Diffusion in the mean for a periodic Schrödinger equation perturbed by a fluctuating potential

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1 Diffusion in the mean for a periodic Schrödinger equation perturbed by a fluctuating potential Jeffrey Schenker, F. Zak Tilocco, and Shiwen Zhang arxiv:9.6598v [math-ph] 9 Jan 9 Abstract We consider the evolution of a quantum particle hopping on a cubic lattice in any dimension and subject to a potential consisting of a periodic part and a random part that fluctuates stochastically in time. If the random potential evolves according to a stationary Markov process, we obtain diffusive scaling for moments of the position displacement, with a diffusion constant that grows as the inverse square of the disorder strength at weak coupling. More generally, we show that a central limit theorem holds such that the square amplitude of the wave packet converges, after diffusive rescaling, to a solution of a heat equation. Introduction and the Main Results Proving that diffusive motion emerges from the Schrödinger wave equation with a weakly disordered potential (in dimension d 3) remains one of the outstanding open problems of mathematical physics. Both at a conceptual and technical level, recurrence in space presents is one of the fundamental obstacles to the proof. Technically, in any sort of perturbation or multi-scattering expansion potential terms corresponding to the same region of space are identical at all times, preventing a direct application of central limit theorem type arguments. At a conceptual level, and viewing the motion semi-classically, when the quantum particle described by the wave function visits a given region of space it samples the same potential. Based on the above considerations, it is reasonable to expect that diffusion occurs more readily for a model in which recurrence is eliminated or reduced. This was the idea behind prior work of the first author and collaborators [7],[], [5], in which diffusive propagation was shown to occur for solutions to a tight binding Schrödinger equation with a random potential evolving stochastically in time. In the papers [7, ], the following stochastic Schrödinger equation on l (Z d ) was considered: i t ψ t (x) = H ψ t (x)+λv(x,t)ψ t (x), (.) with H a (non-random) translation invariant Schrödinger operator, λ is a real coupling constant, and V(x, t) a zero-mean random potential with time dependent stochastic fluctuations. These models had been considered previously by Tcheremchantsev [6, 7], who obtained diffusive bounds for position moments up to logarithmic corrections. In [7, ], diffusive scaling for all moments (without logarithms) was proved, under suitable hypotheses on H and V. Furthermore it was observed that at weak disorder λ, the corresponding diffusion constant D has the asymptotic form D C λ. (.) The divergence of D as λ seen in eq. (.) is to be expected, since the translation invariant Schrödinger operator H on its own leads to ballistic transport. In ref. [5], the first author considered the more subtle situation in which the environment is a superposition of two parts: i t ψ t (x) = H ψ t (x)+u(x)ψ t (x)+λv(x,t)ψ t (x), (.3)

2 where u is a static random potential that, at λ =, gives rise to Anderson localization (absence of transport). In [5], it was observed that the diffusion constant in this case has the asymptotic form D Cλ. (.4) Taken together, the results in [7,, 5] suggest that solutions to (.3) with a general potential u should satisfy diffusion with a diffusion constant whose asymptotic behavior in the small λ limit is governed by the dynamics of the static Schrödinger operator H + u. In this paper, we study this idea in the context of models of the form of eq. (.3) but with periodic u that leads to ballistic transport. We will obtain diffusive propagation for the evolution, and more generally, a central limit theorem for the square amplitude. Furthermore, we prove that in this case the asymptotic relation eq. (.) does indeed hold. We consider below solutions to eq. (.3) with {u(x)} x Z d a real valued p-periodic potential. Recall that given p = {p j } d j= Zd >, a function u : Z d R is called p-periodic. if u(x+p j e j ) = u(x) (.5) for all j d and x Z d, where e j denotes the standard basis of Z d. Throughout this paper, we denote by U the multiplicative operator, (Uψ)(x) = u(x)ψ(x) for ψ(x) l (Z d ). The analysis below is applicable to a broad class of operators H and V(x,t). To avoid technicalities in this introduction, let us state the main results in terms of the following so-called Markovian flip process, which is a non-trivial, and somewhat typical, example of operators satisfying the general requirements. Let the hopping H be the standard discrete Laplacian on Z d. The random potential is given by V(x,t) = v x (ω(t)) as follows, where ω(t) is an evolving point in an auxiliary state space Ω. For the flip model, we take the state space Ω = {,} Zd, and v x (ω) = ω x, the x th coordinate of ω. At any time t, the potential v x (ω(t)) takes only the values ±. Now suppose the process ω(t) is obtained by putting independent, identical Poisson processes at each site x, and allowing each coordinate ω x to flip sign at the times t (x) t (x) of the Poisson process. Now the general equation (.3) becomes: i t ψ t (x) = ψ t (y)+u(x)ψ t (x)+λv x (ω(t))ψ t (x). (.6) y x = A hallmark of diffusion is the existence of the diffusion constant for eq. (.6) D := lim t t x E( ψ t (x) ), (.7) x characterized by the relationship x t. Here, and throughout this introduction, E( ) denotes averaging with respect to the Poisson fliping times t (x) t (x) and the initial values {ω x } x Z d, taken independent and uniform in {,}. We will show below that the limit in eq. (.7) exists for any p-periodic potential u and λ >, and furthermore D >. To give an unambiguous definition, one may take the initial value ψ (x) = δ (x) = when x = and otherwise. However, as we will show, the limit remains the same for any other choice of (normalized) ψ with x x ψ (x) <. We refer to the existence of a finite, positive diffusion constant as in eq. (.7) as diffusive scaling. More generally, we have the following Without loss of generality, we assume that p j for some j. Otherwise, u is constant and the problem reduces to that studied in [7].

3 Theorem. (Central limit theorem). For any periodic potential u and λ >, there is a positive definite d d matrix D = D(λ,u) such that for any bounded continuous function f : R d R and any normalized ψ l (Z d ) we have lim f t x Z d ( x t ) ( E ψ t (x) ) = R d f(r) ( )d e r, D r dr, (.8) π where ψ t (x) is the solution to eq. (.6). If furthermore x (+ x ) ψ (x) <, then diffusive scaling eq. (.7) holds with the diffusion constant ( D(λ) = lim x E ψ t (x) ) = trd(λ). (.9) t t x Z d Morover, eq. (.8) extends to quadratically bounded continuous f with sup x (+ x ) f(x) <. It is well known that if λ = in (.6), the free periodic Schrödinger equation has Bloch-wave solutions and exhibits ballistic motion by the Floquet theory, see [, 8], lim t t x δ x, e it( +U) δ (, ). (.) x Z d If we extend the definition of D(λ) in (.9) to λ =, then D() =. We are primarily interested here in the regime λ, although we will demonstrate diffusion for all λ >. Dynamical randomness prevents ballistic motion and furthermore induces diffusion whenever λ >. However, for small λ the diffusion constant will be large and will blow up at λ =. In fact, we have the following asymptotic behavior of D(λ) and D(λ). Theorem.. Under the hypotheses of Thm.., there is a positive definite d d matrix D such that D(λ) = λ ( D +o() ) and D(λ) = trd(λ) = λ ( trd +o() ) as λ. (.) The conclusions of Theorems. and. are true for periodic equation (.3) under much more general assumptions on the hopping H and the time dependent stochastic potential V(x,t). We will state the general assumptions and results in Section. Before turning to the general framework, let us discuss a history of related work on diffusive scaling, explain the relation of prior works to the present one, and finally describe several conjectures for more general tight binding models. These conjectures are closely related to, but do not follow from, the work presented here. A brief history of related studies is as follows. Ovchinnikov and Erikman obtained diffusion for a Gaussian Markov ( white noise ) potential [3]. Pillet obtained results on transience of the wave in related models and derived a Feynman-Kac representation [4] which is the basis of the present work. Using Pillet s Feynman-Kac formula, Tchermentchansev[6, 7] showed that position moments exhibit diffusive scaling, up to logarithmic corrections for any bounded potential u(x) in (.3): t s (lnt) ( x s E ψ t (x) ) t s (lnt) ν +, t. (.) ν x Let J be a periodic block Jacobi matrix on m j= l (Z), which includes + U on l (Z) as a special case. Let X be the position operator and let X(t) = e it( +U) Xe it( +U) be its Heisenberg time evolution. Strong ballistic motion was obtained for J in [8] : There is a bounded self-adjoint operator Q with ker(q) = {} such that for any ψ with Xψ l (Z), lim t X(t)ψ = Qψ. t 3

4 The case u(x) (or equivalently, p = (,,)) was considered in the previous work [7], where (.) was shown to hold for s = with ν = ν + =. Moreover, the central limit theorem (.8) and the asymptotic behavior (.) were also obtained in [7]. The proof in [7] was revisited in [] to obtain diffusive scaling for all position moments of the mean wave amplitude. The models studied in [7] are special cases of those considered here. For a certain class of random potentials u(x), including the case of an i.i.d. potential, diffusive scaling and the central limit theorem were proved in [5]. If moreover H + u exhibits Anderson localization, then O(λ ) asymptotics (.4) were proved for the diffusion constant. The arguments in [5] do not require strict independence of the static potential at different sites, however the Equivalence of Twisted Shifts assumption taken in [5] excludes the p-periodic case as well as all almost-periodic cases. The periodic case can be viewed as lying in the intermediate regime of the period-free case and the i.i.d. case. This is another motivation for us to revisit the proofs in [7] and [5] and develop the current approach in the p-periodic case, for both diffusive scaling and limiting behavior. In[6], Fröhlichandthe firstauthorused the techniquesof[5] tostudy the diffusion foralattice particle governed by a Lindblad equation describing jumps in momentum driven by interaction with a heat bath. In some sense, this is the quantum analogue of the classical dynamics of a disordered oscillator systems perturbed by noise in the form of a momentum jump process, considered in [, 3] and reviewed in [4]. In those works, heat transport is considered in the limit of weak noise in a regime for which transport is known to vanish for the disordered oscillator system without noise. A key feature ofthe noise in [, 3] is that energyis conservedin the system with noise; this is necessary so that one can speak about heat flux. By contrast, in the present work, and in [7,, 5, 6], energy conservation is broken by the noise. Indeed the only conserved quantity for the evolution we consider is quantum probability; and it is this quantity which is subject to diffusive transport. There are also parallel works on the diffusion for the continuum Schrödinger equation with Markovian forcing and periodic boundary conditions in space, e.g. [5]. One physical interpretation of this continuous model is as a rigid rotator coupled to a classical heat bath. In [5], the H s norm of the wave function is shown to behave as t s/4. It is interesting to point out that, as in the present work, the existence of a spectral gap for the Markov generator is essential both for their analysis and the results. In many models with Markovian forcing, the potential V(x, t) is quite rough. However, Bourgain studied the case where V(x, t) is analytic/smooth in x and quasi-periodic/smooth in t. In [5], he showed that energy may grow logarithmically. We refer readers to, e.g., [,, 8], for more work on Sobolev norm growth and controllability of Schrödinger equations with time-dependent potentials. Theproofwepresenthereisageneralizationofthatin[7]. Someoftheargumentsareessentially standard fare and parallel the work of[7] closely. However, there are three places in the proof where some substantially new arguments were needed. First, in the Fourier analysis (see Sec. 3.) in our work is more subtle and requires careful consideration due to the periodic potential. The extension developed here is of independent interest and may benefit the future study of the limit-periodic and quasi-periodic cases. Secondly, the spectral gap Lemma 4.8 and the proof of the main results in Sec.5 are technically more involved in the current work. The interaction between the periodic part and the hopping terms complicates the block decomposition on the augmented space. Finally, in the present proof, the analysis of the asymptotic behavior of the diffusion constant is quite a bit more involved. In [7], (.) essentially follows from a formula derived for the diffusion constant in the midst of the proof of diffusion. Unfortunately, Theorem. in the p-period case does not have such a simple proof and is obtained by a new approach. The proof is based on an interesting observation linking the ballistic motion of the unperturbed part to the diffusion scaling. This observation is part of the motivation behind our conjecture below on the more general situations, linking the transport exponent to the limiting behavior of the diffusion scaling. In light of the present work, it is natural to ask what can be said about eq. (.3) with u a general ergodic/deterministic potentials. In particular, 4

5 . Under which hypotheses on u do we have diffusive propagation over long time scales?. When diffusion holds, what is the limiting behavior of the diffusion constant with respect to the disorder coupling constant? Based on the limiting behavior of the diffusion constant in the periodic case and in the i.i.d case, it is natural to make the following Conjecture.3. For any bounded potential u(x) on Z d and any λ >, there exist positive, and finite, upper and lower diffusion constants, D(λ),D(λ) (, ) such that ( D(λ) := liminf x E ψ t (x) ) ( limsup x E ψ t (x) ) =: D(λ). (.3) t t x Z d t t x Z d Suppose + U exhibits ballistic motion, then D(λ),D(λ) O(λ ) for λ. Suppose + U exhibits dynamical localization, then D(λ),D(λ) O(λ ) for λ. Remark.4. ) Similar conjectures can be made for general equations which will be introduced in Section. ) More generally, if the unperturbed equation has transport exponent ρ [, ], then we expect D(λ),D(λ) O(λ ρ ). 3) Here, we also want to bring readers attention to the recent work [9], though not directly relevant to our current paper, on the ballistic transportation for the Schrödinger operator with limit-periodic or quasi-periodic potential in dimension two. If the unperturbed part is given by the almost Mathieu operators with parameters g R,θ,α [,], we have the following AMO-Markovian equation on l (Z): i t ψ t (x) = ψ t (x+)+ψ t (x )+gcosπ(θ +xα)ψ t (x)+λv(ω x (t))ψ t (x). (.4) Conjecture.5. For almost every θ, α [, ], the AMO-Markovian equation has a diffusion constant D(g,λ) (, ) which is a smooth function for all (g,λ) R R +. Moreover, D(g,λ) O(λ ) for all g > and D(g,λ) O(λ ) for all g <. The rest of the paper is organized as follows: In Sec., a more general class of operators is introduced and the main result Theorem., which generalizes Theorems. and., is formulated. In Sec.3 the basic analytic tools of augmented space analysis are developed. In Sec. 4, we use a block decomposition to study the spectral gap of the induced operator on the augmented space, which is the heart of the entire analysis. Sec. 5 is devoted to a proof of the main result. Certain technical results used below are collected in appendices. General assumptions and results We may study a more general class of equations in which hopping terms other than nearest neighbor and the perturbing potential V is not necessarily the flip process. More precisely, we shall consider equation (.3) in the form i t ψ t (x) = H ψ t (x)+u(x)ψ t (x)+λv ω (x)ψ t (x) (.) Here u is the real-valued, p-periodic potential as in (.5) for some p Z d >; H is a self-adjoint, short-ranged, hopping operator with span(supp H ) = Z d ; V ω(t) is time-dependent random potential that fluctuates according to a stationary Markov process ω(t); and λ is a coupling constant used to set the strength of the disorder. These assumptions will be made precise below. Some assumptions are similar to those in [7] and [5]. They are repeated here for convenience. In particular, our assumptions on the probability space and Markov dynamics remain largely unchanged, while modifications to the hopping term must be made to ensure the particle is not confined to motion along a sub-lattice. 5

6 . Assumptions Assumption. (Probability space). Throughout, let (Ω, µ) be a probability space. For each x Z d there is a µ-measure preserving map, τ x : Ω Ω, and the collection of these maps form a µ-measure preserving group of translations, i.e., τ is the identity map and τ x τ y = τ x+y for each x,y Z d. Assumption. (Markov dynamics). The space Ω is a compact Hausdorff space, µ is a Borel measure and for each α Ω there is a probability measure P α on the Σ-algebra generated by Borelcylinder subsets of the path space P(Ω) = Ω [, ). Furthermore the collection of these measures has the following properties. Right continuity of paths: For each α Ω, with P α probability one, every path t ω(t) is right continuous and has initial value ω() = α.. Stationary strong Markov property: There is a filtration {F t } t on the Borel σ-algebra of P(A) such that ω(t) is F t measurable and ) P a ({ω(t+s)} t E F s = P ω(s) (E) for any measurable E P(Ω) and any s >. 3. Invariance of µ: For any Borel measurable E Ω and each t >, P α (ω(t) E)µ(dα) = µ(e). Ω Invariance of µ under the dynamics is equivalent to the identity E(f(ω(t))) = E(f(ω())) for f L (Ω). An important tool for studying Markov processes is conditioning on the value of a process at a given time. The proper definition can be found in, e.g. [5]. Conditioning on the value of the processes at t = determines the initial value: E( ω() = α) = E( ). To the process {ω(t)} t, there is associated a Markov semigroup, obtained by averaging over the initial value conditioned on the value of the process at later times: S t f(α) := E(f(ω()) ω(t) = α). As is well known, S t is a strongly continuous contraction semi-group on L p (Ω) for p <. 3 The semigroup S t has a generator Bf := lim t t (f S tf), (.) defined on the domain D(B) where the right hand side exists in the L -norm. By the Lumer- Phillips theorem, B is a maximally accretive operator. The invariance of µ under the process {ω(t)} t implies that S t ½ = S t ½ = ½, where ½(α) = for all α Ω. It follows that } L {f (Ω) := L (Ω) f(α)µ(dα) = is invariant under the semi-group S t and its adjoint S t. We assume that B is sectorial and strictly dissipative on this space. 3 The semi-group property follows from the Markov property, while strong continuity follows from the right continuity of paths. The adjoint of S t is the backward semigroup S tf(α) := E(F(ω(t)) ω() = α). Ω 6

7 Assumption.3 (Sectoriality of B). There are b,γ such that Imf, Bf γref, Bf+b f (.3) for all f D(B). Here f, g = fgdµ denotes the inner product on L (Ω). Assumption.4 (Gap condition for B). There is T > such that Ref, Bf T f Ω fdµ L (Ω) (.4) for all f D(B). Remark.5. ) The resolvent of the semigroup e tb is the operator valued analytic function R(z) := (B z) = e tz e tb dt, which is defined and satisfies R(z) Rez when Rez <. Sectoriality is equivalent to the existence of a analytic continuation of R(z) to z C\K b,γ with the bound R(z) dist (z,k b,γ ) where K b,γ is the sector {Rez } { Imz b+γ Rez } (see [8, Theorem V.3.]). In particular Assumption.3 holds (with b = and γ = ) if the Markov dynamics is reversible, in which case B is self-adjoint. ) The gap assumption implies that the restriction of B to L (Ω) is strictly accretive, and thus that St L (Ω) e t T. Assumption.6 (Translation covariance, boundedness and non-degeneracy of the potential). The potentials V ω (x) appearing in the Schrödinger equation (.3) are given by V ω (x) = v(τ x ω) where v L (Ω). We assume that v =, Ωv(ω)µ(dω) =, and v is non-degenerate in the sense that there is χ > such that B (v(τ x ) v(τ y )) χ (.5) L (Ω) for all x,y Z d, x y. Remark.7. Since the Markov process is translation invariant, B commutes with the translations T x f(α) = f(τ x α) of L (Ω). Thus (.5) is equivalent to B (v(τ x ) v( )) L χ. (.6) (Ω) for all x Z d, x. The non-degeneracy amounts essentially to requiring that B (vτ x ) are uniformly non-parallel to B (v) for x. In particular, the condition is trivially satisfied if for example if the processes v(τ x ω(t)) and v(ω(t)) are independent for x, as in the flip process. Assumption.8 (Translation invariance and non-degeneracy of the hopping terms). The hopping operator, H, on l (Z d ) is defined by H ψ(x) = ξ xh(x ξ)ψ(ξ). (.7) Additionally, the hopping kernel h : Z d \{} C is. Self-adjoint:. Short range: h( ξ) = h(ξ); ξ h(ξ) < ; (.8) ξ Z d \{} 7

8 3. Non-degeneracy of ĥ : For each non-zero k R d, k ξ h(ξ) > ; (.9) ξ Z d \{} 4. Non-degeneracy of the support of h: Let supph = { ξ Z d : h(ξ) }, span Z (supph) = Z d. (.) Remark.9. ) It follows that ĥ(k) = x e ik x h(x) is a C function on the torus [,π) d. In particular, H is a bounded operator with H l (Z d ) l (Z d ) = maxk ĥ(k) and ĥ, ĥ, ĥ (+ ξ ) h(ξ) <. (.) ξ Z d \{} ) It is natural to assume that supph can generate the entire Z d lattice, otherwise the system can always be reduced a direct sum of systems over several sub-lattices.. General results The main result is the following Theorem. (Central limit theorem). For any periodic potential u and λ >, there is a positive definite d d matrix D = D(λ,u) such that for any bounded continuous function f : R d R and any normalized ψ l (Z d ) we have lim f t x Z d ( x t ) ( E ψ t (x) ) = R d f(r) ( )d e r, D r dr, (.) π where ψ t (x) is the solution to eq. (.). If furthermore x (+ x ) ψ (x) <, then diffusive scaling eq. (.7) holds with the diffusion constant ( D(λ) = lim x E ψ t (x) ) = trd(λ). (.3) t t x Z d Moreover, eq. (.) extends to quadratically bounded continuous f with sup x (+ x ) f(x) <. Assume further that lim T T 3 e t T δ x, e it(h+u) δ dt >, j =,d, (.4) x Z d x j then there is a positive definite d d matrix D such that D(λ) = λ ( D +o() ) and D(λ) = trd(λ) = λ ( trd +o() ) as λ. (.5) Remark.. ) In the case with the short range hoping H and periodic U, the strong limit of all the j-th velocity operators lim t t X j (ψ t ) always exist, which implies the existence of the limit in (.4). We say H +U has ballistic motion if the limit in (.4) is positive. ) δ in in (.4) can be replaced by any ψ with compact support. 3) There always exists a semi-positive definite d d matrix D such that (.5) holds true regardless of (.4). If (.4) holds true for j S on a sub-lattice S {,,,d}, then the restriction of D on S S is positive definite, and we still have D(λ) λ since trd >. 8

9 3 Augmented space analysis 3. The Markov semigroup on augmented spaces and the Pillet-Feynman- Kac formula The term augmented spaces refers here to certain spaces of functions F : X Ω C where X is an auxiliary space in the examples below X will be Z d or Z d Z d. The spaces we consider will be of the following form. Definition 3.. Let (B(X), B(X) ) be a Banach space of functions on X whose norm satisfies. If g B(X) and f(x) g(x) for every x X, then f B(X) and f B(X) g B(X).. For every x X, the evaluation x f(x) is a continuous linear functional on B(X). For p, the augmented space B p (X Ω) is the set of maps F : X Ω C such that F(,x) L p (Ω) B(X). When it is clear from the context, we will sometimes write B for B(X) and B p for B p (X Ω). Proposition 3.. The augmented space B p (X Ω) becomes a Banach space when equipped with the norm ( ) F Bp (X Ω) := F(x,ω) p p µ(dω). Ω B(X) Furthermore, ( ) F Bp (X Ω) F(x,ω) p B(X) µ(dω) Ω Denote the space of all strongly-measurable maps F : Ω B for which F p B is integrable by L p (Ω;B). Then the inequality in proposition 3. says L p (Ω;B) B p (X Ω). When φ B(X) and f L p (Ω) we denote their product by φ f L p (Ω;B) and define (φ f)(x,ω) := φ(x)f(ω). As a concrete example, let p,q < and take B(X) = l p (X). Then the augmented space l p;q (X Ω) is the set of all maps F : X Ω C such that q F(x,ω) p p µ(dω) x X Ω is finite. In particular, if p = q then by proposition 3. the augmented space l p;p becomes a Banach space with norm ( ) p F l p;p (X Ω) = F(x,ω) p µ(dω). This observation leads to the following: Proposition 3.3. If p <, then x X l p;p (X Ω) = L p (Ω;l p (X)) = L p (X Ω), where we take product measure counting measure µ on X Ω. In particular, l ; (X Ω) is a Hilbert space with inner product F, G = F(x, ω)g(x, ω)µ(dω). x X Ω Ω p. 9

10 Throughout, we will use e tb to denote the Markov semigroup lifted to B p (X Ω), with B the corresponding generator. This semigroup is defined by e tb F(x,α) := E Ω (F(ω(),x) ω(t) = α), (3.) for F B p (X Ω). That is, e tb is defined on B p (X Ω) so that the following diagram is commutative for each x X: B p (X Ω) J x e tb B p (X Ω) J x (3.) L P (Ω) e tb L p (Ω) where J x F( ) = F(x, ) is the evaluation map from B p (X Ω) to L p (Ω). Proposition 3.4. The semigroup e tb is contractive and positivity preserving on B p (X Ω) and B is sectorial on L (X Ω), with the same constants b and γ as appear in Assumption.3. The starting point for the analysisof eq. (.) is a formulafor E(ρ t ), where ρ t (x,y) = ψ t (x)ψ t (y) is the density matrix corresponding to a solution ψ t to eq. (.). The formula, due in this context to Pillet [4], expresses the expectation E(ρ t ) in terms of a contraction semi-group on the augmented Hilbert space H := L (Ω;HS(Z d )), (3.3) where HS(Z d ) denotes the Hilbert-Schmidt ideal in the bounded operators on l (Z d ). Since HS(Z d ) can be identified with l (Z d Z d ) by taking R(x,y) := δ x,rδ y for R HS(Z d ), we see that H is the augmented space (see Prop. 3.3): where H = l ; (Z d Z d Ω) = L (M), M := Z d Z d Ω (3.4) with the product measure m = ( counting measure on Z d Z d) µ. Depending on context we will think of an element F H either as a C-valued map on M or as a HS(Z d )-valued map on Ω, via the identification F(x,y,ω) := δ x,f(ω)δ y. (3.5) We define operators K, U and V that lift H, U and V ω to H respectively. More precisely, we lift the commutators with these operators on Hilbert-Schmidt operators: KF(ω) := [H,F(ω)], UF(ω) := [U,F(ω)], and VF(ω) := [V ω,f(ω)]. (3.6) Proposition 3.5. The operators K, U and V are self-adjoint and bounded. Lemma 3.6 (Pillet s formula [4]). Let L := ik+iu +iλv +B (3.7) on the domain D(B) L (M). Then L is maximally accretive and sectorial and if ρ t = ψ t ψ t, is the density matrix corresponding to a solution ψ t to eq. (.6) with ψ l (Z d ), then E(ρ t ω(t) = α) = e tl (ρ ½), (3.8) where ½(ω) = for all ω. Consequently, we have [ E(ρ t ) = e tl (ρ ½) ] (ω)µ(dω). (3.9) Ω

11 This equation relates the mean square amplitude of the time dependent dynamics (.6) to spectral properties of the non-self adjoint operator L. Taking matrix elements of various expressions above gives the following Lemma 3.7. The operators K, U and V are given by the following explicit expressions KF(x,y,ω) = ξ h(ξ)[f(x ξ,y,ω) F(x,ξ y,ω)], (3.) and UF(x,y,ω) = [u(x) u(y)]f(x,y,ω) (3.) VF(x,y,ω) = [v(τ x ω) v(τ y ω)]f(x,y,ω), (3.) for any F L (M). Furthermore, for a solution ψ t to eq. (.), we have the Pillet s Feynman-Kac formula (PFK): ( ) E ψ t (x)ψ t (y) = δ x δ y ½, e tl( ψ ψ ½ ) L (M). (3.3) In particular, we have E(ρ t (x,x)) = δ x δ x ½,e tl ρ ½ H. (3.4) Remark 3.8. Here and below we will use tensor product notation for elements of l (Z d Z d ), [φ ψ](x,y) = φ(x)ψ(y). Thus a rank one operator ψφ, HS(Z d ) corresponds to ψ φ. 3. Vector valued Fourier Analysis For each ξ Z d, define the (simultaneous position and disorder) shift operator for any function Ψ defined on Z d Z d Ω. S ξ Ψ(x,y,ω) := Ψ(x ξ,y ξ,τ ξ ω) (3.5) Proposition 3.9. The map ξ S ξ is a representation of the additive group Z d and, for each ξ, S ξ Ψ L (M) = Ψ L (M). In particular ξ S ξ H is a unitary representation of Z d. Lemma 3.. For every ξ Z d, [S ξ,k] = [S ξ,u] = [S ξ,v] = [S ξ,b] = Because of Lemma 3., a suitable generalized Fourier transform will give a fibre decomposition of the various operators K, U, V and B. Let T d = [,π) d denote the torus and M := Z d Ω. We use the following notation throughout. Let m Z, then Z m := {n Z : n m} and Z d m is the d-fold Cartesian product of Z m; similar definitions apply for Z >m. If p Z d >, let p := p p d Z; C p := C p C p d ; Z p := Z p Z pd ; and pz := p Z p d Z. Let π σ : C p C be the projection onto lattice points modulo σ = (σ,,σ d ) Z p. For f,g L ( M;C p ), we use the following inner product on L ( M;C p ) throughout the paper f, g L ( M;C p ) = π σ f, π σ g L ( M;C). (3.6) σ Z d p

12 Given Ψ L (M) and k T d, the Fourier transform 4 of Ψ at k is defined to be a map Ψ k : M C p as follows: for all σ Z d p π σ Ψk (x,ω) := n pz d +σ e ik n S n Ψ(x,,ω) = n pz d +σ e ik n Ψ(x n, n,τ n ω). (3.7) One may easily compute π σ(kψ)k (x,ω) = ξ h(ξ) [ ] π σ Ψk (x ξ,ω) e ik ξ π σ ξ Ψk (x ξ,τ ξ ω) ; π σ (UΨ)k (x,ω) =(u(x σ) u( σ))π σ Ψk (x,ω); π σ (VΨ)k (x,ω) =(v(τ x ω) v(ω))π σ Ψk (x,ω); π σ (BΨ)k (x,ω) =Bπ σ Ψk (x,ω), where B is understood to act as a multiplication operator with respect to x Z d and k T d. With the above computations in mind, let K k, Û, and V denote the following operators on functions φ : M C p : π σ ( K k φ)(x,ω) = ξ h(ξ) [ π σ φ(x ξ,ω) e ik ξ π σ ξ φ(x ξ,τ ξ ω) ]. (3.8) Û is diagonal on C p : V and B is multipication for each x Z d on C p : π σ (Û φ)(x,ω) = (u(x σ) u( σ))π σφ(x,ω); (3.9) ( Vφ)(x,ω)(v(τ x ω) v(ω)) φ(x,ω). (3.) The following lemmas (Lemma 3.-Lemma 3.4) essentially can be found in [5]. The only difference is here we consider vector valued space L ( M;C p ) instead of L ( M;C). We omit the details here. Lemma 3.. Let M = Z d Ω, K k, Û and V be given as above, then. Kk, Û and V are bounded on l ; ( M;C p ).. Kk, Û and V are bounded and self-adjoint on L ( M;C p ) with the following bounds: L K k ĥ, Û u, V ( M;C p ) L ( M;C p ) L ( M;C p ) 4 Initially we define this Fourier transform on the augmented space W (M) := {F : M C sup F(x+y,y,ω) µ(dω) < }. x y The basic results of Fourier analysis are naturally extended to this generalized Fourier transform. In particular, Plancherel s Theorem holds for F W (M) L (M). Thus, the generalized Fourier transform extends naturally to L (M). Throughout the rest of the paper, we assume that the generalized Fourier transform is properly defined on L (M) and most formulas hold true for F L ( M) almost surely. For more details of this extension, we refer readers to Sec. 3 in [5].

13 3. If Ψ L (M;C) and let Ψ k be given as in (3.7), then (KΨ) k = K k Ψk, (ÛΨ) k = Û Ψ k and ( VΨ) k = V Ψ k for ν-almost every k T d. Because of the shift invariance under distribution, the Markov semigroup(as defined in eq. (3.)) commutes with Fourier transformation: Lemma 3. (Lemma 3.4,[5]). Let the Markov semigroup e tb be defined as in eq. (3.). Then, for Ψ L (M) and ν-almost every k T d. [e tb Ψ] k = e tb Ψk Lemma 3.3. Let K k be given as in (3.8) and h satisfies (.8), then the map k K k is C on T d, considered either as a map into the bounded operators on l ; ( M;C p ) or as a map into the bounded operators on L ( M;C p ). Moreover, we have the explicit expression of the derivatives for any φ(x,ω) L ( M;C p ), k T d and i,j d: π σ kj Kk φ(x,ω) = i ξ ξ j h(ξ)e ik ξ π σ ξ φ(x ξ,τ ξ ω), (3.) π σ ki kj Kk φ(x,ω) = ξ ξ i ξ j h(ξ)e ik ξ π σ ξ φ(x ξ,τ ξ ω). (3.) with bounds kj Kk ĥ, ki kj Kk ĥ, (3.3) where ĥ, ĥ are bounded in (.). In particular, let C p be such that 5 π σ = for all σ Z d p, we have kj K δ ½ = i ξ ξ j h(ξ)δ ξ ½, (3.4) ki kj K δ ½ = ξ ξ i ξ j h(ξ)δ ξ ½. (3.5) Putting these results together we obtain Lemma 3.4. For each k T d, let L k := i K k +iû +iλ V +B (3.6) on the domain D(B) L ( M;C p ).. For each k T d, L k generates an exponentially bounded semigroup on l ; ( M;C p ). Furthermore, (a) For t > the map k e t L k is a C map from T d into the bounded operators on l ; ( M;C p ). 5 Throughout the rest of the paper, we will frequently use the notation q C q for any q Z > to indicate the constant vector in C q with all entries and write = p for simplicity. 3

14 (b) If Ψ L (M), then for ν-almost every k T d, e t L k Ψk := [e tl Ψ] k. (3.7). For each k T d, L k is maximally accretive on L ( M;C p ). Furthermore (a) For t >, the map k e t L k is a C map from T d into the contractions on L ( M;C p ). (b) The operators { L k } k T d are uniformly sectorial; that is for every k T d and every f L ( M;C p ) Im f, Lk f γre f, Lk f +b f L (3.8) where γ,b are given as in (.3) and b = b+ ĥ + u +λ. (c) If Ψ L (M) then eq. (3.7) holds for ν-almost every k. Combine (3.7) with Pillet s formula (Lemma 3.6), we obtain the following Fourier transformed Pillet formula in vector form: Lemma 3.5 (Fourier transformed Pillet formula). Let ψ l (Z d ) and define ρ ;k (x) C p for any x Z d,k T d as π σ ρ ;k (x) := e ik n ψ (x n)ψ ( n), σ Z d p. (3.9) n pz d +σ Then y Z d e ik y E ( ) ψ t (x y)ψ t ( y) = δ x ½, e t L k ( ρ ;k ½), (3.3) L ( M;C p ) where ψ t is the solution to eq. (.) with initial condition ψ. Here e t L k ( ρ ;k ½) c ( M;C p ) for each k and is in L ( M;C p ) for ν-almost every k. In particular, for ν-almost every k T d, e ik x ( E ψt (x) ) = δ ½, e t L k ( ρ ;k ½). (3.3) L ( M;C p ) x Z d Proof. Let Ψ(x,y,ω) = ( e tl (ρ ½) ) (x,y,ω) = δ x δ y, e tl (ρ ½). PFK (3.3) L (Z d Z d ) can be rewritten as ( E(Ψ(x,y, )) = e tl (ρ ½) ) (x,y,ω)µ(dω) Ω = δ x δ y ½, e tl (ρ ½) ) (ψ = E L (Z d Z d Ω) t (x)ψ t (y). Recall the definition of π σ ( )k in (3.7), π σ Ψk (x,ω) = n pz d +σ e ik n Ψ(x n, n,τ n ω). Direct computation shows b y := e ik y π σ Ψk (x,ω)dk = πψ(x y, y,τ y ω)δ pz+σ (y). T d 4

15 Fourier inverse formula gives that for a.e. k T d : y pz d +σ e ik y Ψ(x y, y,τ y ω) = e ik y b y π = π σ Ψ k (x,ω), y Z d ) e ik y E(Ψ(x y, y, )) = π σ E( Ψk (x, ) = δ x ½, π σ Ψk y pz d +σ On the other hand, by (3.7), for Φ = ρ ½, we have Ψ k = (e tl Φ) k = e t L k Φk, L ( M;C). where π σ Φk = π σ(ρ ½) k (x,ω) = n pz d +σ e ik n ψ (x n)ψ ( n) ½. Clearly, by the defintion of ρ ;k in (3.9), Φ k = ρ ;k ½. Putting everything together, we have ( ) e ik y E ψ t (x y)ψ t ( y) = δ x ½, π σ e t Lk ρ ;k ½ y pz d +σ Finally, by (3.6), summing over the periodic cell, we have that ( ) e ik y E ψ t (x y)ψ t ( y) = δ x ½, e t Lk ρ ;k ½ y Z d L ( M;C). L ( M;C p ). 4 Spectral analysis on the augmented space 4. Spectral analysis of K The spectral analysis of L k plays an important role in studying the diffusive scaling of this model. We begin by finding the eigenvalues of K. The following three lemmas show that the nondegeneracy of the support of the hopping operator (.), implies that is an eigenvalue of K. This observation allows us to write down a block decomposition and to find a spectral gap for L in the two sections that follow. Lemma 4.. For p Z >, let A p denote the p p right shift matrix, A p := , (4.) p gcd(m,p) on C p, and let A m p := (A p ) m be the m-th power of A p for m Z. Then the matrix A m p has distinct eigenvalues, p e πilm p, l =,,,, (4.) gcd(m,p) each of multiplicity gcd(m, p). 5

16 Proof. It is sufficient to restrict our attention to < m < p, otherwise replace m with m mod p. For m =, we may use that fact that A p is a permutation of order p to find its eigenvalues: λ l = e πi l p, l =,,,p. For < m < p, it follows from the spectral mapping theorem that A m p has eigenvalues λm l for l =,,,p. From here, it is easy to verify that λ m l = λ m l whenever l l np = gcd(m,p) for some integer n. Finally, since l l p < p, it follows that there are gcd(m,p) distinct eigenvalues each of multiplicity gcd(m, p). This result has an immediate extension to the tensor product of right shift operators, Corollary 4.. If p = (p,,p d ) Z d > and m = (m, m d ) Z d, then A m p := d eigenvalues d e πil jm j p j ; l j =,,, j= In particular, if (e j ) d j= is the standard basis on Zd, then j= Amj p j has p j. (4.3) gcd(m j,p j ) Ker(I A ej p ) = C p pj C p d. (4.4) Lemma 4.3. Let m,,m k Z d, n,,n k Z, and M = n m + + n k m k for some k Z. Then, we have k Ker(I A mj p ) Ker(I A M p ). (4.5) j= In particular, if m,,m k generate Z d, then k j= Ker(I A mj p ) = d j= Ker(I A ej p ) = span{ p }. (4.6) Proof. Suppose w k j=ker(i Amj p ), then for each j =,,,k, w = A mj p w = = ( ) A mj nj n w = A jm j p w. (4.7) Repeated application of (4.7) yields w = A nm p p A n km k p w = A M p w. Thus, w Ker(I A M p ). If m,,m k generate Z d, then (4.5) implies the first equality in (4.6). The second equality follows from Corollary 4. since d j= Ker(I A ej p ) = d j= ( C p ) pj C p d = span{ p }. Lemma 4.4. Let x Z d and w C p for some p Z d >. Then K δ x w ½ = ξ h(ξ)δ x ξ (I A ξ p ) w ½. (4.8) 6

17 Proof. This follows from direct computation: π σ K (δ x w ½) = ξ h(ξ)[π σ δ x ξ w ½ π σ ξ δ x ξ w ½] = ξ h(ξ)δ x ξ [π σ π σ ξ ] w ½ = ξ h(ξ)δ x ξ π σ (I A ξ p ) w ½. The non-degenerate support condition (.) guarantees that the hopping kernel, h, is nonzero on a spanning set, {ξ j } j J, of Z d. Combining this fact with lemma (4.3), we can see that (I A ξj p ) w = for each j J if and only if w. In particular, lemma (4.4) allows us to say the following: Corollary 4.5. Let x Z d and w C p for some p Z d >. Then K (δ x w ½) = if and only if w. Moreover, for all ξ supp(h), there exists β(ξ,p) (,) such that for w, K (δ x w ½) w h(ξ) β (ξ,p). (4.9) ξ 4. Block decomposition of L It was shown in the previous section that δ ½ is an eigenvector of K corresponding to the eigenvalue. Using (3.9) and (3.), it is easy to check that this claim also holds true for Û and V. Finally, the Markov generator satisfies B½ = B ½ =. Therefore, L δ ½ = L δ ½ =. (4.) To analyze the matrix element on the right hand side of (3.3) we will use a block decomposition of the generator L k associated to the following direct sum decomposition of L ( M;C p ): Ĥ Ĥ Ĥ Ĥ3 = L ( M;C p ) = l (Z d ) C p L (Ω) (4.) where M = Z d Ω and Ĥj,j =,,,3 are defined as follows: Ĥ := span{δ ½}; Ĥ := δ ½ = span{δ t j ½,j =, p}, where t,, t p are orthogonal basis to in C p. Ĥ := δ C p ½ = l (Z d \{}) C p ½; ) { Ĥ 3 := (Ĥ Ĥ Ĥ = Ψ(x,ω) : Ω } Ψ(x,ω)dµ(ω) =. We will write operators on L ( M;C p ) as 4 4matrices of operatorsacting between the various spaces Ĥj,j =,,,3. Throughout we will use the notation:. P j = the orthogonal projection onto Ĥj, 7

18 . P j = P j. In particular, P = P +P +P 3 is the orthogonal projection of L ( M;C p ) onto the space Ĥ Ĥ Ĥ = l (Z d ) C p ½ of non-random functions, PΨ(x) = Ψ(x, ω)dµ(ω). Then P 3 = P = P is the projection onto the space of mean zero functions Ĥ3. We have the following block decomposition of the components of L as follows: Ω Lemma 4.6. On Ĥ Ĥ Ĥ Ĥ3 the operators K,Û, V, and B have following block decomposition K = P K P P K P P K P, Û = P ÛP, P 3 K P 3 P 3 ÛP 3 V = P VP3, and B =. P 3 VP P 3 VP3 P 3 BP 3 Proof. The eigenvalue equation (4.) gives for all j =,,3 and all T = K,Û, V,B, L, P j T = T P j =. The hopping is off-diagonal on l (Z d ) in the sense h() =, therefore, P K P =. The expressions of Û and V in (3.9),(3.) imply that they vanish on δ : P Û = P V =, ÛP = VP =. Since K,Û are non-random, we have for j =,,, P j K P 3 =, P 3 K P j =, P j ÛP 3 =, P 3 ÛP j =. Since V is mean zero on L (Ω) and B½ = B ½ =, then P 3 VP 3 =, P 3 B = BP 3 =. Corollary 4.7. On Ĥ the operator L = i K k +iû +iλ V +B has block decomposition L = ip K P ip K P P (i K +iû)p iλp VP3. (4.) iλp 3 VP P 3 L P 3 8

19 Imz N + = {z : Rez, Imz ĥ + u +λ+γrez} σ( L )\{} {z : Rez > g } N + σ( L ) Rez g Figure : Spectral gap of L 4.3 Spectral gap With the block decomposition (4.), we are now in a position to prove L has a spectral gap. Lemma 4.8. For each λ > and p Z d >, there is g > such that where. is a non-degenerate eigenvalue, and. Σ + {z : Rez > g}. σ( L ) = {} Σ + For λ small and any p, there is c = c(p, ĥ, u,γ,t,b) > such that g cλ. First note that Re L = ReB in the sense of quadratic forms. Thus by the sectoriality of B ImΦ, L Φ K +Û +λ V + ImΦ,BΦ ĥ + u +λ+γreφ, L Φ, if Φ =. It follows that the numerical range of L, Num( L ) = {Φ, L Φ : Φ = } is contained in N + := {z : Rez and Imz ĥ + u +λ+γrez}. (4.3) Since σ( L ) Num( L ), we may restrict our attention to z N +, see Figure. Consider ip K P J = ip K P P (i K +iû)p iλp VP3 (4.4) iλp 3 VP P 3 L P 3 9

20 acting on Ĥ := Ĥ Ĥ Ĥ3. To prove Lemma 4.8, it is enough to prove the following spectral gap for J since Ĥ = span{δ ½} is one dimensional. Lemma 4.9. There is g >, such that Γ i z are bounded invertible on Ĥi,i =,,3 for Rez < g where ) Γ 3 = P 3 L P 3, Γ (z) = P (i K +iû +λ V(Γ3 z) V P (4.5) Γ (z) = P K P (Γ z) P K P (4.6) Therefore, J z is bounded invertible on Ĥ for Rez < g. In particular, at z =, for any Ψ Ĥ, J Ψ := Π J Π Ψ = Π (Π Γ ()Π ) Π Ψ, (4.7) where Π is the projection onto Ker(P K ) Ĥ. Furthermore, for any φ l (Z d \{}) ½, we have that φ = J φ and Re φ, φ = ReΓ φ, φ g φ >. (4.8) Proof. Now fix z N + and consider the equation (J z) ζ z ip K P φ = ip K P P (i K +iû)p z iλp VP3 ζ φ = Φ iλp 3 VP P 3 L P 3 z Φ ζ φ, (4.9) Φ for (ζ,φ,φ) Ĥ Ĥ Ĥ3 given ( ζ, φ, Φ) Ĥ Ĥ Ĥ3. By the gap condition (.4) on B, ReP 3 L P 3 = Re(i K +iû +B +iλp 3 VP 3 ) T P 3. Therefore, Γ 3 z = P 3 L P 3 z is bounded invertible on Ĥ3 provided Rez < T. The third equation of (4.9) is which can be solved to give Thus the second equation of (4.9) is reduced to iλp 3 VP φ+[γ 3 z]φ = Φ (4.) Φ = (Γ 3 z) Φ (Γ3 z) iλ Vφ. (4.) [Γ (z) z] φ = φ ip K ζ iλp V(Γ3 z) Φ (4.) where ) Γ (z) = P (i K +iû +λ V(Γ3 z) V P. (4.3) Now take ϕ ½ Ĥ = L (Z d \{};C p ), in view of (4.3), direct computation shows that Reϕ ½, Γ (z)ϕ ½Ĥ =λ P 3 (Γ 3 z) V ϕ ½, (ReB Rez)P3 (Γ 3 z) Vϕ ½ Ĥ

21 λ ( T Rez) (Γ3 z) V ϕ ½ =λ ( T Rez) ( B P 3 ( L z)p 3 ) B Vϕ ½ Ĥ Ĥ (4.4) where the inverse of B is well defined since Vϕ Ĥ3 = RanP 3. Furthermore B is bounded on Ĥ 3, with B P 3 T. Thus B P 3 ( L z)p 3 is bounded for z N + {Rez < T } by, B Ĥ P 3 ( L z)p 3 + B P 3 ( K +Û +λ V) + B P 3 z Let +T( ĥ + u +λ+ z ) Therefore, by putting (4.5) and (.6) together, we get +γ +4T( ĥ + u +λ). (4.5) (4.4) λ ( B V ϕ ½ T Rez) Ĥ B P 3 ( L z)p 3 λ ( T Rez) σ Z d x p Ĥ χ π σ ϕ(x) ( +γ +4T( ĥ + u +λ) λ χ = ( ) ( +γ +4T( ĥ T Rez) ϕ ½ Ĥ. + u +λ) c = T ) λ χ ( ( λ χ + +γ +4T( ĥ + u +λ) ) ), (4.6) which solves λ χ ( (+γ+4t( ĥ + u +λ)) T g) = g. Then for Rez c T, we have ReΓ (z) c Rez > Rez. (4.7) This implies that Γ (z) z is bounded invertible. Therefore, (4.) can be solved on Ĥ as φ = (Γ z) φ (Γ z) ip K ζ (Γ z) iλp V (Γ3 z) Φ. (4.8) Now we reduce the first equation of (4.9) as follows: [Γ (z) z]ζ = ζ ip K (Γ z) φ λp K (Γ z) P V (Γ3 z) Φ, (4.9) where Γ (z) = P K (Γ z) P K P. We are going to use the same strategy to show that Γ (z) z is bounded invertible. Take ζ = δ w ½ Ĥ = span{δ t j ½,j =, p}. By the gap for Γ in (4.7), we have that Rew, Γ (z)wĥ = (Γ z) K ζ, (ReΓ Rez)(Γ z) K ζ Ĥ K ζ Ĥ c Γ z. Ĥ

22 First, for ζ = δ w ½ Ĥ, w, ζ Ĥ = w C p and by (4.9), we have that K ζ w h(ξ) β (ξ,p) = h(ξ) β (ξ,p) ζ Ĥ. (4.3) Ĥ ξ/ W(ξ) ξ/ W(ξ) On the other hand, for z N + {Rez < T }, by (4.3) and (.4), Γ (z) z Ĥ ĥ + u +4λ (P3 L P 3 z) + z Ĥ3 ( 4 ĥ +4 u +4λ T ) +λ+(γ +)Rez T Put (4.3) and (4.3) together, we obtain Reζ, Γ (z)ζĥ = 4 ĥ +4 u +8Tλ +λ+(γ +)(T) (4.3) c ξ/ W(ξ) h(ξ) β (ξ,p) (4 ĥ +4 u +8Tλ +λ+(γ +)(T) ) ζ Ĥ =: c ζ Ĥ. Therefore, ReΓ (z) > Rez on Ĥ providedz N + and Rez < min{c, T,c } =: g, i.e. Γ (z) z is bounded invertible on Ĥ. The asymptotic behavior of g for small λ follows from the definition of c in (4.6). Now (4.9) can be solved on Ĥ as: ζ = (Γ z) ζ (Γ z) ip K (Γ z) φ (Γ z) λp K (Γ z) P V (Γ z) Φ. (4.3) Therefore, (4.9) is explicitly solvable on Ĥ = Ĥ Ĥ Ĥ3, i.e., J z is bounded invertible for all z {z : Rez < g} N +. Let Ψ = (ζ,φ,φ) and Ψ = (, φ,), where φ l (Z d \{}) ½. It remains to prove (4.7) and (4.8) at z = by solving JΨ = Ψ. The first equation of JΨ = Ψ is reduced to P K φ =, which implies φ Ker(P K ). It is also easy to check that K φ = by Corollary 4.5. Let Π be the projection onto the kernel of P K. Then φ = Π φ and φ = Π φ. In view of the second equation of JΨ = Ψ, we have i K ζ +Γ φ = φ. Notice that Π K =, then Therefore, φ and iπ K ζ +Π Γ φ = Π φ Π Γ Π φ = φ. φ = Π φ = Π J φ = Π (Π Γ Π φ) φ =: J Ψ. Direct computation shows that Re φ, φ = Re i K ζ +Γ φ, φ = ReΓ φ, φ c φ g φ >. The spectral gap g of L has consequences for the dynamics of the semi-group. Lemma 4.. Let Q = orthogonal projection onto Ĥ = spanδ in L ( M;C p ). Then e t L ( Q ) is a contraction semi-group on Ran( Q ), and for all sufficiently small ǫ > there is C ǫ > such that e t L ( Q ) C ǫ e t(g ǫ) (4.33) L ( M;C p )

23 Lemma 4.. There is c > such that L k L L ( M;C p ) c k. If k is sufficiently small, the spectrum of L k consists of:. A non-degenerate eigenvalue E(k) contained in S = {z : z < c k }.. The rest of the spectrum is contained in the half plane S = {z : Rez > g c k } such that S S =. Furthermore, E(k) is C in a neighborhood of, E() =, E() =. (4.34) Denote j = kj and ϕ = p δ ½ for simplicity where p = p p d, then i j E() = j K ϕ, J i K ϕ + i K ϕ, J j K ϕ (4.35) where J = Π J Π is defined in (4.7) in Lemma 4.9. Remark 4.. Let D := (D i,j ) d d = ( i j E()) d d. It is clear from (4.35) that D is symmetric. Furthermore, for any k T d, in view of the expression of i K in (3.4), i k i i K ϕ l (Z d ) ½. It is non-zero due the non-degeneracy of h. Therfore, by (4.8) in Lemma 4.9, Rek, Dk = Re i k i i K ϕ, J k i i K ϕ > g J k i i K ϕ >. i In the next section, we will relate the matrix element of D with limits of diffusively scaled moments. From the real valued moments, we will see that i j E() R and then D is positive definite. Similar to Lemma 4., dynamical information about the semi-group e t L k follows from the spectral gap of L k in Lemma 4.: Lemma 4.3. If ǫ is sufficiently small, then there is C ǫ < such that e t L k ( Q k ) C ǫe t(g ǫ c k ) L ( M;C p ) for all sufficiently small k. Notice that p = p p p d. The case where d = and p = p = is equivalent to the free case considered in [7], where the above lemmas were proved. The proof follows from the standard perturbation theory of analytic semi-group see for instance [],[8]. There are no essential difference in the proof when p >. We omit the proof for Lemma 4.-Lemma 4.3 here. We only sketch the proof for (4.34) and (4.35), which plays the most important role for the explicit expression of the diffusion constant in the next section. Proof of (4.34) and (4.35). Write j = kj for short. Let E(k) be the non-degenerate eigenvalue of L k, and the associated normalized eigenvector ϕ k. Let Q k be the orthogonal projection onto ϕ k. Clearly E() =, ϕ = p δ ½ and L ϕ = L ϕ =. Since L k ϕ k = E(k)ϕ k (4.36) i 3

24 Direct computation shows j Lk ϕ k + L k j ϕ k = j E(k)ϕ k +E(k) j ϕ k (4.37) = j L ϕ + L j ϕ = j E()ϕ. (4.38) Notice that j L = i j K maps Ĥ = RanQ to Ĥ, therefore, Q j L = and j E() = ϕ, j L ϕ + ϕ, L j ϕ = Q ϕ, j L ϕ + L ϕ, j ϕ =. Differential (4.37) agian, we have j j Lk ϕ k + j Lk i ϕ k + i Lk j ϕ k + L k j j ϕ k = j j E(k)ϕ k + j E(k) i ϕ k + i E(k) j ϕ k +E(k) j j ϕ k. (4.39) Evaluating (4.39) at k = and using E() =, we have that j j L ϕ + j L i ϕ + i L j ϕ + L j j ϕ = j j E()ϕ. We also have Q j j L = for the same reason as Q j L. Therefore, j j E() = ϕ, j L i ϕ + ϕ, i L j ϕ = Π j L ϕ, Π i ϕ Π i L ϕ, Π j ϕ, since j L = i j K = j L and j L ϕ l (Z d \{}) ½ Ker(P K ) = Ran(Π ) Ĥ in view of (3.4). It remains to solve Π i ϕ from j L ϕ + L j ϕ = in (4.38). Recall the block form of L in (4.), it is enough to consider the following equations: ip K P ip K P P (i K +iû)p iλp VP3 Π j ϕ = iλp 3 VP P 3 L P 3 By (4.7) in Lemma 4.9, we have Π j ϕ = J j L ϕ = (Π Γ Π ) j L ϕ, j L ϕ where Γ,J are given as in (4.5),(4.7). Therefore, j j E() = j L ϕ, J i L ϕ + i L ϕ, J j L ϕ, which gives (4.35).. 5 Proof of the main results 5. Central limit theorem We first prove (.) for bounded continuous f and normalized ψ l (Z d ). The extension to quadratically bounded f follows from some standard arguments combing (.) for bounded continuous f and diffusive scaling for second moments Lemma 5.. We refer readers to Section 4.5 in [5] for more details about this extension. We omit the proof for the extension here. 4

25 To prove (.) for bounded continuous f, it suffices, by Levy s Continuity Theorem and a limiting argument, to prove ( lim t E ψt (x) ) = e k, Dk, (5.) e ik x t x Z d where ψ t (x) l (Z d ;C) is the solution to eq. (.) with initial condition ψ l (Z d ;C). As pointed out in Sec. 4., [5], it is enough to establish eq. (5.) for ψ l (Z d ;C); it then extends to all of ψ l (Z d ;C) by a limiting argument. So throughout this section, we assume that ψ l =, and ψ l := x Z d ψ (x) <. (5.) We also denote for simplicity ϕ := ϕ (x,ω) = p δ ½, Φ k := Φ k (x,ω) = p ρ ;k (x) ½, (5.3) where, ρ ;k (x) C p are defined in (3.9). Recall that for any σ Z d p π σ =, πσ ρ ; (x) = ψ (x n)ψ ( n). n pz d +σ By (3.3), we have x Z d e i k t x E ( ψ t (x) ) = ϕ, e t L k t Φ k t. L ( M;C p ) Letting Q k denote the Riesz projection onto the eigenvector of L k near zero, we have x Z d e i k x t E ( ψ t (x) ) = ϕ, e t L k tq k t Φ k t + ϕ, e t L k t ( Q k t )Φ k t =e te( k t ) ϕ, Q t k Φ t k + ϕ, e t L k t ( Q k t )Φ k t. (5.4) By Lemma 4.3, the second term in (5.4) is exponentially small in the large t limit, ϕ, e t L k t ) ( Q k t Φ t k ( Q )e t L k t ϕ k Φ t k t C ǫ e t(g ǫ c k t). ϕ Φ t k (5.5) Direct computation shows that lim Φ t k =( p) ρ ; t L ( M;C p l ) (Z d ;C p ) ( p) ψ (x n)ψ ( n) σ Z d p x Z d n pz d +σ ( p) ψ l ψ ( n) σ Z d p n pz d +σ 5

26 ( p) ψ l ψ l <. Therefore, in (5.5), ϕ, e t L k t ) ( Q k t Φ t k as t. Regarding the first term in (5.4), we have by Taylor s formula, ) k E( = j j E() k i k j + o( t t t t ) = j j E()k i k j + o( t t ), i,j since E() = E() =. Thus, Thus, e te(k/ t) = e t t i,j i,j j je()kikj +o() = e i,j j je()kikj +o(). (5.6) Direct compuation shows that ϕ, Φ L δ ( M;C p ) = ½, ρ ; ½ L ( M;C p ) = ψ ( n)ψ ( n) = ψ l (Z d ;C) =. Putting together everything, we have lim e i k t x Z d σ Z d p n pz d +σ Q Φ = Proj ϕ Φ = ϕ, Φ t x E ( ψ t (x) ) = lim ( ) e te k t t =e ϕ ϕ = ϕ. (5.7) ϕ, Q k t Φ k t i,j j je()kikj ϕ, Q Φ = e i,j j je()kikj. Therefore, (5.) holds true with D i,j = j j E() and normalized ψ l (Z d ). 5. Diffusive scaling and reality of the diffusion matrix We proceed to prove the diffusive scaling (.3) under the assumption that ψ (x) =, x ψ (x) <. (5.8) x x Similar to (5.), it is enough to establish the results for xψ l (Z d ;C); it then extends to all of xψ l (Z d ;C) by a limiting argument. We assume that x ψ (x) <. (5.9) x Wecontinuetousethenotationin(5.3)forshort., willstandfor, L ( M;C p ) unlessspecified. We also denote i = ki,i =,,d for short. As pointed out in Sec. 4.4 in [5], x (+ x ) ψ t (x) e Ct for each t >. Thus the second moments of the position M i,j (t) := ( x i x j E ψ t (x) ) (5.) x Z d are well defined and finite. The main task of this section is to show that M i,j (t) D i,j t, where D i,j = i j E() are given in (4.35). More precisely, 6