ,1 = { σ ( ρ, δ ) , s ( ρ, δ ) } = − λ ( ρ, δ ) p · ∇ q (cid:101) β ( q, δ ) , nd to observe that the derivatives of λ ( · , δ ) inherit the boundedness of the deriva-tives of (cid:101) β ( · , δ ), see Theorem 3.1. (cid:3) We will use this germ of symplectic coordinates and the rotation matrix R ( q )introduced in Theorem 3.1 to construct a normal form. By the Darboux Theorem(see [Ho, Theorem 21.1.6]) close to a point ρ = ( q , t , p , τ ) ∈ R d +2 with q ∈ S and p transverse to S at q , there exists a locally defined canonical transform κ δ : ( s, z, σ, ζ ) (cid:55)→ ( q, t, p, τ )with s, σ ∈ R and ( z, ζ ) ∈ R d , such that(18) ( RP R ∗ ) ◦ κ δ = 1 λ ◦ κ δ (cid:18) − σ + (cid:18) s δ ˇ γδ ˇ γ − s (cid:19)(cid:19) , ˇ γ = ( λ (cid:101) γ ) ◦ κ δ , and the function ˇ γ is nonzero everywhere.This local change of coordinates preserves the symplectic structure of the phasespace R d +1 q,t × R d +1 p,τ : The variables σ and ζ are the dual variables of s and z ,respectively. Besides, in the new variables ( s, z, σ, ζ ), the geometry of the conicalcrossing for δ = 0 is simple, since we have E ± = {− σ ± (cid:112) s + δ ˇ γ = 0 } , S = { s = 0 , σ = 0 } . In particular, by Remark 3.3, the ingoing and outgoing trajectories are included inthe sets { s ≥ } and { s ≤ } , respectively. The off-diagonal function ˇ γ satisfiesadditional properties that will be useful later on: Lemma 4.2.
Let δ ∈ I and ρ ∈ S ∩ Ω . Then, δ (ˇ γ ◦ κ − δ )( ρ ) = g ( q, δ ) (cid:16) | det p · ∇ q V ( q, δ ) | − / + O ( δ ) (cid:17) . Proof.
For q ∈ S we have (cid:101) β ( q, δ ) = 0 and g ( q, δ ) = 4 δ (cid:101) γ ( q ) . We also observethat p · ∇ q (cid:101) γ ( q ) = { | p | , (cid:101) γ } = − λ { σ, (cid:101) γ } − σ { λ , (cid:101) γ } = − λ ∂ s ( (cid:101) γ ◦ κ δ ) ◦ κ − δ for ρ = ( q, t, p, τ ) ∈ S , and that ∂ s ( (cid:101) γ ◦ κ δ ) is a derivative normal to S = { s = 0 } .Therefore, by Theorem 3.1, the product p ·∇ q (cid:101) γ ( q ) is bounded. Using equation (17),we then obtain δ (ˇ γ ◦ κ − δ )( ρ ) = δ λ ( ρ, δ ) (cid:101) γ ( q ) = − g ( q, δ ) ( p · ∇ q (cid:101) β ( q, δ )) − = g ( q, δ ) (cid:18)(cid:16) ( p · ∇ q (cid:101) β ( q, δ )) + δ ( p · ∇ q (cid:101) γ ( q )) (cid:17) − / + O ( δ ) (cid:19) = g ( q, δ ) (cid:16) | det p · ∇ q V ( q, δ ) | − / + O ( δ ) (cid:17) . (cid:3) The quantization of the normal form.
We now lift the classical normalform (18) to the quantum level using Fourier integral operator theory. We definethe matrix-valued symbol B ( q, t, p, τ, δ ) = (cid:112) λ ( q, t, p, τ, δ ) R ( q )for δ ∈ I and ( q, t, p, τ ) ∈ Ω ⊂ R d +2 . The following quantization process retainsthe minimal gap size δ as a controlled parameter. roposition 4.3. Consider ρ = ( q , t , p , τ ) ∈ R d +2 with q ∈ S and p ∈ R d transverse to S at q . Then there exist neighborhoods (cid:101) I ⊂ I of δ = 0 and Ω ⊂ Ω of the point ρ , a matrix-valued function B ε = B + εB defined on Ω , a canonicaltransform κ ε which is a perturbation of order ε of the canonical transform κ δ , anda unitary operator K ε of L ( L ( R d )) such that the transformed solution v ε = K ∗ ε op ε ( B ∗ ε ) − ψ εt satisfies (19) op ε ( ϕ ) op ε (cid:18) − σ + s δ ˇ γδ ˇ γ − σ − s (cid:19) v ε = O ( ε ) , for any compactly supported function ϕ ∈ C ∞ c ( R d +2 ) . Proposition 4.3 allows us to microlocally trade the original Schr¨odinger equa-tion op ε ( P ) ψ εt = 0 for the reduced system (19). When saying that the canonicaltransform κ ε is a perturbation of order ε of κ δ , we mean that κ ε is defined on asubset Ω of the open set where κ δ is defined and that for all a ∈ C ∞ c ( κ − ε (Ω )), a ◦ κ ε = a ◦ κ δ + O ( ε ) with respect to the semi-norms of the derivatives of a .We provide the proof of Proposition 4.3 for the sake of completeness. It relieson the Fourier integral operator construction of [FG02, § Proof.
The first step uses [FG02, § K associated with κ δ such that for any a ∈ C ∞ c (Ω , C × ), K ∗ op ε ( a ) K − op ε ( a ◦ κ δ ) = O ( ε )in L ( L ( R d +1 )), where the O ( ε ) contains semi-norms of derivatives of a . We notethat the the Fourier integral operator is a diagonal operator with the same scalaroperator on each position of the diagonal.The second steps turns to the classical normal form (18) to obtain K ∗ op ε ( BP B ∗ ) K = op ε (cid:18) − σ + s δ ˇ γδ ˇ γ − σ − s (cid:19) + O ( ε ) . This suggests the change of unknown ψ ε (cid:55)→ K ∗ op ε ( B ∗ ) − ψ ε . However, this changeof unknown generates unsatisfactory terms of order ε , since we have by symboliccalculus op ε ( BP B ∗ ) = op ε ( B )op ε ( P )op ε ( B ∗ ) + ε op ε ( R ) + O ( ε ) , where R is the self-adjoint matrix defined by R = 12 i ( B { P, B ∗ } + { B, P } B ∗ ) . More precisely, we have K ∗ op ε ( B )op ε ( P )op ε ( B ∗ ) K = op ε (cid:18) − σ + s δ ˇ γδ ˇ γ − σ − s (cid:19) + ε op ε ( R ◦ κ δ ) + O ( ε ) . We note that here the function δ ˇ γ is treated as a whole. Since it has boundedderivatives, the above remainder is uniform with respect to δ . Next we will removethe term of order ε by modifying B and κ δ at order ε , which will give the statedchange of unknown v ε = K ∗ ε op ε ( B ∗ ε ) − ψ ε . n the rest of this proof, and only here, τ will be a parameter belonging to [0 , κ ( τ ) which is a perturbation of identity andsolves the Hamiltonian equationdd τ κ ( τ ) = H εϕ κ ( τ ) , κ (0) = Id , where ϕ ∈ C ∞ c ( R d +2 ) is a smooth function that we shall fix later on. The canonicaltransform κ is a perturbation of order ε of the identity, so that κ ε := κ δ ◦ κ (1)is the sought-after perturbation of order ε of κ δ . We associate with κ ε ( τ ) a Fourierintegral operator K ε ( τ ) by setting iε dd τ K ε ( τ ) = op ε (1 + εϕ ) K ε ( τ ) , K ε (0) = Id . The solution of our problem will be K ε := K ◦ K ε (1) . Note that this construction method [FG02, § K : Given acanonical transform, one links it to the identity in a differentiable way, therebydefining a function ϕ and the operator K ε ( τ ) as a solution of a differential system.We define the matrix B ε ( τ ) = B + ετ B where B will be fixed later, such B ε := B ε (1) , will be the solution of the Proposition. Let us now investigate how B and ϕ haveto be chosen, which might require to restrict to smaller neighbourhood of δ = 0and ρ . For τ ∈ (0 , L ε ( τ ) := K ε ( τ ) ∗ K ∗ [op ε ( B ε ( τ ))op ε ( P )op ε ( B ε ( τ ) ∗ ) − ε (1 − τ )op ε ( R )] K K ε ( τ ) . We have L ε (0) = op ε (cid:18) − σ + s δ ˇ γδ ˇ γ − σ − s (cid:19) + O ( ε )and we are going to prove that we can find B and ϕ such that dd τ L ε ( τ ) = O ( ε ),so that we shall get L ε (1) = L ε (0) + O ( ε ) which will conclude our proof.A simple computation shows thatdd τ L ε ( τ ) = ε K ε ( τ ) ∗ K ∗ (cid:104) op ε ( B P B ∗ + B P B ∗ ) + op ε ( R )+ 1 i [op ε ( ϕ ) , op ε ( B P B ∗ )] (cid:105) K K ε ( τ ) + O ( ε )= ε K ε ( τ ) ∗ K ∗ op ε [ B P B ∗ + B P B ∗ + R + { ϕ, B P B ∗ } ] K K ε ( τ )+ O ( ε ) . The choice of B and ϕ such that B P B ∗ + B P B ∗ + R + { ϕ, B P B ∗ } = 0 ispossible by [CdV1, Lemma 5]. (cid:3) .3. The off-diagonal components.
Our final step towards the Landau–Zenermodel is to remove the dependence of the off-diagonal function ˇ γ ( s, z, σ, ζ, δ ) on thecoordinates s and σ , following the method proposed in [FG03, Lemma 5 and 6], seealso [FG02, Proposition 8]. From now on, we restrict ourselves to0 < δ ≤ R √ ε, R = R ( ε ) = ε − / , see also Remark 3.5. Moreover, since the scattering result for the Landau–Zenersystem, that we use in Section 5.4.2, has an error estimate of the order R √ ε/s ,we also start focusing on regions, where rR √ ε ≤ | s | ≤ rR √ ε for some 1 (cid:28) r ≤ R , that will be chosen as r = r ( ε ) = ε − / later on. Thesechoices of r and R imply that δ ≤ | s | as soon as ε / ≤ / Lemma 4.4. On κ δ (Ω ) × (cid:101) I , there exist matrix-valued functions M ε ( s, z, σ, ζ, δ ) and (cid:102) M ε ( s, z, σ, ζ, δ ) , such that M ε = M ε + δM ε , (cid:102) M ε = (cid:102) M ε + δ (cid:102) M ε , with M ε and (cid:102) M ε unitary matrices, and for all ϕ ∈ C ∞ c ( R d +2 ) supported in a setwith s = O ( rR √ ε ) , with (cid:28) r ≤ R , one has op ε ( ϕ ) op ε ( (cid:102) M ε ) op ε (cid:18) − σ + s δ ˇ γδ ˇ γ − σ − s (cid:19) =op ε ( ϕ ) op ε (cid:18) − σ + s δ ˇ γ δ ˇ γ − σ − s (cid:19) op ε ( M ε ) + O ( r ε / ) in L ( L ( R d +1 )) , where ˇ γ ( z, ζ, δ ) = ˇ γ (0 , z, , ζ, δ ) . Moreover, the families (op ε ( ϕ ) op ε ( M ε )) ε,δ> and (op ε ( ϕ ) op ε ( (cid:102) M ε )) ε,δ> are uni-formly bounded in L ( L ( R d +1 )) .Remark . The proof below shows that if ˇ γ ( · , δ ) had bounded derivatives uni-formly with respect to δ , then Lemma 4.4 would hold with a remainder estimate ofthe order εδ . Remark . To estimate the norm of operators such as op ε (ˇ γ ), we shall use thescaling operator T ε defined by ∀ f ∈ L ( R d +1 ) , T ε f ( u ) = ε d +14 f ( √ εu ) . This unitary operator is such that ∀ a ∈ C ∞ ( R d +2 ) , T ε op ε ( a ) T ∗ ε = op ( a ( √ ε · , √ ε · )) . The Calder´on–Vaillancourt theorem yields the existence of N ∈ N and C N > ∀ a ∈ C ∞ c ( R d +2 ) , (cid:107) op ε ( a ) (cid:107) L ( L ( R d +1 )) ≤ C N sup β ∈ N d +2 , | β |≤ N sup ρ ∈ R d +2 (cid:16) ε | β | (cid:12)(cid:12) ∂ βρ a ( ρ ) (cid:12)(cid:12)(cid:17) holds. As a consequence, Theorem 3.1 implies for all ϕ ∈ C ∞ c ( R d +2 ) (cid:107) op ε ( ϕ )op ε (ˇ γ ) (cid:107) L ( L ( R d +1 )) ≤ C (1 + √ ε δ − sup s ∈ supp( ϕ ) | s | ) . roof. We consider the three matrices J = (cid:18) − (cid:19) , K = (cid:18) − (cid:19) , L = (cid:18) (cid:19) , that satisfy JL = K = − LJ, JK = L = − KJ, J = Id . We write (cid:18) − σ + s δ ˇ γδ ˇ γ − σ − s (cid:19) = − σ Id + sJ + δ ˇ γL and proceed in two steps.We first remove the s -dependence of ˇ γ and construct matrix-valued functions D εj = D εj ( s, z, σ, ζ, δ ), j = 0 ,
1, withop ε ( D ε + δD ε ) op ε ( − σ Id + sJ + δ ˇ γL )= op ε ( − σ Id + sJ + δ ˇ γ ∗ L ) op ε ( D ε + δD ε ) + o ( ε ) , where ˇ γ ∗ ( z, σ, z, δ ) = ˇ γ (0 , z, σ, ζ, δ ) . Symbolic calculus provides that the above equation is equivalent toop ε (( D ε + δD ε )( − σ Id + sJ + δ ˇ γL )) + ε i op ε ( { D ε + δD ε , − σ Id + sJ + δ ˇ γL } )= op ε (( − σ Id + sJ + δ ˇ γ ∗ L )( D ε + δD ε )) + ε i op ε ( {− σ Id + sJ + δ ˇ γ ∗ L, D ε + δD ε } )+ op ε ( ρ ε ) . where the remainder symbol ρ ε consists of second order derivatives terms times afactor ε . Since any derivatives of δ ˇ γ and δ ˇ γ ∗ are of the order | s | , Remark 4.6 yieldsthat op ε ( ϕ )op ε ( ρ ε ) = O ( | s | · | s | δ ) = O ( r ε / ) , if the a priori estimate(20) ∂ α ( D ε + δD ε ) = O (( | s | δ/ε ) | α | ) , α ∈ N d +2 , holds, that we will justify later during the proof. We neglect the term ε i { D ε , δ ˇ γL } + ε i { δD ε , − σ Id + sJ + δ ˇ γL }− ε i { δ ˇ γ ∗ L, D ε } − ε i {− σ Id + sJ + δ ˇ γ ∗ L, δD ε } , which by the same argument produces an error of the order O ( | s | · | s | δ ) = O ( r ε / )in the region of observation. Then, we obtain the three relations D ε ( − σ Id + sJ ) = ( − σ Id + sJ ) D ε ,D ε ( − σ Id + sJ ) + ˇ γD ε L = ( − σ Id + sJ ) D ε + ˇ γ ∗ LD ε , ˇ γD ε L + ε iδ { D ε , − σ Id + sJ } = ˇ γ ∗ LD ε + ε iδ {− σ Id + sJ, D ε } . We make the ansatz D ε = (cid:18)(cid:101) a ε (cid:101) d ε (cid:19) , D ε = (cid:18) (cid:101) b ε (cid:101) c ε (cid:19) , nd rewrite the second of the three relations as s (cid:18) − (cid:101) b ε (cid:101) c ε (cid:19) = (cid:32) − ˇ γ (cid:101) a ε + ˇ γ ∗ (cid:101) d ε − ˇ γ (cid:101) d ε + ˇ γ ∗ (cid:101) a ε (cid:33) . This requires (cid:101) a ε (0 , z, ζ, δ ) = (cid:101) d ε (0 , z, ζ, δ )and (cid:101) b ε = 12 s (ˇ γ (cid:101) a ε − ˇ γ ∗ (cid:101) d ε ) , (cid:101) c ε = 12 s (ˇ γ ∗ (cid:101) a ε − ˇ γ (cid:101) d ε ) . The third relation can be rewritten as (cid:32) ˇ γ (cid:101) b ε − ˇ γ ∗ (cid:101) c ε
00 ˇ γ (cid:101) c ε − ˇ γ ∗ (cid:101) b ε (cid:33) = εiδ (cid:18) − ∂ s (cid:101) a ε − ∂ σ (cid:101) a ε − ∂ s (cid:101) d ε + ∂ σ (cid:101) d ε (cid:19) . We define (cid:101) ϑ ε ( s, z, σ, ζ, δ ) = iδ εs (cid:0) ˇ γ ∗ ( z, σ, ζ, δ ) − ˇ γ ( s, z, σ, ζ, δ ) (cid:1) = − iδ ε (cid:90) ∂ s (ˇ γ )( sr, z, σ, ζ, δ ) dr and observe that all s -derivatives of (cid:101) ϑ ε are of the order δ /ε , while any derivativewith respect to ( z, σ, ζ ) is of the order δ/ε in view of Theorem 3.1. We obtain theequations ( ∂ s + ∂ σ ) (cid:101) a ε = (cid:101) ϑ ε (cid:101) a ε , ( ∂ s − ∂ σ ) (cid:101) d ε = − (cid:101) ϑ ε (cid:101) d ε , that can be solved by (cid:101) a ε ( s, z, σ, ζ, δ ) = exp (cid:18)(cid:90) s (cid:101) ϑ ε ( τ, z, σ − s + τ, ζ, δ )d τ (cid:19) , (cid:101) d ε ( s, z, σ, ζ, δ ) = exp (cid:18) − (cid:90) s (cid:101) ϑ ε ( τ, z, σ + s − τ, ζ, δ )d τ (cid:19) such that (cid:101) a ε (0 , z, σ, ζ, δ ) = (cid:101) d ε (0 , z, σ, ζ, δ ) = 1. We observe that ∂ αz,σ,ζ (cid:101) a ε , ∂ αz,σ,ζ (cid:101) d ε = O (( | s | δ/ε ) | α | )for any α ∈ N d +1 , while the s -derivatives satisfy ∂ ks (cid:101) a ε , ∂ ks (cid:101) d ε = O ( δ /ε ) + O ( | s | δ/ε )for any k ≥
1. We now write δ (cid:101) b ε = δ (cid:101) a ε (cid:90) ∂ s (ˇ γ )( sr, z, σ, ζ, δ ) dr + δ s γ ∗ ( (cid:101) a ε − d ε )and derive a similar expression for δ (cid:101) c ε , such that ∂ β ( δD ε ) = O (( | s | δ/ε ) | β | ) + O (( δ /ε ) | β | )for all β ∈ N d +2 .This implies the claimed a priori estimate (20).We now remove the σ -dependence of the scalar function ˇ γ ∗ , taking advantage ofthe boundedness of any derivatives of the function ˇ γ ∗ = ˇ γ | { s =0 } , see Theorem 3.1. e look for two matrix-valued functions C εj = C εj ( σ, z, ζ, δ ), j = 0 ,
1, with thefollowing properties. First, they satisfy the intertwining relationop ε ( J ( C ε + δC ε ) J ) op ε ( − σ Id + sJ + δ ˇ γL )= op ε ( − σ Id + sJ + δ ˇ γ ∗ L ) op ε ( C ε + δC ε ) + o ( ε )in L ( L ( R d +1 )), whereˇ γ ( s, z, ζ, δ ) = ˇ γ ∗ ( z, , ζ, δ ) = ˇ γ (0 , z, , ζ, δ ) . This relation is equivalent toop ε ( C ε + δC ε ) op ε ( − σJ + s Id + δ ˇ γ ∗ K )= op ε ( − σJ + s Id + δ ˇ γ K ) op ε ( C ε + δC ε ) + O ( ε δ ) , using the growth bound ∀ α ∈ N d +2 ∃ c α > ∀ ε, δ > (cid:107) ∂ α C εj ( · , δ ) (cid:107) ∞ < c α ( δ /ε ) | α | . Symbolic calculus yieldsop ε (( C ε + δC ε )( − σJ + s Id + δ ˇ γ ∗ K )) + ε i op ε ( { C ε + δC ε , − σJ + s Id + δ ˇ γ ∗ K } )= op ε (( − σJ + s Id + δ ˇ γ K )( C ε + δC ε )) + ε i op ε ( {− σJ + s Id + δ ˇ γ K, C ε + δC ε } )+ O ( ε δ ) , where the neglected terms of the form ε times derivatives of the order ≥ O ( ε δ ) remainder. We now sort in powers of δ and obtain the following threerelations, C ε ( − σJ + s Id) = ( − σJ + s Id) C ε , (21) C ε ( − σJ + s Id) + ˇ γ ∗ C ε K = ( − σJ + s Id) C ε + ˇ γ KC ε , (22) ˇ γ ∗ C ε K + ε iδ { C ε , − σJ + s Id } = ˇ γ KC ε + ε iδ {− σJ + s Id , C ε } . (23)We denote the components of the two matrices by C ε = (cid:18) a ε b ε c ε d ε (cid:19) , C ε = (cid:18) a ε b ε c ε d ε (cid:19) . The first relation (21) is equivalent to [
J, C ε ] = 0, that is, b ε = c ε = 0 . The second relation (22) is equivalent to − σ [ C ε , J ] = − ˇ γ ∗ C ε K + ˇ γ KC ε , that is, σ (cid:18) b ε − c ε (cid:19) = (cid:18) − ˇ γ ∗ a ε + ˇ γ d ε ˇ γ ∗ d ε − ˇ γ a ε (cid:19) . This requires a ε ( z, , ζ, δ ) = d ε ( z, , ζ, δ )and b ε = 12 σ ( − ˇ γ ∗ a ε + ˇ γ d ε ) , c ε = 12 σ ( − ˇ γ ∗ d ε + ˇ γ a ε ) . The third relation (23) is equivalent toˇ γ ∗ C ε K − ˇ γ KC ε = εiδ (cid:0) − ∂ σ C ε − ( J∂ s C ε + ∂ s C ε J ) (cid:1) , hat is, ˇ γ ∗ (cid:18) − b ε a ε − d ε c ε (cid:19) − ˇ γ (cid:18) c ε d ε − a ε − b ε (cid:19) = εiδ (cid:18) − ∂ σ a ε − ∂ σ d ε (cid:19) . This can be satisfied by a ε = d ε = 0 , and requires a ε σ (cid:0) ˇ γ ∗ − ˇ γ (cid:1) = − εiδ ∂ σ a ε ,d ε σ (cid:0) ˇ γ − ˇ γ ∗ (cid:1) = − εiδ ∂ σ d ε . We set ϑ ε ( z, σ, ζ, δ ) = iδ εσ (cid:0) ˇ γ ( z, ζ, δ ) − ˇ γ ∗ ( z, σ, ζ, δ ) (cid:1) and rewrite the above equations as ϑ ε a ε = ∂ σ a ε , ϑ ε d ε = − ∂ σ d ε . The functions a ε ( z, σ, ζ, δ ) = exp (cid:18)(cid:90) σ ϑ ε ( z, τ, ζ, δ ) d τ (cid:19) ,d ε ( z, σ, ζ, δ ) = exp (cid:18)(cid:90) σ ϑ ε ( z, τ, ζ, δ ) d τ (cid:19) solve these equations and satisfy a ε ( z, , ζ, δ ) = d ε ( z, , ζ, δ ) = 1 . We conclude that the constructed matrices C ε and C ε have the desired properties. (cid:3) Arriving at the Landau–Zener model.
We now use Proposition 4.3 andLemma 4.4 to introduce (cid:101) v ε = op ε ( M ε ) v ε with M ε = M ε + δM ε . Sinceop ε (cid:18) − σ + s δ ˇ γ δ ˇ γ − σ − s (cid:19) (cid:101) v ε = op ε ( (cid:102) M ε ) (cid:18) − σ + s δ ˇ γδ ˇ γ − σ − s (cid:19) v ε + O ( r ε / ) , we obtain for all ϕ ∈ C ∞ c ( R d +2 ) the doubly reduced system(24) op ε ( ϕ ) op ε (cid:18) − σ + s δ ˇ γ δ ˇ γ − σ − s (cid:19) (cid:101) v ε = O ( r ε / ) in L ( R d +1 ) . The estimate of [FG02, Proposition 7] also implies, that (op ε ( ϕ ) (cid:101) v ε ) ε> is a boundedsequence in L ∞ ( R s , L ( R dz )), and we compare the new function (cid:101) v ε with the solu-tion ˇ v ε of the Landau–Zener type system(25) εi ∂ s ˇ v ε = op ε (cid:18) s δ ˇ γ δ ˇ γ − s (cid:19) ˇ v ε , ˇ v ε | s =0 = (cid:101) v ε | s =0 . The order r ε / right hand side of the doubly reduced system (24) is small enoughto be treated as a perturbation and we obtain a positive constant C > s ∈ R and ε > (cid:107) (cid:101) v ε ( s ) − ˇ v ε ( s ) (cid:107) L ( R dz ) ≤ C | s | r ε − / . his implies that for all ϕ ∈ C ∞ c ( R d +2 ) that are supported in a region, where s ∼ rR √ ε = rε / , we obtain(27) op ε ( ϕ ) ( (cid:101) v ε − ˇ v ε ) = O ( r ε / ) in L ( R d +1 s,z ) , and we have established the microlocal link of the original Schr¨odinger equationop ε ( P ) ψ εt = 0to the Landau–Zener system (25), provided we choose r ≤ ε − κ with 0 < κ < / Remark . If ˇ γ ( · , δ ) had uniformly bounded derivatives, then, in view of Re-mark 4.5, the term | s | r ε − / in (26) would be replaced by δ | s | . For boundedtimes, this remainder would be of order ε / .5. The proof of the main result
For proving our main result Theorem 2.3, we now analyse the dynamics of theblock diagonal components of the Wigner transform w ε ± ( ψ εt ), that is, (cid:90) R d +1 χ ( t ) a ( q, p ) w ε ± ( ψ εt )( q, p, δ ) d( q, p, t ) = (cid:10) op ε ( χa Π ± ) ψ εt , ψ εt (cid:11) for scalar observables a ∈ C ∞ c ( R d ) with supp( a ) ⊂ Ω and χ ∈ C ∞ c ([0 , T ]). Bythe assumption (A1) δ , the observables have support away from the set of smalleigenvalue gap, providing the derivative bounds(28) ∀ α, β ∈ N d ∃ c α,β > ∀ ε, δ > (cid:13)(cid:13) ∂ βq ∂ αp ( a Π ± ( · , δ )) (cid:13)(cid:13) ∞ < c α,β ( R √ ε ) −| β | . In the following, the scale R √ ε will play an important role. We note that, ofcourse, ( R √ ε ) −| β | ≤ ( R √ ε ) −| β | for β ∈ N d . Our first step in the proof is now thereplacement of the eigenprojectors Π ± ( q, δ ) in op ε ( χa Π ± ) ψ εt by the localisation onthe corresponding energy shell E ± = (cid:8) ( q, t, p, τ ) ∈ R d +2 , τ + Λ ± ( q, p, δ ) = 0 (cid:9) , that is a subset of the space-time phase space.5.1. Localization in energy.
The localization is implemented by a smooth cut-offfunction θ ∈ C ∞ c ( R ) satisfying0 ≤ θ ( u ) ≤ , θ ( u ) = 0 for | u | > , θ ( u ) = 1 for | u | < / . We combine it with the energy function and set θ ± ε,R ( q, p, τ, δ ) = θ (cid:18) τ + Λ ± ( q, p, δ ) R √ ε (cid:19) , ( q, p, τ ) ∈ R d +1 . Lemma 5.1.
For all symbols χ ∈ C ∞ c ( R ) and a ∈ C ∞ c ( R d ) satisfying the derivativebounds (28) , we have op ε (cid:0) χa Π ± (cid:1) ψ εt = op ε (cid:16) χaθ ± ε,R (cid:17) ψ εt + O ( R − ) + O ( R − √ ε ) in L ( R d +1 t,q ) . roof. Following the lines of the proof of [FL08, Lemma 5.1], we observe that, since1 − θ vanishes identically close to 0, we can write1 − θ ± ε,R ( q, p, τ, δ ) = τ + Λ ± ( q, p, δ ) R √ ε G (cid:18) τ + Λ ± ( q, p, δ ) R √ ε (cid:19) for some smooth function G . Since( τ + Λ ± ( q, p, δ ))Π ± ( q, δ ) = Π ± ( q, δ ) P ( q, p, τ, δ ) , we have (1 − θ ± ε,R )Π ± = 1 R √ ε G (cid:18) τ + Λ ± R √ ε (cid:19) Π ± P. We can use symbolic calculus to bring into play that ψ εt solves the Schr¨odingerequation op ε ( P ) ψ εt = 0. The derivative bounds (28) imply thatop ε (cid:16) χa (1 − θ ± ε,R )Π ± (cid:17) ψ εt = O ( R − ) + O ( R − √ ε ) in L ( R d +1 t,q ) . Now it remains to remove the matrix Π ± ( q, δ ) from the right hand side of theequation op ε (cid:0) χa Π ± (cid:1) ψ εt = op ε (cid:16) χaθ ± ε,R Π ± (cid:17) ψ εt + O ( R − ) + O ( R − √ ε ) . In view of χaθ ± ε,R = χaθ ± ε,R Π ± + χaθ ± ε,R Π ∓ , we only need to prove that(29) op ε ( χaθ ± ε,R Π ∓ ) ψ εt = O ( R − ) + O ( R − √ ε ) in L ( R d +1 t,q ) . We observe that, for ε small enough, θ ± ε,R = θ ± ε,R (1 − θ ∓ ε,R ) , since (cid:12)(cid:12) τ + | p | + α ( q ) ± g ( q, δ ) (cid:12)(cid:12) ≤ R √ ε on the support of θ ± ε,R . Using again the Schr¨odinger equation, symbolic calculusand the estimate (28), we obtain the desired relation (29). (cid:3) We now reconsider the Landau–Zener transformation of Section 4 in terms ofexpectation values. By Lemma 5.1, (cid:10) op ε ( χa Π ± ) ψ εt , ψ εt (cid:11) = (cid:10) op ε ( χaθ ± ε,R ) ψ εt , ψ εt (cid:11) + O ( R − ) + O ( R − √ ε ) . Next, we rewrite the expectation value using the function v ε = K ∗ ε op ε ( B ∗ ε ) − ψ εt that has been introduced in Proposition 4.3. Since B ε = B + εB with B ∗ B = λ ,symbolic calculus implies (cid:10) op ε ( χa Π ± ) ψ εt , ψ εt (cid:11) = (cid:68) op ε ( χaλθ ± ε,R )op ε ( B ∗ ε ) − ψ εt , op ε ( B ∗ ε ) − ψ εt (cid:69) + O ( R − ) + O ( R − √ ε ) . In the presence of the symbol θ ± ε,R that loses a factor R √ ε per derivative, theapplication of the Fourier integral operator K ε yields (cid:10) op ε ( χa Π ± ) ψ εt , ψ εt (cid:11) = (cid:10) op ε (cid:0) ( χaλθ ± ε,R ) ◦ κ δ (cid:1) v ε , v ε (cid:11) + O ( R − ) + O ( R − √ ε ) . n the next step, we move towards (cid:101) v ε = op ε ( M ε + δM ε ) v ε . Since the matrix M ε is unitary and | δ | ≤ R √ ε , we have (cid:10) op ε ( χa Π ± ) ψ εt , ψ εt (cid:11) = (cid:10) op ε (cid:0) ( χaλθ ± ε,R ) ◦ κ δ (cid:1)(cid:101) v ε , (cid:101) v ε (cid:11) + O ( R − ) + O ( R √ ε ) . We then arrive at the solution ˇ v ε of the Landau–Zener system (25) by (cid:10) op ε ( χa Π ± ) ψ εt , ψ εt (cid:11) = (cid:10) op ε (( χaλθ ± ε,R ) ◦ κ δ )ˇ v ε , ˇ v ε (cid:11) + O ( R − ) + O ( R √ ε ) + O ( r ε / ) . (30)At this stage of the proof of Theorem 2.3, we have rewritten the expectation valuesfor the Schr¨odinger solution ψ εt in terms of the Landau–Zener solution ˇ v ε . Next,we reformulate the Markov process in the new coordinates.5.2. The Markov process for the normal form.
Let us now introduce a newMarkov process for effectively describing the dynamics of the reduced Landau–Zenerproblem (25). We use the analogous building blocks as for the original process thatdefines the semigroup ( L tε ) t ≥ . We shall prove that this new process is close to theimage of the original Markov process by the canonical transform κ δ .5.2.1. The image of the classical trajectories by the canonical transform.
We ob-serve that by the transformation in (18), the eigenvalues of the reduced system (19)satisfy − σ + j (cid:112) s + δ ˇ γ ( s, z, σ, ζ ) = (cid:0) λ (cid:0) τ + Λ ∓ ( q, p, δ ) (cid:1)(cid:1) ◦ κ δ , where the sign j = ± § H − σ ± √ s + δ ˇ γ ( s,z,σ,ζ ) are mapped by the canonical transform κ δ to those of H λ ( τ +Λ ∓ ( q,p,δ )) = λH τ +Λ ∓ ( q,p,δ ) + (cid:0) τ + Λ ∓ ( q, p, δ ) (cid:1) H λ . Since the energy τ + Λ ∓ ( q, p, δ ) is of order R √ ε in our zone of observation, thesetrajectories are those of λH τ +Λ ∓ ( q,p,δ ) up to some term of order R √ ε . Sincethe function λ does not vanish in our zone of observation, the integral curvesof λH τ +Λ ∓ ( q,p,δ ) are those of H τ +Λ ∓ ( q,p,δ ) up to a change of parametrization ofthe curve. We therefore consider the image of the Hamiltonian curves of the ini-tial Markov process by the canonical transform κ δ as being close to those of thefunctions (cid:101) Λ ± ( s, z, σ, ζ ) = − σ ∓ (cid:112) s + δ ˇ γ ( s, z, σ, ζ ) . The relation of the energies.
For discussing the relation of the different en-ergies occuring in our analysis, we also introduce the functions(31) (cid:101) Λ ± ( s, z, σ, ζ ) = − σ ∓ (cid:112) s + δ ˇ γ ( z, ζ ) that belong to the Landau–Zener system (25). We observe that both energies (cid:101) Λ ± and (cid:101) Λ ± satisfy (cid:101) Λ ± , (cid:101) Λ ± = − σ ∓ | s | + O ( δ | s | − ) , where we have used (cid:101) Λ ± ( s, z, σ, ζ ) ± | s | = ∓ δ ˇ γ ( z, ζ ) | s | + (cid:112) s + δ ˇ γ ( z, ζ ) nd a similar relation for (cid:101) Λ ± . Therefore, since δ ≤ R √ ε , we obtain (cid:101) Λ ± − (cid:101) Λ ± = O ( R ε | s | − )and for any smooth cut-off function (cid:101) θ ∈ C ∞ c ( R ) (cid:101) θ (cid:32) (cid:101) Λ ± ( s, z, σ, ζ ) R √ ε (cid:33) = (cid:101) θ (cid:32) (cid:101) Λ ± ( s, z, σ, ζ ) R √ ε (cid:33) + O ( R √ ε | s | − ) . Hence a change in the energy localisation from (cid:101) Λ ± to (cid:101) Λ ± causes a deviation of theorder O ( R − r − ) = O ( R − ), when choosing s ∼ rR √ ε provided r (cid:29) The numbering of the eigenvalues.
The classical trajectories of the Landau–Zener system (25) are generated by the eigenvalues − σ ± (cid:112) s + δ ˇ γ ( z, ζ ) . For enumerating these eigenvalues such that the classical trajectories in in theoriginal and the new coordinates can be naturally linked, we consider the case δ = 0 and use the vectors H and H (cid:48) that have been defined in (14) and (15).We recall that H and H (cid:48) are associated with ingoing trajectories for Λ + and Λ − ,respectively, and satisfy ω ( H, H (cid:48) ) <
0. Up to some perturbation term of order R √ ε ,the canonical transformation κ − sends H and H (cid:48) on vectors that are collinear to − ∂ s − ∂ σ and − ∂ s + ∂ σ above the singular set { s = 0 } . Since ω ( − ∂ s − ∂ σ , − ∂ s + ∂ σ ) = 2 > , the vector H is collinear to − ∂ s + ∂ σ and H (cid:48) to − ∂ s − ∂ σ above { s = 0 } . Since − ∂ s + ∂ σ = H − σ −| s | , − ∂ s − ∂ σ = H − σ + | s | on { s > } and s > H − σ −| s | corresponds to theplus mode, while H − σ + | s | belongs to the minus mode. We therefore number theeigenvalues in the new coordinates according to (31).5.2.4. The Hamiltonian trajectories.
The eigenvalues (cid:101) Λ ± generate the Hamiltoniansystems˙ s = − , ˙ z = ∓ δ ˇ γ ∂ ζ ˇ γ (cid:112) s + δ ˇ γ , ˙ σ = ± s (cid:112) s + δ ˇ γ , ˙ ζ = ± δ ˇ γ ∂ z ˇ γ (cid:112) s + δ ˇ γ , with corresponding flow maps (cid:101) Φ (cid:105) , ± : R d +2 → R d +2 . Using that | ˙ z | + | ˙ ζ | = O ( δ | s | − ) and in view of (31), we have(32) (cid:101) Φ (cid:105) , ± ( s, z, σ, ζ ) = ( s − (cid:105) , z, ∓| s − (cid:105) | , ζ ) + O ( δ | s | − )for all points ( s, z, σ, ζ ) in our zone of observation and propagation times (cid:105) > s ∼ rR √ ε , we obtain an error of the order R − r − , which is smallerthan R − . .2.5. The non-adiabatic transitions.
Monitoring the gap function (cid:101) g ( s, z, ζ ) = 2 (cid:112) s + δ ˇ γ ( z, ζ ) along the Hamiltonian trajectories associated with (cid:101) Λ ± we look for points in R d +2 ,where a local minimum is attained. We obtain the condition0 = ∂ s (cid:101) g ∂ σ (cid:101) Λ ± + ∇ z (cid:101) g · ∇ ζ (cid:101) Λ ± − ∂ σ (cid:101) g ∂ s (cid:101) Λ ± − ∇ ζ (cid:101) g · ∇ z (cid:101) Λ ± = − ∂ s (cid:101) g , that is equivalent to s = 0. Hence, the new jump manifold is the set (cid:101) Σ ε = (cid:8) ( s, z, σ, ζ ) ∈ R d +2 | s = 0 , δ | ˇ γ ( z, ζ ) | ≤ R √ ε (cid:9) . The Landau–Zener formula of [FG03, Proposition 7], see § (cid:101) T ε ( z, ζ ) := exp (cid:16) − πε δ ˇ γ ( z, ζ ) (cid:17) when reaching the jump set (cid:101) Σ ε . By the energy localization of our observables, wehave s = 0 and σ = O ( R √ ε ) on the jump manifold (cid:101) Σ ε . By Theorem 3.1, a Taylorexpansion around σ = 0 readsˇ γ (0 , z, σ, ζ ) = ˇ γ ( z, ζ ) + σ∂ σ ˇ γ (0 , z, , ζ ) + O ( σ /δ ) . Using that δ/δ is bounded, we obtain δ ε (ˇ γ (0 , z, σ, ζ ) − ˇ γ ( z, σ )) = O ( R √ ε )and (cid:101) T ε ( z, ζ ) = exp (cid:16) − πε δ ˇ γ ( s, z, σ, ζ ) (cid:17) + O ( R √ ε ) . Lemma 4.2 then provides(33) (cid:101) T ε = T ε ◦ κ δ + O ( R √ ε ) , where the original transition rate T ε ( q, p, δ ) has been defined in (16). Besides, theoriginal jump condition p · ∇ q g ( q, δ ) = 0 is equivalent to (cid:101) β ( q, δ )( p · ∇ q (cid:101) β ( q, δ )) + δ (cid:101) γ ( q )( p · ∇ q (cid:101) γ ( q )) = 0 , that is, s = O ( δ ). Hence, the non-adiabatic transitions of the new and the originalMarkov process mostly differ due to the transition rate estimate (33).5.2.6. The drift. At s = 0, we observe for all δ ≥ (cid:101) Λ ± (0 , z, σ ∓ δ | ˇ γ ( z, ζ ) | , ζ ) = (cid:101) Λ ∓ (0 , z, σ, ζ ) . The best drift is, of course, the exact one and is given by (cid:101) J ± : σ (cid:55)→ σ ∓ δ | ˇ γ ( z, ζ ) | . This exact drift is performed in the direction of ∂ σ that is collinear to the differenceof the two Hamiltonian vector fields in the particular case δ = 0, motivating thegeometric underpinning of the original drift construction, see Remark 2.1. We alsonote that the size of the drift 2 δ | ˇ γ ( z, ζ ) | is precisely the gap size for points in thejump manifold. .3. The semigroup for the normal form.
The Markov process described inthe previous section § (cid:101) L ε acting on functions in the space (cid:101) B = (cid:8) f : R d +2 × {− , } → C | f is measurable, bounded (cid:9) . Following the normal form transformation of the expectation values given in (30),we consider a symbol c outε,R ∈ (cid:101) B whose plus-minus-components are defined by c ± ,outε,R ( s, z, σ, ζ ) = b ± ( s, z, σ, ζ ) (cid:101) θ (cid:32) (cid:101) Λ ± ( s, z, σ, ζ ) R √ ε (cid:33) , where the functions b ± and (cid:101) θ have the following properties: b ± ( s, z, σ, ζ ) are twosmooth functions compactly supported in the outgoing region { s < } such thatthe random trajectories reaching their support have only one transition duringthe observation time that is of length (cid:105) >
0. The functions b ± ( s + (cid:105) , z, σ, ζ ) aresupported in the incoming region { s > } . The cut-off function (cid:101) θ ∈ C ∞ c ( R ) satisfies0 ≤ (cid:101) θ ≤ , (cid:101) θ ( u ) = 0 for | u | > , (cid:101) θ ( u ) = 1 for | u | < / . We now analyse the pull back of the symbol c outε,R ∈ (cid:101) B by the semigroup for a suitablychosen time (cid:105) > c inε,R := (cid:101) L (cid:105) ε c outε,R . By Assumption ( A δ of § c + ,inε,R by c + ,inε,R, + and c + ,inε,R, − , where the subscript depends onthe outgoing mode. Our aim is to relate (cid:68) op ε ( c ± ,outε,R )ˇ v ε , ˇ v ε (cid:69) with (cid:68) op ε ( c + ,inε,R, ± )ˇ v ε , ˇ v ε (cid:69) . If the incoming trajectories are associated with the minus mode, the arguments areanalogous.5.3.1.
Transport without transitions.
The component c + ,inε,R, + takes into account theclassical transport along the plus trajectories and the probability of staying on thesame mode. By conservation of energy along classical trajectories, we have (cid:101) Λ +0 ( (cid:101) Φ − (cid:105) , + ( s − (cid:105) , z, σ, ζ )) = (cid:101) Λ +0 ( s − (cid:105) , z, σ, ζ ) . By (32), we therefore deduce c + ,inε,R, + ( s − (cid:105) , z, σ, ζ ) = (cid:16) − (cid:101) T ε ( z, ζ ) (cid:17) ( c + ,outε,R ◦ (cid:101) Φ − (cid:105) , + )( s − (cid:105) , z, σ, ζ )= (cid:16) − (cid:101) T ε ( z, ζ ) (cid:17) b + ( s, z, −| s | , ζ ) (cid:101) θ (cid:32) (cid:101) Λ +0 ( s − (cid:105) , z, σ, ζ ) R √ ε (cid:33) + O ( R √ ε ) , so that c + ,inε,R, + ( s, z, σ, ζ )= (cid:16) − (cid:101) T ε ( z, ζ ) (cid:17) b + ( s + (cid:105) , z, −| s + (cid:105) | , ζ ) (cid:101) θ (cid:32) (cid:101) Λ +0 ( s, z, σ, ζ ) R √ ε (cid:33) + O ( R √ ε ) . .3.2. Transport and transitions with drift.
The component c + ,inε,R, − is more intricate,since it incorporates classical transport through both modes, application of thetransfer coefficient and of the drift. Indeed, the branches of minus trajectories whichreach the support of c − ,outε,R result from plus trajectories that have been drifted. Moreprecisely, we have( (cid:101) Φ s , − ◦ (cid:101) J + ◦ (cid:101) Φ − s − (cid:105) , + )( s − (cid:105) , z, σ, ζ ) = ( s, z, | s | , ζ ) + O ( R √ ε ) . for ( s, z, σ, ζ ) ∈ supp( c − ,outε,R ). By conservation of energy and the drift relation (34)we have (cid:101) Λ − (cid:16)(cid:101) Φ s , − ◦ (cid:101) J + ◦ (cid:101) Φ − s − (cid:105) , + ( s − (cid:105) , z, σ, ζ ) (cid:17) = (cid:101) Λ +0 ( s − (cid:105) , z, σ, ζ ) . Applying the transfer coefficient, we obtain c + ,inε,R, − ( s − (cid:105) , z, σ, ζ ) = (cid:101) T ε ( z, ζ ) ( c − ,outε,R ◦ (cid:101) Φ s , − ◦ (cid:101) J + ◦ (cid:101) Φ − s − (cid:105) , + )( s − (cid:105) , z, σ, ζ )= (cid:101) T ε ( z, ζ ) b − ( s, z, | s | , ζ ) (cid:101) θ (cid:32) (cid:101) Λ +0 ( s − (cid:105) , z, σ, ζ ) R √ ε (cid:33) + O ( R √ ε ) , that is, c + ,inε,R, − ( s, z, σ, ζ ) = (cid:101) T ε ( z, ζ ) b − ( s + (cid:105) , z, | s + (cid:105) | , ζ ) (cid:101) θ (cid:32) (cid:101) Λ +0 ( s, z, σ, ζ ) R √ ε (cid:33) + O ( R √ ε ) . The transitions.
We now prove that the semigroup (cid:101) L ε effectively describesthe dynamics of the Landau–Zener system (25) in the sense that(35) (cid:68) op ε ( c ± ,outε,R )ˇ v ε , ˇ v ε (cid:69) = (cid:68) op ε ( c + ,inε,R, ± )ˇ v ε , ˇ v ε (cid:69) + O ( η ε ) , where the error term is obtained as O ( η ε ) = O ( r − ) + O ( R − ) + O ( εR ln( rR )) . By Theorem 3.1 the function ˇ γ is a smooth function with bounded derivatives.Thus, we can follow the argumentation developed in [FL08, § Using energy localization.
We work with the eigenprojectors (cid:101) Π ± ( s, z, ζ ) = 12 (cid:32) Id ∓ (cid:112) s + δ ˇ γ ( z, ζ ) (cid:18) s δ ˇ γ ( z, ζ ) δ ˇ γ ( z, ζ ) − s (cid:19)(cid:33) of the Landau–Zener system, numbered consistently with the eigenvalues in (31).We observe that for | s | ∼ rR √ ε (cid:101) Π +0 ( s, z, ζ ) = (cid:18) (cid:19) + O ( R − ) in { s > } , (cid:101) Π +0 ( s, z, ζ ) = (cid:18) (cid:19) + O ( R − ) in { s < } , (36)and obtain similar asymptotics for (cid:101) Π − since Id = (cid:101) Π +0 + (cid:101) Π − . By Lemma 5.1, wethen have (cid:68) op ε ( c ± ,outε,R )ˇ v ε , ˇ v ε (cid:69) = (cid:68) op ε ( b ± (cid:101) Π ± )ˇ v ε , ˇ v ε (cid:69) + O ( R − ) + O ( R − √ ε ) , nd consequently, (cid:68) op ε ( c + ,outε,R )ˇ v ε , ˇ v ε (cid:69) = (cid:68) op ε ( b + ,out )ˇ v ε , ˇ v ε (cid:69) + O ( R − ) , (cid:68) op ε ( c − ,outε,R )ˇ v ε , ˇ v ε (cid:69) = (cid:68) op ε ( b − ,out )ˇ v ε , ˇ v ε (cid:69) + O ( R − )with b ± ,out ( s, z, σ, ζ ) = b ± ( s, z, σ, ζ )supported in the outgoing region {− rR √ ε ≤ s ≤ − r R √ ε } . Similarly, we have (cid:68) op ε ( c + ,inε,R, + )ˇ v ε , ˇ v ε (cid:69) = (cid:68) op ε (cid:16) (1 − (cid:101) T ε ) b + ,in (cid:17) ˇ v ε , ˇ v ε (cid:69) + O ( R − ) + O ( R − √ ε ) , (cid:68) op ε ( c + ,inε,R, − )ˇ v ε , ˇ v ε (cid:69) = (cid:68) op ε (cid:16) (cid:101) T ε b − ,in (cid:17) ˇ v ε , ˇ v ε (cid:69) + O ( R − ) + O ( R − √ ε )with b ± ,in ( s, z, σ, ζ ) = b ± ( s + (cid:105) , z, ∓| s + (cid:105) | , ζ ) . supported in the incoming region { r R √ ε ≤ s ≤ rR √ ε } .5.4.2. The Landau–Zener formula.
Following [FG03, Proposition 7], we rewrite theLandau–Zener system (25) as(37) εi ∂ s ˇ v ε = (cid:18) s √ εG √ εG ∗ − s (cid:19) ˇ v ε with G = δ √ ε op ε (ˇ γ ( z, ζ )) . Then there exist two vector-valued functions k ε, ± ∈ L ( R d , C ) such that for anycut-off function χ ∈ C ∞ c ([0 , R ]) and for ± s > χ ( GG ∗ )ˇ v ε ( z, s ) = χ ( GG ∗ )e is / (2 ε ) (cid:12)(cid:12)(cid:12) s √ ε (cid:12)(cid:12)(cid:12) i GG ∗ k ε, ± ( z ) + O ( R √ ε/s ) ,χ ( G ∗ G )ˇ v ε ( z, s ) = χ ( G ∗ G )e − is / (2 ε ) (cid:12)(cid:12)(cid:12) s √ ε (cid:12)(cid:12)(cid:12) − i G ∗ G k ε, ± ( z ) + O ( R √ ε/s ) , where k ε, + = S ε k ε, − with S ε = (cid:18) a ( GG ∗ ) − b ( GG ∗ ) Gb ( G ∗ G ) G ∗ a ( G ∗ G ) (cid:19) . The functions defining the scattering matrix satisfy a ( λ ) = e − πλ/ , a ( λ ) + λ | b ( λ ) | = 1 , λ ∈ R . Moreover, the asymptotics of [FG03, Lemma 8 & 9] provide for any smooth andcompactly supported symbol φ ∈ C ∞ c ( R d +2 )(38) (cid:12)(cid:12)(cid:12) s √ ε (cid:12)(cid:12)(cid:12) ± i G ∗ G op ε ( φ ) (cid:12)(cid:12)(cid:12) s √ ε (cid:12)(cid:12)(cid:12) ∓ i G ∗ G = op ε ( φ ) + O ( R ε | ln( s/ √ ε ) | ) . These asymptotics yield an error of the order 1 /r , which motivates to choose r = r ( ε ) = ε − / , so that the error r ε / in equations (27) and (30) is of the same size as 1 /r . Remark . If (cid:101) γ ( · , δ ) had uniformly bounded derivatives, then, in view of Re-marks 4.5 and 4.7, we could choose r = R and obtain an overall remainder of theorder 1 /R = ε / . .4.3. Applying the Landau–Zener formula.
Using the Landau–Zener formalism de-scribed in § (cid:68) op ε ( b + ,out )ˇ v ε , ˇ v ε (cid:69) = (cid:68) op ε (cid:16) (1 − (cid:101) T ε ) b + ,in (cid:17) ˇ v ε , ˇ v ε (cid:69) + O ( η ε ) , since the proof for the second estimate in (35) is analogous.We first use the relation between ˇ v ε and k ε, − on the outgoing region { s < } for s ≤ − rR √ ε to obtain (cid:10) op ε ( b + ,out )ˇ v ε , ˇ v ε (cid:11) = (cid:10) op ε ( b + ,out ) k ε, − , k ε, − (cid:11) + O ( r − ) + O ( εR ln( rR )) . Then, we perform the change of variable s (cid:55)→ s − (cid:105) , (cid:10) op ε ( b + ,out )ˇ v ε , ˇ v ε (cid:11) = (cid:10) op ε ( b + ,in ) k ε, − , k ε, − (cid:11) + O ( r − ) + O ( εR ln( rR )) . Since we have assumed that the incoming minus contributions are negligibly small,we neglect the scattering contribution from ˇ v ε and consequently from k ε, +1 . Wetherefore deduce from the scattering relation k ε, − = S ∗ ε k ε, + that k ε, − = − G b ( GG ∗ ) k ε, +2 + O ( η ε )and (cid:68) op ε ( b + ,out )ˇ v ε , ˇ v ε (cid:69) = (cid:68) op ε (cid:16) (1 − (cid:101) T ε ) b + ,in (cid:17) k ε, +2 , k ε, +2 (cid:69) + O ( η ε )= (cid:68) op ε (cid:16) (1 − (cid:101) T ε ) b + ,in (cid:17) ˇ v ε , ˇ v ε (cid:69) + O ( η ε ) . Numerical simulations
We consider four specific examples of avoided crossings in one space dimension.The corresponding eigenvalues surfaces are plotted in Figure 1, while the detaileddefinition of the four model systems is given in Tables 1 and 2. Three examples aretaken from Tully’s 1994 paper [Tu1] on the surface hopping algorithm of the fewestswitches: the simple, the dual and the extended crossing. The arctangent crossingis included as an example, which meets the assumptions of our main Theorem 2.3.Its eigenvalues are defined by smooth functions, and the coefficients except forthe minimum gap parameter δ are of order one with respect to the semiclassicalparameter ε . In all simulations, the initial data ψ ( q ) = ( πε ) − / exp (cid:0) − ε ( q − q ) + iε p ( q − q ) (cid:1) e ± ( q )are multiples of a Gaussian wave packet with phase space centers ( q , p ) ∈ R anda real-valued eigenvector e ± ( q ) of the matrix V ( q ). Following [Tu1], the three Tullyexamples have the semiclassical parameter ε = 1 / √ ≈ . , which roughly corresponds to the mass of the hydrogen atom of 1836 atomic units.For the arctangent crossing we have chosen ε = 10 − . The time interval [0 , t fin ] ofall simulations allows that the wave packet passes the crossing region once. Simple −4 −2 0 2 4−101
Arctangent −10 −5 0 5 10−0.06−0.04−0.0200.020.04
Dual −10 −5 0 5 10−0.2−0.100.10.2
Extended
Figure 1.
The eigenvalue surfaces of the potentials considered forour numerical simulations. Simple Arctangent ε − / − δ .
005 10 − / initial level minus plus( q , p ) ( − ,
1) ( − , t fin
10 2 β ( q ) 0 .
01 sgn( q ) (cid:0) − e − . | q | (cid:1) arctan( q ) γ ( q ) δ e − q δ α ( q ) 0 0 Table 1.
Functions and parameters defining the simple and thearctangent crossings. In both cases, the eigenvalue surfaces havetheir minimal gap at q = 0.6.1. A surface hopping algorithm.
Our analysis of the dynamics through anavoided eigenvalue crossing suggests a surface hopping algorithm formulated interms of Wigner functions. Such an algorithm can either treat the effective Landau-Zener transitions by a deterministic branching scheme or by a probabilistic accept-reject mechanism. The probabilistic version, which will be discussed here, keepsthe number of trajectories constant, which is to the best advantage for the memoryrequirements of the algorithm, see the simulations for a model of pyrazine and ofthe ammonia cation [LS, BDLT]. ual Extended ε − / − / δ .
015 6 · − initial level minus plus( q , p ) ( − ,
1) (0 , − t fin
10 10 β ( q ) 0 . − . q − . δ γ ( q ) δ e − . q . q )(1 − e − . | q | ) + 0 . α ( q ) − β ( q ) 0 Table 2.
Functions and parameters defining the dual and the ex-tended crossing. The dual crossing surfaces have their minimal gapat q ≈ ± .
6. The surface gap of the extended crossing decreasesmonotonically as q → −∞ .For notational simplicity, we restrict ourselves to the case, that the initial dataare associated with the upper level. The same reasoning applies for initial dataassociated to the lower level with the obvious alterations. The probabilistic surfacehopping algorithm works as follows:6.1.1. Initial sampling.
Draw N ∈ N pseudorandom phase space samples( q , p ) + , . . . , ( q N , p N ) + , which are independent and identically distributed according to w ε + ( ψ ). If theinitial data are a Gaussian wave packet, then the Wigner function is given by theexplicit formula w ε + ( ψ )( q, p ) = ( πε ) − exp (cid:0) − ε | ( q, p ) − ( q , p ) | (cid:1) , that is the densitiy function of a bivariate normal distribution.6.1.2. Transport.
Propagate the sample points along the Hamiltonian curves of Φ t + .6.1.3. Non-adiabatic transitions.
If a trajectory t (cid:55)→ ( q + j ( t ) , p + j ( t )) attains a localminimal gap of size smaller than √ ε at time t ∗ in the phase space point ( q ∗ , p ∗ ), thendraw a pseudrandom number ζ uniformly distributed in [0 , T ε ( q ∗ , p ∗ ) > ζ ,then a hop occurs according to( q ∗ , p ∗ , +) −→ ( q ∗ , p ∗ + ω ( q ∗ , p ∗ ) , − ) . Otherwise, the trajectory continues on the upper level.6.1.4.
Computation of expectation values.
If at time t there are N + trajectories onthe upper and N − trajectories on the lower level, then the expectation values forobservables a ( q, p ) = a + ( q, p )Π + ( q ) + a − ( q, p )Π − ( q )are approximated as(39) (op ε ( a ) ψ εt , ψ εt ) ≈ N + N + (cid:88) j =1 a + ( q + j ( t ) , p + j ( t )) + 1 N − N − (cid:88) j =1 a − ( q − j ( t ) , p − j ( t )) . The overall accuracy of the approximation is then determined by the initialsampling, the discretization of the Hamiltonian flows, and the asymptotic accuracy f the surface hopping semigroup. If the flow discretization is a symplectic order p method with time step ∆ t , then the error of the approximation (39) is O (1 / √ N ) + O (∆ pt ) + O ( ε γ ) ,γ = 1 / ε -dependent error of the algorithm is dominant.Our reference values have been obtained from numerically converged solutionsof the Schr¨odinger equation (1), which have been computed by a Strang splittingscheme with Fourier collocation. All figures show the reference values as solid lines,while the little stars and circles mark values computed by the surface hoppingalgorithm. We note, that the space grid for the Fourier collocation must resolvethe oscillations of the wave function, which is easily achieved in one space dimension.For higher dimensional problems, however, such discretizations suffer from the curseof dimensionality, which is not the case for our surface hopping algorithm.6.2. The simple and the arctangent crossing.
For both examples the sur-face hopping algorithm produces meaningful approximations of the dynamics eventhough the simple crossing has a non-smooth potential matrix and a surface gapjust varying by a factor two. Figure 2 shows population transfer away from theinitial energy level and a corresponding change of the average momentum on theinitial level. The final populations are approximated within an accuracy of 0 .
04 to0 .
05. The error of the momentum expectation is even smaller.6.3.
The dual and the extended crossing.
These two examples illustrate thelimitations of our approximation. Since the dual crossing model has two subsequentcrossings at q ≈ − . q ≈ .
6, one has to expect interferences between theupper and the lower level for the passage of the second crossing, which cannotbe resolved by the surface hopping semigroup. The numerical simulations confirmthis expectation. Figure 3(a) shows that the surface hopping algorithm correctlyresolves the first passage, while the non-adiabatic transfer for the second passage isdefinitely wrong. Since the mean size of the eigenvalue gap and the mean Landau-Zener rate computed by the surface hopping approach qualitatively reflect the truedynamical situation also for the passage of the second crossing, the failure of theapproximation must be due to unresolved interlevel interferences.Also the extended crossing case is not covered by our analysis, since the eigen-value surfaces do not have a minimal gap but a distance which monotonically de-creases as q → −∞ . We have therefore also considered a modified surface hoppingalgorithm, which allows non-adiabatic transitions at any time step of the numericalsimulation if the trajectory’s Landau-Zener coefficient is larger than a random num-ber uniformly distributed within the interval [0 , P opu l a t i on M o m en t u m Time E rr o r PopulationMomentum (a) Simple: ε ≈ . δ = 0 . P opu l a t i on M o m en t u m Time E rr o r PopulationMomentum (b) Arctangent: ε = 10 − , δ = √ ε Figure 2.
The simple and the arctangent crossing. The initialwave function is associated with the lower (a) and the upperlevel (b). The results of the surface hopping algorithm, markedwith stars and circles, are in good agreement with the reference. P opu l a t i on G ap Time R a t e (a) Dual: ε ≈ . δ = 0 . P opu l a t i on −4 −2 G ap R a t e Time (b) Extended: ε ≈ . δ = 6 · − Figure 3.
The dual and the extended crossing. The initial wavefunction is associated with the lower (a) and the upper eigenvec-tor (b). In both cases, the surface hopping algorithm expectedlyfails to reproduce the dynamics. on-adiabatic transfer much too early but finally arrives at an upper level popula-tion, which is rather close to the true solution. Appendix A. Proof of classical transport
Here we prove classical transport in the zone of large gap, where g ( q ) > R √ ε .More precisely, as long as the trajectories of Φ t ± which reach the support of theobservable a ( q, p ) stay in the region of large gap, we have (cid:90) R d +1 χ ( t ) (cid:0) w ε ± ( t ) − w ε ± (0) ◦ Φ − t ± (cid:1) ( q, p ) a ( q, p ) d( q, p, t )= O ( R − ) + O ( R − ε − / ) + O ( R − √ ε ) . (40)Note that we shall assume in what follows that R √ ε (cid:28)
1. The proof of equation (40)relies on symbolic calculus. We shortly recall the two main estimates we are goingto use:
Proposition A.1.
For a, b ∈ C ∞ c ( R d , C N × N ) , N ∈ N , we have (41) (cid:107) op ε ( a ) (cid:107) L ( L ( R d )) ≤ C sup | β |≤ d +1 sup x ∈ R d (cid:90) R d (cid:12)(cid:12)(cid:12) ∂ βξ a ( x, ξ ) (cid:12)(cid:12)(cid:12) d ξ. for some constant C > independent of a and ε . Moreover, (42) op ε ( a )op ε ( b ) = op ε ( ab ) + ε i op ε ( { a, b } ) + ε R ε , with { a, b } = ∇ ξ a · ∇ x b − ∇ x a · ∇ ξ b and (cid:107) R ε (cid:107) L ( L ( R d )) ≤ C sup | α | =2 , ≤| β |≤ d +3 sup x ∈ R d (cid:107) ∂ αx ∂ βξ a ( x, · ) (cid:107) L ( R d ) (cid:107) ∂ αx ∂ βξ b ( x, · ) (cid:107) L ( R d ) for some constant C > independent of a , b and ε . For a proof of Proposition A.1 we refer to § § L ε = 1 iε (cid:20) op ε ( c ε,R Π + ) , − ε q + V ( q ) (cid:21) , for c ε,R ( q, p ) = a ( q, p ) θ (cid:18) g ( q ) R √ ε (cid:19) where a ∈ C ∞ c ( R d , R ) and θ ∈ C ∞ ( R , R ) is a bounded function with boundedderivatives and support outside 0. We observe that the symbol c ε,R Π + satisfies theestimate ∀ α, β ∈ N d , ∃ C α,β > , ∀ p, q ∈ R d , (cid:12)(cid:12) ∂ βq ∂ αp ( c ε,R Π + ) (cid:12)(cid:12) ≤ C α,β ( R √ ε ) −| β | , where the constant C α,β depends on derivatives of a , θ and V . We write the poten-tial matrix V = λ + Π + + λ − Π − in terms of the eigenvalues and the eigenprojectorsand use Proposition A.1 to obtain L ε = op ε (cid:0) ( p · ∇ q c ε,R − ∇ λ + · ∇ p c ε,R )Π + (cid:1) + op ε ( B ) + O ( R − ) + O ( √ ε ) , ith B = c ε,R p · ∇ Π + − (cid:0) λ + ( { c ε,R Π + , Π + } − { Π + , c ε,R Π + } ) (cid:1) − (cid:0) λ − ( { c ε,R Π + , Π − } − { Π − , c ε,R Π + } ) (cid:1) = c ε,R p · ∇ Π + − g (cid:0) { c ε,R Π + , Π + } − { Π + , c ε,R Π + } (cid:1) = c ε,R p · ∇ Π + − g (cid:0) Π + ∇ p c ε,R · ∇ Π + + ∇ p c ε,R · ∇ Π + Π + (cid:1) = c ε,R p · ∇ Π + − g ∇ p c ε,R · ∇ Π + . where the last equation uses that ∇ Π + = Π + ∇ Π + + ∇ Π + Π + . As a consequence, B is an off-diagonal symbol, in the sense that B = Π + B Π − + Π − B Π + . Moreover, we have B = B + B with B = c ε,R p · ∇ Π + , B = − g ∇ p c ε,R · ∇ Π + . Our next step for proving (40) is therefore to investigate time averages of off-diagonal observables.
Lemma A.2.
For any χ ∈ C ∞ c ( R , R ) , j ∈ Z and any off-diagonal B j ∈ C ∞ c ( R d , C × ) satisfying the bound (43) ∀ α, β ∈ N d , ∃ C α,β > , ∀ p, q ∈ R d , (cid:12)(cid:12) ∂ βq ∂ αp B j (cid:12)(cid:12) ≤ C α,β ( R √ ε ) −| β |− j , we have (cid:90) R χ ( t ) (cid:104) op ε ( B j ) ψ εt , ψ εt (cid:105) d t = O ( ε ( j − R − j ) + O ( R − √ ε ) . Note that in view of and R √ ε (cid:28)
1, we have ε ( j − R − j = ε ( R √ ε ) j − (cid:29) R − √ ε as soon as j < Remark
A.3 . The previous Lemma shows in particular that in the large gap region { g ( q ) > R √ ε } , the contribution of the off-diagonal part of the Wigner transformis negligible. Indeed, for all a ∈ C ∞ c ( R d +2 , C × ), for all χ ∈ C ∞ c ( R ) and for all θ ∈ C ∞ ( R , R ) bounded, with bounded derivatives and support outside 0, we have (cid:90) R χ ( t ) (cid:28) op ε (cid:18) Π ± ( q ) a ( q, p )Π ∓ ( q ) θ (cid:18) g ( q ) R √ ε (cid:19)(cid:19) ψ εt , ψ εt (cid:29) d t = O ( R − ) . This relation comes from the fact that the off-diagonal symbolΠ ± ( q ) a ( q, p )Π ∓ ( q ) θ (cid:18) g ( q ) R √ ε (cid:19) satisfies Lemma A.2 with j = 1. Proof.
Since B j is off-diagonal, we can write B j = [(Π − B j Π + − Π + B j Π − ) g − , V ]= [(Π − B j Π + − Π + B j Π − ) g − , P ] , here we have used that V Π ± = ± g Π ± . After quantization, we getop ε ( B j ) = (cid:2) op ε ((Π − B j Π + − Π + B j Π − ) g − ) , op ε ( P ) (cid:3) + O ( ε ( j − R − j ) + O ( R − √ ε ) . Once applied to ψ εt which satisfies the Schr¨odinger equation op ε ( P ) ψ εt = 0, weobtain the announced relation. (cid:3) Applying Lemma A.2 to B = − g ∇ p c ε,R · ∇ Π + , we have (cid:90) R χ ( t ) (cid:104) op ε ( B ) ψ εt , ψ εt (cid:105) d t = O ( R − ) + O ( R − √ ε ) . However, for B = c ε,R p · ∇ Π + the result of Lemma A.2 only provides (cid:90) R χ ( t ) (cid:104) op ε ( B ) ψ εt , ψ εt (cid:105) d t = O ( ε − R − ) , an estimate that we want to ameliorate in order to prove (40). Therefore, we goone step further in the symbolic calculus and writeop ε ( B ) = (cid:2) op ε ((Π − B Π + − Π + B Π − ) g − ) , op ε ( P ) (cid:3) + εi op ε (cid:0)(cid:8) (Π − B Π + − Π + B Π − ) g − , τ + | p | (cid:9)(cid:1) + ε i op ε (cid:0) ( (cid:8) (Π − B Π + − Π + B Π − ) g − , V (cid:9)(cid:1) − ε i op ε (cid:0)(cid:8) V, (Π − B Π + − Π + B Π − ) g − (cid:9)(cid:1) + O ( R − ) + O ( R − ε / )= (cid:2) op ε ((Π − B Π + − Π + B Π − ) g − ) , op ε ( P ) (cid:3) − εi op ε (cid:0) p · ∇ q (cid:0) (Π − B Π + − Π + B Π − ) g − (cid:1)(cid:1) + O ( R − ) + O ( R − √ ε ) . We set B − = p · ∇ q (cid:0) (Π − B Π + − Π + B Π − ) g − (cid:1) , and we observe that B − satisfies (43) with j = −
2. We claim moreover that B − is off-diagonal, so that, Lemma A.2 gives (cid:90) R χ ( t ) (cid:104) op ε ( B − ) ψ εt , ψ εt (cid:105) d t = O ( R − ε − / ) + O ( R − √ ε )and (cid:90) R χ ( t ) (cid:104) op ε ( B ) ψ εt , ψ εt (cid:105) d t = O ( R − ε − / ) + O ( R − √ ε )It remains to prove the off-diagonal claim. A simple calculation shows B − = − g − ( p · ∇ q g )(Π − B Π + − Π + B Π − ) + g − p · ∇ q (Π − B Π + − Π + B Π − ) . Therefore,Π ± B − Π ± = g − Π ± (cid:0) p · ∇ Π − B Π + + Π − B p · ∇ Π + − p · ∇ Π + B Π − − Π + B p · ∇ Π − (cid:1) Π ± = 2 g − Π ± (cid:0) − p · ∇ Π + B Π + + Π + B p · ∇ Π + (cid:1) Π ± = 2 g − Π ± (cid:2) B , p · ∇ Π + (cid:3) Π ± = 0 ince B = c ε,R p · ∇ Π + . Acknowledgements.
We thank the anonymous referees for their help in improvingthe presentation of our results.
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