Ballistic Transport for Schrödinger Operators with Quasi-periodic Potentials
aa r X i v : . [ m a t h - ph ] F e b BALLISTIC TRANSPORT FOR SCHR ¨ODINGER OPERATORSWITH QUASI-PERIODIC POTENTIALS
YULIA KARPESHINA, LEONID PARNOVSKI AND ROMAN SHTERENBERG
Abstract.
We prove the existence of ballistic transport for a Schr¨odinger operatorwith a generic quasi-periodic potential in any dimension d > Introduction
Prior results on ballistic transport.
It is well known that the spectral anddynamical properties of Schr¨odinger operators H = − ∆ + V acting in H = L ( R d )are related. A general correspondence of this kind is given by the RAGE theorem,e.g. [36]. Stated briefly, it says that solutions Ψ( · , t ) = e − iHt Ψ of the time-dependentSchr¨odinger equation are ‘bound states’ if the spectral measure µ Ψ of the initial stateΨ is pure point, while Ψ( · , t ) is a ‘scattering state’ if µ Ψ is (absolutely) continuous.However, knowing the spectral type is not sufficient to quantify transport propertiesmore precisely, for example in terms of diffusion exponents β . These exponents, ifthey exist, characterize how time-averaged m -moments hh X m Ψ ii T := 2 T Z ∞ exp (cid:18) − tT (cid:19) k X m/ Ψ( · , t ) k H dt, m > X grow as a power T mβ of time T , where ( Xu )( x ) := | x | u ( x )( x ∈ R d and m is a positive real number). The special cases β = 1, β = 1 / β = 0are interpreted as ballistic transport, diffusive transport, and dynamical localization,respectively.In general, due to the possibility of fast travelling small tails, β may depend on m .In this paper, we will restrict our attention to the most frequently considered case ofthe second moment m = 2. The ballistic upper bound k X Ψ( · , t ) k H ≤ C (Ψ ) T + C (Ψ ) , (1.2)and thus also its averaged version hh X ii T ≤ C (Ψ ) T + C (Ψ ), is known tohold for general potentials V with relative ∆-bound less than one (in particular allbounded potentials) and initial statesΨ ∈ S := { f ∈ L ( R d ) : | x | f ∈ L ( R d ) , |∇ f | ∈ L ( R d ) } , (1.3)see [35]. As most authors, we will work with the Abel mean used in (1.1), but notethat the existence of a ballistic upper bound can be used to show that Abel meansand Cesaro means T − R T . . . dt lead to the same diffusion exponents (see for exampleTheorem 2.20 in [13]). Date : February 9, 2021.
In the late 1980s and 1990s methods were developed which led to more concretebounds on diffusion exponents by also taking fractal dimensions of the associatedspectral measures into account and showing that this gives lower transport bounds.In particular, again for the special case of the second moment, the Guarneri-Combestheorem [18, 19, 8, 31] says that hh X ii T ≥ C Ψ T α/d . (1.4)for initial states Ψ with uniformly α -H¨older continuous spectral measure (and sat-isfying an additional energy bound in the continuum case [8]). In dimension d = 1this says that states with an absolutely continuous spectral measure ( α = 1) alsowill have ballistic transport (as by (1.2) the transport can not be faster than bal-listic). In particular, this means that in cases where the spectra of one-dimensionalSchr¨odinger operators with limit or quasi-periodic potentials were found to have ana.c. component, e.g. [2, 7, 14, 15, 32, 33, 34], one also gets ballistic transport.The bound (1.4) does not suffice to conclude ballistic transport from the existenceof a.c. spectrum in dimension d ≥
2. In fact, examples of Schr¨odinger operators withabsolutely continuous spectrum, but slower than ballistic transport have been found:A two-dimensional ‘jelly-roll’ example with a.c. spectrum and diffusive transport isdiscussed in [30], while [3] provides examples of separable potentials in dimension d ≥ α < d ≥
2. Ballistic lower bounds and thusthe existence of waves propagating at non-zero velocity were known only for V = 0,where this is classical, e.g. [36], and for periodic potentials [1]. Scattering theoreticmethods show that this extends to potentials of sufficiently rapid decay, or sufficientlyrapidly decaying perturbations of periodic potentials. In [26] two results on ballisticlower bounds in dimension d = 2 were obtained, one for limit-periodic and one forquasi-periodic potentials. Our goal here is to generalize these result to any dimension d ≥ The Main Result.
We study the initial value problem i ∂ Ψ ∂t = H Ψ , Ψ( x ,
0) = Ψ ( x ) (1.5) ALLISTIC TRANSPORT 3 for multidimensional Schr¨odinger operator H acting on L ( R d ), d ≥
2, defined in thefollowing way. Let ω , . . . , ω l ∈ R d , l > d , be a collection of vectors that we willcall the basic frequencies . It will be convenient to form a ‘vector’ out of the basicfrequencies: ~ ω := ( ω , . . . , ω l ). We consider the operator H := − ∆ + V, (1.6)where V := X | n |≤ Q V n e n ~ ω . (1.7)The last sum is finite and taken over all vectors n = ( n , . . . , n l ) ∈ Z l with | n | := max j =1 ,...,l | n j | < Q, Q ∈ N . (1.8)We have also denoted e θ ( x ) := e i h θ , x i , θ , x ∈ R d (1.9)and n ~ ω := l X j =1 n j ω j ∈ R d ; (1.10)these vectors n ~ ω being called the frequencies . For convenience and without loss ofgenerality, we assume that the basic frequencies ω j ∈ [ − / , / d and thus ~ ω ∈ [ − / , / dl (so that the Lebesgue measure of this set is one; obviously, we canalways achieve this by rescaling). We assume the frequencies ω , ..., ω l are linearlyindependent over rationals. We also assume V − n = ¯ V n . Clearly, V is real valued.Consider the evolution equation (1.5) for operators H described above. Clearly, theballistic upper bound of [35] can be applied and we have (1.2) for initial conditions Ψ satisfying (1.3). We prove that for these operators there are corresponding ballisticlower bounds for a large class of initial conditions. To formulate our main result,we use the infinite-dimensional spectral projection E ∞ for H whose construction isdescribed in Section 2 below. Theorem 1.1.
For any given set of Fourier coefficients { V n } , V − n = ¯ V n , | n | ≤ Q , Q ∈ N , there exists a subset Ω ∗ = Ω ∗ ( { V n } ) ⊂ [ − / , / dl of basic frequencies withmeas (Ω ∗ ) = 1 such that for any ~ ω ∈ Ω ∗ there is an infinite-dimensional projection E ∞ = E ∞ ( V ) in L ( R d ) (described in Section 2) with the following property: Forany Ψ ∈ C ∞ with E ∞ Ψ = 0 (1.11) there are constants c = c (Ψ ) > and T = T (Ψ ) such that the solution Ψ( x , t ) of (1.5) satisfies the estimate T Z ∞ e − t/T (cid:13)(cid:13) X Ψ( · , t ) (cid:13)(cid:13) L ( R d ) dt > c T (1.12) for all T > T . Y. KARPESHINA, L. PARNOVSKI, R. SHTERENBERG
Remark 1.2.
The set Ω ∗ in the formulation of the Theorem is implicit. Morespecifically, it is the very same set for which the results from [27] are valid. Inparticular, the frequencies in this set satisfy Strong Diophantine Condition (see [27]for more details). In what follows, we will assume that the potential V is fixedand corresponding frequencies belong to Ω ∗ . We also remark that the notation inthis paper, while following in most symbols the notation of [26] and [27], sometimesdiffers slightly from it. For example, the projection E ∞ is denoted by E ( ∞ ) in [27],etc.In Section 2 we show that E ∞ is close in norm to F ∗ χ ( G ∞ ) F , where F is theFourier transform and χ ( G ∞ ) the characteristic function of a set G ∞ , which hasasymptotically full measure in R d , see (2.5) and (2.34).As already remarked in Section 1.1, due to the validity of the ballistic upper bound(1.2) for all initial conditions Ψ ∈ C ∞ ⊂ S , Theorem 1.1 remains true if the Abelmeans are replaced by Cesaro means.Theorem 1.1 will be proven in two steps. First we will show Proposition 1.3. If Ψ ∈ E ∞ C ∞ , Ψ = 0 , E ∞ being defined as in [27] , then thesolution Ψ( x , t ) of (1.5) satisfies the ballistic lower bound (1.12). Note that Proposition 1.3 differs from Theorem 1.1 by the fact that the initialcondition Ψ for which the ballistic lower bound is concluded is in the image of C ∞ under the projection E ∞ (but that Ψ itself is not in C ∞ here). This propositiontakes the role of our core technical result, i.e. most of the technical work towardsproving Theorem 1.1 will go into the proof of the proposition. Theorem 1.1 givesa more explicit description of initial conditions for which ballistic transport can beestablished. In fact, one easily combines Theorem 1.1 with the ballistic upper bound(1.2) to get ballistic transport in form of a two-sided bound for many initial conditions: Corollary 1.4.
There is an L ( R d ) -dense and relatively open subset D of C ∞ ( R d ) such that for every Ψ ∈ D there are constants < c ≤ C < ∞ such that theballistic upper bound (1.2) and the ballistic lower bound (1.12) holds for T > T (Ψ ) . This follows by an elementary argument using only that E ∞ is not the zero projec-tion and C ∞ ( R d ) is dense in L ( R d ) (and that C ∞ ( R d ) functions also satisfy (1.3)).It is certainly desirable to go beyond this corollary and to more explicitly char-acterize classes of initial conditions for which (1.11) holds. This requires to muchbetter describe and exploit the nature of the projection E ∞ . While we believe that E ∞ Ψ = 0 for any non-zero Ψ ∈ C ∞ ( R d ), we do not have a proof of this. Wewill return to this question later, see Remark 3.1, where we will more explicitlyconstruct initial conditions which lead to both upper and lower ballistic transportbounds. These will have the form of suitably regularized generalized eigenfunctionexpansions.As mentioned above, the proof of Theorem 1.1 is very similar to the two-dimensionalproof [26]. We just need to use the recent results from [27] instead of those in [29].In what follows, we present the main steps in the proof and explain the changes weneed to make in the proof due to the increase in dimension. ALLISTIC TRANSPORT 5
Acknowledgments.
The authors would like to dedicate this paper to the memoryof Jean Bourgain.The results were partially obtained during the programme Periodic and ErgodicSpectral Problems in January–June 2015, supported by EPSRC Grant EP/K032208/1.YK is grateful to Mittag-Leffler Institute for their support and hospitality (April,2019). The research of YK and RS was partially supported by NSF grant DMS–1814664. The research of LP was partially supported by EPSRC grants EP/J016829/1and EP/P024793/1.2.
Spectral Properties of the Operator H Our proofs of Proposition 1.3 and Theorem 1.1 are based on the results and prop-erties of quasi-periodic Schr¨odinger operators derived in the paper [27]. While thatwork has derived, in particular, the existence of an absolutely continuous componentof the spectrum, we will show now how the bounds obtained in [27] for the spectralprojections can be used to prove the existence of ballistic transport. In this sectionwe give a thorough discussion of the results and some of the methods from [27]. Inparticular, we give a detailed construction of the spectral projection E ∞ used in ourmain results. Unless stated otherwise, all statements in this section have been provedin [27].2.1. Prior results.
For any given set of Fourier coefficients { V n } , V − n = ¯ V n , | n | ≤ Q , Q ∈ N , there exists a subset Ω ∗ = Ω ∗ ( { V n } ) ⊂ [ − / , / dl of basic frequencieswith meas (Ω ∗ ) = 1 such that for any ~ ω ∈ Ω ∗ the following statements hold, forsufficiently small positive number σ , depending on V , l and d only.(1) The spectrum of the operator (1.6) contains a semi-axis.(2) There are generalized eigenfunctions U ∞ ( k , x ), corresponding to the semi-axis, which are close to the unperturbed exponentials. More precisely, forevery k in an extensive (in the sense of (2.5) below) subset G ∞ of R d there isa solution U ∞ ( k , x ) of the equation HU ∞ = λ ∞ U ∞ that satisfies the following properties: U ∞ ( k , x ) = e i h k , x i (1 + u ∞ ( k , x )) , (2.1) k u ∞ k L ∞ ( R d ) = | k |→∞ O ( | k | − γ ) , γ = 1 − σ > , (2.2)where u ∞ ( k , x ) is a quasi-periodic function: u ∞ ( k , x ) := X r ∈ Z l c r ( k ) e r ~ ω ( x ) , (2.3)the series converging in L ∞ ( R d ). The eigenvalue λ ∞ ( k ) corresponding to U ∞ ( k , x ) is close to | k | : λ ∞ ( k ) = | k |→∞ | k | + O ( | k | − γ ) , γ = 2 − σ > . (2.4)The “non-resonant” set G ∞ of the vectors k , for which (2.1) to (2.4) hold, canbe expressed as G ∞ = ∩ ∞ n =1 G n , where {G n } ∞ n =1 is a decreasing sequence of sets Y. KARPESHINA, L. PARNOVSKI, R. SHTERENBERG in R d . Each G n has a finite number of holes in each bounded region. Typically,as n increases, more holes of smaller sizes appear in the intersection. As aresult, the overall intersection G ∞ is, typically, a Cantor type set (i.e., it hasempty interior). This set satisfies the estimate:meas ( G ∞ ∩ B R )meas( B R ) = R →∞ O ( R − cσ ) , σ > , c = c ( l, d, ~ ω ) , (2.5)where B R is the ball of radius R centred at the origin.(3) The set D ∞ ( λ ), defined as a level (isoenergetic) set for λ ∞ ( k ), D ∞ ( λ ) = { k ∈ G ∞ : λ ∞ ( k ) = λ } , is a slightly distorted sphere, typically with infinite number of holes. It canbe described by the formula: D ∞ ( λ ) = { k : k = κ ∞ ( λ, ~ν ) ~ν, ~ν ∈ B ∞ ( λ ) } , (2.6)where B ∞ ( λ ) is a subset of the unit sphere S d − . The set B ∞ ( λ ) can beinterpreted as the set of possible directions of propagation for the almostplane waves (2.1). The set B ∞ ( λ ) typically has a Cantor type structure andhas an asymptotically full measure on S d − as λ → ∞ :meas ( B ∞ ( λ )) = λ →∞ meas ( S d − ) + O (cid:0) λ − cσ (cid:1) . (2.7)The value κ ∞ ( λ, ~ν ) in (2.6) is the “radius” of D ∞ ( λ ) in a direction ~ν . Thefunction κ ∞ ( λ, ~ν ) − λ / describes the deviation of D ∞ ( λ ) from the perfectsphere of radius λ / . It is proven that the deviation is asymptotically small,uniformly in ~ν ∈ B ∞ ( λ ): κ ∞ ( λ, ~ν ) = λ →∞ λ / + O (cid:0) λ − γ (cid:1) , γ = (3 − σ ) / > . (2.8)(4) The part of the spectrum corresponding to { U ∞ ( k , x ) } k is absolutely contin-uous. Remark 2.1.
While parameter σ can be chosen arbitrary small, all constants in O ( · )depend on σ . For the purposes of this paper, we will not need to impose any additionalassumptions on σ on top of those assumed in [27] (in particular, σ < (100 d ) − ).2.2. Description of the method.
To prove the results formulated in previous sub-section, in [27] we have considered the sequence of operators H n = H n ( k ), eachbeing restriction of the operator H onto the linear subspace of Z l spanned by theexponentials e k + n ~ ω , | n | ≤ | k | r n . Here, r n is a super exponentially growing sequenceof numbers of the formEach operator H n , n ≥
0, is considered as a perturbation of the previous operator H n − ( H − = − ∆). For every operator H n , there is one eigenvalue located sufficientlyfar (at least ∼ | k | − r n away) from the rest of the spectrum of H n . Correspondingeigenvector is close to the unperturbed exponential. More precisely, for every k ina certain subset G n of R d , there is a solution U n ( k , x ) of the differential equation H n U n = λ n U n that satisfies the following asymptotic formula: U n ( k , x ) = e i h k , x i (1 + u n ( k , ~x )) , k u n k L ∞ ( R d ) = | k |→∞ O ( | k | − γ ) , (2.9) ALLISTIC TRANSPORT 7 where u n ( k , · ) is quasi-periodic, a finite combination of e r ~ ω ( x ): u n ( k , x ) := X r ∈ Z l , | r | To study them, one needs properties of the limit B ∞ ( λ ) of B n ( λ ): B ∞ ( λ ) = ∞ \ n =0 B n ( λ ) , B n ( λ ) ⊂ B n − ( λ ) . This set has asymptotically full measure, as (2.7) follows from (2.14). The sequence κ n ( λ, ~ν ), n = 0 , , , ... , describing the isoenergetic sets D n ( λ ), quickly converges as n → ∞ . Hence, D ∞ ( λ ) can be described as the limit of D n ( λ ) in the sense (2.6),where κ ∞ ( λ, ~ν ) = lim n →∞ κ n ( λ, ~ν ) for every ~ν ∈ B ∞ ( λ ). The derivatives of thefunctions κ n ( λ, ~ν ) (with respect to the angle variable ~ϕ ) have a limit as n → ∞ forevery ~ν ∈ B ∞ ( λ ). We denote this limit by ∂ κ ∞ ( λ,~ν ) ∂ ~ϕ . We also have ∂ κ ∞ ( λ, ~ν ) ∂ ~ϕ = O (cid:0) λ − γ (cid:1) . (2.16)Thus, the limit set D ∞ ( λ ) takes the form of a slightly distorted sphere with, possibly,infinite number of holes.Let G ′ n be a bounded Lebesgue measurable subset of G n . We consider the spectralprojection E n ( G ′ n ) of H n , corresponding to functions U n ( k , x ), k ∈ G ′ n . By [16], E n ( G ′ n ) : L ( R d ) → L ( R d ) can be represented by the formula: E n (cid:0) G ′ n (cid:1) F = 1(2 π ) d Z G ′ n (cid:0) F, U n ( ~k ) (cid:1) U n ( ~k ) d k (2.17)for any F ∈ C c ( R d ), the space of continuous, compactly supported functions on R d .Here and below, (cid:0) · , · (cid:1) is the integral corresponding to the canonical scalar product in L ( R d ), i.e., (cid:0) F, U n ( k ) (cid:1) = Z R d F ( x ) U n ( k , x ) d x . The above formula can be rewritten in the form E n (cid:0) G ′ n (cid:1) = S n (cid:0) G ′ n (cid:1) T n (cid:0) G ′ n (cid:1) , (2.18) T n : C c ( R d ) → L (cid:0) G ′ n (cid:1) , S n : L ∞ (cid:0) G ′ n (cid:1) → L ( R d ) , ( T n F )( k ) := 1(2 π ) d/ (cid:0) F, U n ( ~k ) (cid:1) for any F ∈ C c ( R d ) , (2.19) T n F being in L ∞ ( G ′ n ), and( S n f )( x ) := 1(2 π ) d/ Z G ′ n f ( k ) U n ( ~k, x ) d k for any f ∈ L ∞ ( G ′ n ). (2.20)By [16], k T n F k L ( G ′ n ) ≤ k F k L ( R d ) (2.21)and k S n f k L ( R d ) ≤ k f k L ( G ′ n ) . (2.22)Hence, T n and S n can be extended by continuity from C c ( R d ) and L ∞ ( G ′ n ) to L ( R d )and L ( G ′ n ), respectively. Obviously, T ∗ n = S n . Thus, the operator E n ( G ′ n ) is de-scribed by (2.18) in the whole space L ( R d ). ALLISTIC TRANSPORT 9 In what follows we will use these operators for the case where G ′ n is given by G ′ n = G n,λ := { k ∈ G n : λ n ( k ) < λ } . (2.23)for finite sufficiently large λ . This set is Lebesgue measurable since G n is open and λ n ( k ) is continuous on G n .Let G ∞ ,λ = { k ∈ G ∞ : λ ∞ ( k ) < λ } . (2.24)The function λ ∞ ( k ) is a Lebesgue measurable function, since it is a pointwise limitof a sequence of measurable functions. Hence, the set G ∞ ,λ is measurable. The sets G n,λ and G ∞ ,λ are also bounded. The measure of the symmetric difference of the twosets G ∞ ,λ and G n,λ converges to zero as n → ∞ , uniformly in λ in every boundedinterval: lim n →∞ meas( G ∞ ,λ ∆ G n,λ ) = 0 . Next, we consider the sequence of operators S n ( G ∞ ,λ ) given by (2.20) and with G ′ n = G ∞ ,λ ; S n ( G ∞ ,λ ) : L ( G ∞ ,λ ) → L ( R d ) . (2.25)This sequence has a limit S ∞ ( G ∞ ,λ ) in operator norm sense as n → ∞ , uniform in λ .Moreover, the estimate k S ∞ ( G ∞ ,λ ) − S − ( G ∞ ,λ ) k < cλ − γ ∗ (2.26)holds for λ > λ ∗ , c not depending on λ, λ ∗ . Here we put U − = e i h k , x i and define S − by (2.20). The operator S ∞ ( G ∞ ,λ ) satisfies k S ∞ k = 1 and can be described by theformula ( S ∞ f )( ~x ) = 1(2 π ) d/ Z G ∞ ,λ f ( k )Ψ ∞ ( ~k, x ) d k (2.27)for any f ∈ L ∞ ( G ∞ ,λ ).Similarly, we consider the sequence of operators T n ( G ∞ ,λ ) which are given by (2.19)and act from L ( R d ) to L ( G ∞ ,λ ). Since T n = S ∗ n , the sequence T n ( G ∞ ,λ ) has a limit T ∞ ( G ∞ ,λ ) = S ∗∞ ( G ∞ ,λ ) in operator norm sense. The operator T ∞ ( G ∞ ,λ ) satisfies k T ∞ k ≤ T ∞ F )( k ) = π ) d/ (cid:0) F, Ψ ∞ ( k ) (cid:1) forany F ∈ C c ( R d ). The convergence is uniform in λ and k T ∞ ( G ∞ ,λ ) − T − ( G ∞ ,λ ) k < cλ − γ ∗ . (2.28)Spectral projections E n ( G ∞ ,λ ) converge in norm to E ∞ ( G ∞ ,λ ) in L ( R d ) as n tendsto infinity, since T n = S ∗ n . The operator E ∞ ( G ∞ ,λ ) is a spectral projection of H . Itcan be represented in the form E ∞ ( G ∞ ,λ ) = S ∞ ( G ∞ ,λ ) T ∞ ( G ∞ ,λ ). For any F ∈ C c ( R d )we have E ∞ ( G ∞ ,λ ) F = 1(2 π ) d Z G ∞ ,λ (cid:0) F, Ψ ∞ ( ~k ) (cid:1) Ψ ∞ ( ~k ) d k , (2.29) HE ∞ ( G ∞ ,λ ) F = 1(2 π ) d Z G ∞ ,λ λ ∞ ( k ) (cid:0) F, Ψ ∞ ( k ) (cid:1) Ψ ∞ ( k ) d k . (2.30) Since E ∞ is a projection, one has the Parseval formula k E ∞ ( G ∞ ,λ ) F k = 1(2 π ) d Z G ∞ ,λ | ( F, Ψ ∞ ( k ) | d k . (2.31)It is easy to see that k E ∞ ( G ∞ ,λ ) − S − T − ( G ∞ ,λ ) k < cλ − γ ∗ , (2.32) S − T − ( G ∞ ,λ ) = F ∗ χ ( G ∞ ,λ ) F . (2.33)Projections E ∞ ( G ∞ ,λ ) are increasing in λ and have a strong limit E ∞ ( G ∞ ) as λ goesto infinity. Hence, the operator E ∞ ( G ∞ ) is a projection. The projections E ∞ ( G ∞ ,λ ), λ ≥ λ ∗ , and E ∞ ( G ∞ ) reduce the operator H . The family of projections E ∞ ( G ∞ ,λ )is the resolution of the identity of the operator HE ∞ ( G ∞ ) acting in E ∞ ( G ∞ ) L ( R d ).Let us denote E ∞ := E ∞ ( G ∞ ) and use k E ∞ − F ∗ χ ( G ∞ ) F k < cλ − γ ∗ . (2.34)Obviously, the r.h.s. can be made arbitrarily small by an appropriate choice of G ∞ .The restriction of H to the range of E ∞ has purely absolutely continuous spectrum.In addition to the above mentioned convergence of the spectral projections of H n tothose of H , uniform in λ ≥ λ ∗ for sufficiently large λ ∗ = λ ∗ ( V ), this requires ananalysis of the continuity properties of the level sets D ∞ ( λ ) with respect to λ .2.3. Extension of λ ∞ ( k ) from G ∞ to R d . Let M be a large natural number; forthe purposes of this paper, taking M := [3 d/ λ ∞ ( k ) from G ∞ to R d , the result being a C M ( R d ) function. Note that theextended function is not going to be a generalised eigenvalue outside of G ∞ .The first step is representing λ ∞ ( k ) − k , k := | k | , k ∈ G ∞ , in the form: λ ∞ ( k ) − k = λ ( k ) − k + ∞ X n =1 ( λ n ( k ) − λ n − ( k )) . Let m = ( m , ...m d ) be a multi-index and put D mk := ∂ m · · · ∂ m d d . We have (see [27],Lemma 11.3): (cid:12)(cid:12) D mk (cid:0) λ ( k ) − k (cid:1)(cid:12)(cid:12) < Ck − γ + σ | m | , γ = 2 − σ, (2.35)when k is in the k − σ -neighborhood of G ⊃ G ∞ and | D mk ( λ n ( k ) − λ n − ( k )) | < Ck − k rn − + | m | k σrn − (2.36)in the k − k σrn − -neighbourhood of G n for all m . Here, the constants depend only on V and m .Now, we introduce a function η ( k ) ∈ C ∞ ( R d ) with support in the (real) k − σ -neighbourhood of G , satisfying η = 1 on G and | D m η ( k ) | < Ck σ | m | . (2.37)This is possible since we can take a convolution of the characteristic function ofthe k − σ -neighbourhood of G with w (2 k σ k ), where ω ( k ) is a non-negative C ∞ ( R d )-function with a support in the unit ball centred at the origin and integral one. Simi-larly, let η n ( k ), n ≥ 1, be a C ∞ function with support in the k − k σrn − -neighbourhood ALLISTIC TRANSPORT 11 of G n , satisfying η n = 1 on G n and | D m k η n ( k ) | ≤ Ck | m | k σrn − . (2.38)Next, we extend λ ∞ ( k ) − k from G ∞ to R d using the formula λ ∞ ( k ) − k = ( λ ( k ) − k ) η ( k ) + ∞ X n =1 ( λ n ( k ) − λ n − ( k )) η n ( k ) . (2.39)It follows from (2.35) – (2.38) that the series converges in C M ( R d ). Taking intoaccount that σ > λ ∗ increases and G ∞ is getting smaller when σ decreases) gives the following lemma: Lemma 2.2. For every natural number M , there exists λ ∗ ( V, M ) > such that thefunction λ ∞ ( k ) − k can be extended, as a C M function, from G ∞ to R and it satisfies (cid:12)(cid:12) D mk (cid:0) λ ∞ ( k ) − k (cid:1)(cid:12)(cid:12) < C M k − γ + σ | m | , (2.40) for any m ∈ N d with | m | ≤ M < σ − . Extension of U ∞ ( k , x ) from G ∞ to R d . We now define U ∞ ( k , x ) for arbitrary k ∈ R d by a formula analogous to (2.39): U ∞ ( k , x ) − e i h k , x i = (cid:16) Ψ ( k , x ) − e i h k , x i (cid:17) η ( k ) + ∞ X n =1 ( U n ( k , x ) − U n − ( k , x )) η n ( k ) , (2.41)here U n are described by (2.9), (2.10), and (see [27], Lemma 11.3) k D m k (cid:16) u (0) − e k (cid:17) k L ∞ ( R d ) < Ck − γ + σ | m | , γ = 1 − σ, (2.42) k D m k (cid:16) u ( n ) − u ( n − (cid:17) k L ∞ ( R d ) < Ck − k rn − + | m | k σrn − , (2.43)Thus, the series (2.41) is convergent in L ∞ ( R d ). Using the last formula and (2.27),we define S ∞ ( e G ∞ ) for any e G ∞ ⊃ G ∞ : (cid:16) S ∞ ( e G ∞ ) f (cid:17) ( x ) := 1(2 π ) d/ Z e G ∞ f ( k ) U ∞ ( k , x ) d k . (2.44)It is easy to see that S ∞ ( e G ∞ ) = S − ( e G ∞ ) + ∞ X n =0 (cid:0) S n ( e G ∞ ) − S n − ( e G ∞ ) (cid:1) η n , (2.45)where S − ( e G ∞ ) is defined by S − ( e G ∞ ) f = 12 π Z e G ∞ f ( k ) e − i h k , x i d k ,η n is multiplication by η n ( k ) and S n ( e G ∞ ) is given by (2.20) with G ′ n being the intersec-tion of e G ∞ with the k − k σrn − -neighborhood of G n for n ≥ k − σ -neighborhoodof G for n = 0. Similarly to (2.26), we show that k S ∞ ( e G ∞ ) − S − ( e G ∞ ) k < c ( V ) λ − γ ∗ . (2.46)In what follows we assume that λ ∗ is chosen so that, in particular, c ( V ) λ − γ ∗ ≤ / k S − ( e G ∞ ) k = 1.Thus we have k S ∞ ( e G ∞ ) k ≤ . (2.47)Similarly, with T − being the Fourier transform,( T ∞ F )( k ) := 1(2 π ) d/ ( F ( · ) , U ∞ ( k , · ))= ( T F )( k ) + ∞ X n =0 (cid:0) ( T n − T n − ) F (cid:1) ( k ) η n ( k ) . (2.48). Lemma 2.3. For any given L ∈ N there exists λ ∗ ( V, L ) such that for any F ∈C ∞ ( R d ) , the function T ∞ F as defined above is in C L ( R d ) . Moreover, if ≤ j ≤ L and m ∈ N d , | m | ≤ L , then (cid:12)(cid:12) | k | j D m ( T ∞ F )( k ) (cid:12)(cid:12) < C ( L, F ) , (2.49) for all k ∈ R d . We prove the lemma using (2.48) and then (2.19) for each T n . Integrating by parts j times and considering (2.38), (2.42), and (2.43), we arrive at (2.49). Remark 2.4. For our needs L = M = [3 d/ L and M are fixed.3. Proofs of Proposition 1.3 and Theorem 1.1 Let S := T ∞ C ∞ ( R d ), see (2.48). Let b Ψ ∈ S . As shown in Lemma 2.3, then (cid:12)(cid:12) | k | j D m ( b Ψ )( k ) | < C ( j, m, b Ψ ) (3.1)for any k ∈ R d .Now we defineΨ( x , t ) := 1(2 π ) d/ Z G ∞ U ∞ ( k , x ) e − itλ ∞ ( k ) b Ψ ( k ) d k , (3.2)then this function solves the initial value problem (1.5), whereΨ ( x ) = 1(2 π ) d/ Z G ∞ U ∞ ( k , x ) b Ψ ( k ) d k (3.3)and Ψ ( x ) ∈ S ∞ S = E ∞ C ∞ . Obviously, S ∞ S is dense in E ∞ L ( R d ).The next step of the proof is replacing G ∞ by a small neighbourhood ˜ G ∞ and toestimate the resulting errors in the integrals. This is an important step, since G ∞ isa closed Cantor-type set, while ˜ G ∞ is an open set. Then we would like to integrateby parts in the integral over ˜ G ∞ with the purpose of obtaining (1.12); the fact that˜ G ∞ is open being used for handling the boundary terms. ALLISTIC TRANSPORT 13 To get the lower bound (1.12), we first note that k X Ψ k L ( R d ) ≥ k X Ψ k L ( B R ) ≥ k Xw k L ( B R ) − k X (Ψ − w ) k L ( B R ) , where B R is the open disc with radius R centred at the origin, R = c T , c to bechosen later, and w ( x , t ) is an approximation of Ψ when G ∞ is replaced by its smallneighbourhood ˜ G ∞ . Namely, w ( x , t ) := 1(2 π ) d/ Z e G ∞ U ∞ ( k , x ) e − itλ ∞ ( k ) b Ψ ( k )˜ η δ ( k ) d k , (3.4)˜ η δ being a smooth cut-off function with support in a δ -neighbourhood e G ∞ of G ∞ and˜ η δ = 1 on G ∞ . The parameter δ (0 < δ < 1) will be chosen later to be sufficientlysmall and depend only on b Ψ . We take ˜ η δ to be a convolution of a function ω ( k / δ )with the characteristic function of the δ/ G ∞ , where ω is a smoothcut-off function defined in the previous section. Then, ˜ η δ ∈ C ∞ ( R d ),0 ≤ ˜ η δ ≤ , ˜ η δ ( k ) = 1 when k ∈ G ∞ , ˜ η δ ( k ) = 0 when k e G ∞ , k D m ˜ η δ k L ∞ < C m δ −| m | . (3.5)To prove (1.12), we will show that there exist a positive constant c and constants c and c such that2 T Z ∞ e − t/T (cid:13)(cid:13) Xw ( · , t ) (cid:13)(cid:13) L ( B R ) dt ≥ c T − c T − c , (3.6)as long as c in the definition of R exceeds a certain value depending only on b Ψ . Informula (3.6), the constant c = c ( b Ψ ) depends on b Ψ , but not δ or c , while theconstants c = c ( b Ψ , δ ) and c = c ( b Ψ , δ ) depend on b Ψ and δ , but not c .We also prove that2 T Z ∞ e − t/T (cid:13)(cid:13) X (Ψ − w )( · , t ) (cid:13)(cid:13) L ( B R ) dt ≤ γ ( δ, b Ψ ) c T , (3.7) γ ( δ, b Ψ ) = o (1) as δ → c .The proofs of (3.6) and (3.7) are completely analogous to those from [26]. Theonly difference is in the estimate of the integral of the form˜ φ ( ~zt, t ) := 1(2 π ) d/ Z e G ∞ ∩{ k : | k − k | < } e it ( h k ,~z i− λ ∞ ( k )) g ( k ) (cid:0) − ˆ η ( k ) (cid:1) d k , ~z := x t , (3.8)where g ( k ) := ∇ λ ∞ ( k ) b Ψ ( k )˜ η δ ( k ), ˆ η is a smooth cut-off function satisfyingˆ η ( k ) = ( , | k − k | ≤ , | k − k | ≥ k = k ( ~z ) = 12 ~z + O ( | ~z | − γ ) , γ > , is the unique solution (see (2.39) and Lemma 2.2) of the equation for a stationarypoint ~z − ∇ λ ∞ ( k ) = 0 , | ~z | > λ ∗ . As in [26], we apply Theorem 7.7.5 in [20] but for arbitrary d > 1. The number ofderivatives required depends on the dimension ( M := [3 d/ φ ( ~zt, t ) = 1(2 i ) d/ e it ( h k ,~z i− λ ∞ ( k )) (cid:0) O ( | ~z | − γ ) (cid:1) g ( k ) t − d/ + ǫ ( g ) t − d/ − (3.9)for | ~z | > λ ∗ and 0 otherwise. Here, | ǫ ( g ) | ≤ c X | m |≤ d +3 sup | k − k | < | D m g ( k ) | ≤ c (cid:13)(cid:13)(cid:13) | k | d/ b Ψ ( k ) (cid:13)(cid:13)(cid:13) C d +3 ( R d ) δ − d − | ~z | − d/ − . Now, the end of the proof of Proposition 1.3 follows as in Section 3 in [26]. Theproof of Theorem 1.1 is identical to the proof in Section 4 of [26]. Remark 3.1. (a) The above proofs show that Theorem 1.1 remains true if we replace C ∞ in (1.11) with S d := { f : | x | s D m f ( x ) ∈ L ( R d ) , ≤ s, | m | ≤ C ( d ) } i.e. for initial conditions which are sufficiently smooth and of sufficiently rapid powerdecay.(b) Using the constructions in the above proofs, we can now also describe moreexplicitly how to choose initial conditions Ψ for the solution of (1.5) which givesimultaneous ballistic upper and lower bounds. Essentially, one has to regularizeelements in the range of E ∞ in two different ways, first at the boundary of G ∞ , usingthe cut-off function ˜ η δ as in (3.5) above, and then at high momentum k . For thelatter, let ϕ ∈ S d on R d and such that ϕ does not vanish identically on G ∞ .Choose Ψ ( x ) := 1(2 π ) d/ Z e G ∞ ϕ ( k ) ˜ η δ ( k ) U ∞ ( k , x ) d k . (3.10)As δ → F ( x ) = π ) d/ R G ∞ ϕ ( x ) U ∞ ( k , x ) d k in the range of E ∞ with k F k = R G ∞ | ϕ | d k / (2 π ) d = 0. 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