Almost Eigenvalues and Eigenvectors of Almost Mathieu Operators Thomas Strohmer and Timothy Wertz Department of Mathematics University of California, Davis Davis CA 95616 {strohmer,tmwertz}@math.ucdavis.edu Abstract ThealmostMathieuoperatoristhediscreteSchro¨dingeroperatorH on(cid:96)2(Z)defined α,β,θ via(H f)(k) = f(k+1)+f(k−1)+βcos(2παk+θ)f(k). Wederiveexplicitestimates α,β,θ for the eigenvalues at the edge of the spectrum of the finite-dimensional almost Mathieu op- (n) eratorH . Wefurthermoreshowthatthe(properlyrescaled)m-thHermitefunctionϕ is α,β,θ m (n) anapproximateeigenvectorofH ,andthatitsatisfiesthesamepropertiesthatcharacterize α,β,θ (n) thetrueeigenvectorassociatedtothem-thlargesteigenvalueofH . Moreover,aproperly α,β,θ (n) translated and modulated version of ϕ is also an approximate eigenvector of H , and it m α,β,θ satisfies theproperties that characterize thetrue eigenvector associated to them-th largest (in modulus)negativeeigenvalue. Theresultsholdattheedgeofthespectrum,foranychoiceof θ and under very mild conditions on α and β. We also give precise estimates for the size of the“edge”,andextendsomeofourresultstoH . Theingredientsforourproofscomprise α,β,θ Taylor expansions, basic time-frequency analysis, Sturm sequences, and perturbation theory foreigenvaluesandeigenvectors. Numericalsimulationsdemonstratethetightfitofthetheo- reticalestimates. 1 Introduction WeconsiderthealmostMathieuoperatorH on(cid:96)2(Z),givenby α,β,θ (H f)(x) = f(x+1)+f(x−1)+2βcos(2παk +θ)f(x), (1.1) α,β,θ with x ∈ Z, β ∈ R,α ∈ [−1, 1), and θ ∈ [0,2π). This operator is interesting both from a 2 2 phyiscal and a mathematical point of view [18,19]. In physics, for instance, it serves as a model for Bloch electrons in a magnetic field [13]. In mathematics, it appears in connection with graph theory and random walks on the Heisenberg group [6,11] and rotation algebras [9]. A major part 1 of the mathematical fascination of almost Mathieu operators stems from their interesting spectral properties, obtained by varying the parameters α,β,θ, which has led to some deep and beautiful mathematics, see e.g [3,5,7,15,17]. For example, it is known that the spectrum of the almost Mathieu operator is a Cantor set for all irrational α and for all β (cid:54)= 0, cf. [4]. Furthermore, if β > 1thenH exhibitsAndersonlocalization,i.e.,thespectrumispurepointwithexponentially α,β,θ decayingeigenvectors[14]. A vast amount of literature exists devoted to the study of the bulk spectrum of H and α,β,θ its structural characteristics, but very little seems be known about the edge of the spectrum. For instance, what is the size of the extreme eigenvalues of H , how do they depend on α,β,θ and α,β,θ what do the associated eigenvectors look like? These are exactly the questions we will address in thispaper. While in general the localization of the eigenvectors of H depends on the choice of β, α,β,θ it turns out that there exist approximate eigenvectors associated with the extreme eigenvalues of H which are always exponentially localized. Indeed, we will show that for small α the m-th α,β,θ Hermitian function ϕ as well as certain translations and modulations of ϕ form almost eigen- m m √ vectors of H regardless whether α is rational or irrational and as long as the product α β is α,β,θ small. ThereisanaturalheuristicexplanationwhyHermitianfunctionsemergeinconnectionwithal- most Mathieu operators. Consider the continuous-time version of H in (1.1) by letting x ∈ R α,β,θ and set α = 1,β = 1,θ = 0. Then H commutes with the Fourier transform on L2(R). It α,β,θ is well-known that Hermite functions are eigenfunctions of the Fourier transform, ergo Hermite functions are eigenvectors of the aforementioned continuous-time analog of the Mathieu operator. Of course, it is no longer true that the discrete H commutes with the corresponding Fourier α,β,θ transform(nordowewanttorestrictourselvestoonespecificchoiceofαandβ). Butnevertheless itmaystillbetruethatdiscretized(andperhapstruncated)Hermitefunctionsarealmosteigenvec- tors for H . We will see that this is indeed the case under some mild conditions, but it only α,β,θ √ holds for the first m Hermite functions where the size of m is either O(1) or O(1/ γ), where √ γ = πα β, depending on the desired accuracy (γ2 and γ, respectively) of the approximation. We will also show a certain symmetry for the eigenvalues of H and use this fact to conclude that α,β,θ a properly translated and modulated Hermite function is an approximate eigenvector for the m-th largest(inmodulus)negativeeigenvalue. The only other work we are aware of that analyzes the eigenvalues of the almost Mathieu operator at the edge of the spectrum is [21]. There, the authors analyze a continuous-time model to obtain eigenvalue estimates of the discrete-time operator H . They consider the case β = α,β,θ 1,θ = 0andsmallα andarriveatanestimatefortheeigenvaluesattherightedgeofthespectrum that is not far from our expression for this particular case (after translating their notation into ours and correcting what seems to be a typo in [21]). But there are several differences to our work. First, [21] does not provide any results about the eigenvectors of H . Second, [21] does α,β,θ not derive any error estimates for their approximation, and indeed, an analysis of their approach yields that their approximation is only accurate up to order γ and not γ2. Third, [21] contains no quantitative characterization of the size of the edge of the spectrum. On the other hand, the scope of[21]isdifferentfromours. 2 The remainder of the paper is organized as follows. In Subsection 1.1 we introduction some notation and definitions used throughout the paper. In Section 2 we derive eigenvector and eigen- value estimates for the finite-dimensional model of the almost Mathieu operator. The ingredients for our proof comprise Taylor expansions, basic time-frequency analysis, Sturm sequences, and perturbation theory for eigenvalues and eigenvectors. The extension of our main results to the infinite dimensional almost Mathieu operator is carried out in Section 3. Finally, in Section 4 we complementourtheoreticalfindingswithnumericalsimulations. 1.1 Definitions and Notation Wedefinetheunitaryoperatorsoftranslationandmodulation,denotedT andM respectivelyby a b (T f)(x) := f(x−a) and (M f)(x) := e2πibxf(x), a b where the translation is understood in a periodic sense if f is a vector of finite length. It will be clear from the context if we are dealing with finite or infinite-dimensional versions of T and M . a b Recallthecommutationrelations(seee.g. Section1.2in[12]) T M = e−2πiabM T . (1.2) a b b a The discrete and periodic Hermite functions we will be using are derived from the standard Hermite functions defined on R (see e.g. [1]) by simple discretization and truncation. We do √ choose a slightly different normalization than in [1] by introducing the scaling terms ( 2γ)2l and √ ( 2γ)2l+1,respectively. Definition1.1 ThescaledHermitefunctionsϕ withparameterγ > 0are m m √ forevenm: ϕ (x) = e−γx2(cid:88)2 (√γ)2lc x2l, where c = m!(2 2)2l(−1)m2−l, (1.3) m m,l m,l (2l)!(m −l)! l=0 2 m−1 √ foroddm: ϕ (x) = e−γx2 (cid:88)2 (√γ)2l+1c x2l+1, where c = m!(2 2)2l+1(−1)m2−1−l, m m,l m,l (2l+1)!(m−1 −l)! l=0 2 (1.4) for x ∈ Z and m = 0,1,.... The discrete, periodic Hermite functions of period n, denoted by ϕ ,aresimilartoϕ ,exceptthattherangeforxisx = −n,..., n −1(withperiodicboundary m,n m 2 2 conditions)andtherangeformism = 0,...,n−1. We denote the finite almost Mathieu operator, acting on sequences of length n, by H(n) . It α,β,θ can be represented by an n×n tridiagonal matrix, which has ones on the two side-diagonals and cos(x + θ) as k-th entry on its main diagonal for x = 2παk, k = −n,..., n − 1, and −n2+k k 2 2 j = 0,...,n−1. Here, we have assumed for simplicity that n is even, the required modification for odd n is obvious. Sometimes it is convenient to replace the translation present in the infinite 3 almostMathieuoperatorbyaperiodictranslation. Inthiscaseweobtainthen×nperiodicalmost MathieuoperatorP(n) whichisalmostatridiagonalmatrix;itisgivenby α,β,θ   cos(x +θ) 1 0 ... 0 1 −n 2  1 cos(x−n+1 +θ) 1 0 ... 0   2  Pα(n,β),θ =  0... 1 cos(x−n2+2 +θ) .1.. ... 0... ,   0 ... 0 1 cos(x +θ) 1  n−2  2 1 0 ... 0 1 cos(x +θ) n−1 2 (1.5) wherex = 2παk fork = −n,..., n −1. Ifθ = 0wewriteP(n) insteadofP(n) . k 2 2 α,β α,β,θ 2 Finite Hermite functions as approximate eigenvectors of the finite-dimensional almost Mathieu operator In this section we focus on eigenvector and eigenvalue estimates for the finite-dimensional model of the almost Mathieu operator. Finite versions of H are interesting in their own right. On the α,β,θ onehand,numericalsimulationsarefrequentlybasedontruncatedversionsofH ,ontheother α,β,θ handcertainproblems,suchasthestudyofrandomwalksontheHeisenberggroup,areoftenmore naturallycarriedoutinthefinitesetting. We first gather some properties of the true eigenvectors of the almost Mathieu operator, col- lectedinthefollowingproposition. Proposition2.1 Consider the finite, non-periodic almost Mathieu operator H(n) . Let ζ ≥ ζ ≥ α,β,θ 0 1 ··· ≥ ζ be its eigenvalues and ψ ,ψ ,...,ψ the associated eigenvectors. The following n−1 0 1 n−1 statementshold: 1. H(n) ψ = ζ ψ forall0 ≤ m ≤ n−1; α,β,θ m m m 2. ThereexistconstantsC ,C ,independentofnsuchthat,forallm 1 2 |ψ (i)| ≤ C e−C2|i|; m 1 3. If ζ is one of the m-th largest eigenvalues (allowing for multiplicity), then ψ changes sign r r exactlymtimes. Proof The first statement is simply restating the definition of eigenvalues and eigenvectors. The second statement follows from the fact that the inverse of a tri-diagonal (or an almost tri-diagonal operator)exhibitsexponentialoff-diagonaldecayandtherelationshipbetweenthespectralprojec- tion and eigenvectors associated to isolated eigenvalues. See [8] for details. The third statement is aresultofLemma2.2below. 4 Lemma2.2 Let A be a symmetric tridiagonal n × n matrix with entries a ,...,a on its main 1 n diagonal and b ,...,b on its two non-zero side diagonals. Let λ ≥ λ ≥ ··· ≥ λ be the 1 n−1 0 2 m eigenvaluesofAwithmultiplicitiesr ,...,r . Letv(1),...,v(r0),v(1),...,v(rm) betheassociated 0 m 0 0 1 m eigenvectors. Then, for each 0 ≤ i ≤ m and each 1 ≤ j ≤ r the entries of the vector v(jm) m m i change signs m times. That is, for all 1 ≤ j ≤ r , the entries of each of the vectors v(j0) all have 0 0 0 thesamesign, whileforall1 ≤ j ≤ r eachof the vectorsv(j1) hasonlya singleindexwherethe 1 1 1 signoftheentryatthatindexisdifferentthantheonebeforeit,andsoon. Proof ThisresultfollowsdirectlyfromTheorem6.1in[2]whichrelatestheSturmsequencetothe sequence of ratios of eigenvector elements. Using the assumption that b > 0 for all i = 1,...,n i yieldstheclaim. The main results of this paper are summarized in the following theorem. In essence we show that the Hermite functions (approximately) satisfy all three eigenvector properties listed in Propo- sition2.1. Thetechnicaldetailsarepresentedlaterinthissection. Theorem2.3 Let A be either of the operators P(n) or H(n) . Let ϕ be as defined in Defini- √ α,β,θ α,β,θ m,n tion 1.1. Set γ = πα β, let 0 < ε < 1, and assume 4 < γ < 1 . Then, for m = 0,...,N n2−ε nε whereN = O(1),thefollowingstatementshold: 1. Aϕ ≈ λ ϕ ,whereλ = 2β +2e−γ −4mγe−γ; m,n m m,n m 2. Foreachm,thereexistconstantsC ,C ,independentofnsuchthat 1 2 |ϕ (i)| ≤ C e−C2|i|; m,n 1 3. Foreach0 ≤ m ≤ n−1,theentriesofϕ changesignsexactlymtimes. m,n Proof Thesecondpropertyisanobviousconsequenceofthedefinitionofϕ ,whilethefirstand m,n third are proved in Theorem 2.6 and Lemma 2.8, respectively. That the theorem applies to both P(n) andH(n) isaconsequenceofCorollary2.7. α,β,θ α,β,θ In particular, the (truncated) Gaussian function is an approximate eigenvector associated with the largesteigenvalueofP(n) (thiswasalsoprovenbyPersiDiaconis[10]). α,β,θ Infact,viathefollowingsymmetryproperty,Theorem2.3alsoappliestothemsmallesteigen- valuesofP(n) andtheirassociatedeigenvectors. α,β,θ Proposition2.4 If ϕ is an eigenvector of P(n) with eigenvalue λ, then M T ϕ is an eigen- α,β,θ 1/2 1/2α vectorofP(n) witheigenvalue−λ. α,β,θ Proof We assume that θ = 0, the proof for θ (cid:54)= 0 is left to the reader. It is convenient to express P(n) as α,β P(n) = T +T +βM +βM . α,β 1 −1 α −α 5 NextwestudythecommutationrelationsbetweenP(n) andtranslationandmodulationbyconsid- α,β ering T M (T +T +βM +βM )(T M )∗. (2.1) a b 1 −1 α −α a b Using(1.2)wehave T M T M T = e2πibT , T M T M T = e−2πibT , a b 1 −m −a 1 a b −1 −m −a −1 and T M M M T = e−2πiαaM , T M M M T = e2πiαaM . a b α −b −a α a b −α −m −a −α Note that e2πib = ±1 if b ∈ 1Z and e−2πiαa = ±1 if aα ∈ 1Z. In particular, if b = 1 and a = 1 2 2 2 2α weobtain T M H (T M )∗ = −H . 1 1 α,β 1 1 α,β 2α 2 2α 2 Hence, since P(n)T M v = −T M P(n)v for any v ∈ Cn, it follows that if ϕ is an eigenvector α,β 1 1 1 1 α,β 2α 2 2α 2 of P(n) with eigenvalue λ, then T M ϕ is an eigenvector of P(n) with eigenvalue −λ. Finally, α,β 1 1 α,β 2α 2 note that if the entries of ϕ change signs k times, then the entries of M ϕ change sign exactly 1 2 n−1−k times,whilethetranslationoperatordoesnotaffectthesignsoftheentries. Theattentivereaderwillnotethatthepropositionabovealsoholdsfortheinfinite-dimensional almostMathieuoperatorH . α,β,θ In the following lemma we establish an identity about the coefficients c in (1.3) and (1.4), m,l andthebinomialcoefficientsb ,whichwewillneedlaterintheproofofTheorem2.6. 2l,k Lemma2.5 Form = 0,...,n−1;l = 0,..., m,thereholds 2 2c b −4c b = −4mc , (2.2) m,l 2l,2 m,l−1 2l−2,1 m,l−1 whereb = (cid:0)j(cid:1) = j! . j,k k k!(j−k)! Proof To verify the claim we first note that b = 2l(2l−1) and b = 2l − 2. Next, note that 2l,2 2 2l−2,1 (2.2)isequivalentto 2l−2−m c = 4 c . (2.3) m,l m,l−1 2l(2l−1) Now,forevenmwecalculate √ √ (2 2)2l(−1)m2−l −8(cid:0)m −l+1(cid:1)m!(2 2)2l−2(−1)m2−l+1 c = m! = 2 m,l (2l)!(m −l)! 2l(2l−1)(2l−2)!(cid:0)m −l+1(cid:1) 2 2 (cid:0) (cid:1) −8 m −l+1 2l−2−m = 2 c = 4 c , m,l−1 m,l−1 2l(2l−1) 2l(2l−1) asdesired. Thecalculationisalmostidenticalforoddm,andislefttothereader. Whilethetheorembelowisstatedforgeneralα,β,θ (withsomemildconditionsonα,β),itis most instructive to first consider the statement for α = 1,β = 1,θ = 0. In this case the parameter n 6 γ appearing below will take the value γ = π. The theorem then states that at the right edge of n the spectrum of P(n) , ϕ is an approximate eigenvector of P(n) with approximate eigenvalue α,β,θ m,n α,β,θ λ = 2β +2e−γ −4mγe−γ, and a similar result holds for the left edge. The error is of the order m 1 andtheedgeofthespectrumisofsizeO(1). Ifweallowtheapproximationerrortoincreaseto n2 √ beoforder 1,thenthesizeoftheedgeofthespectrumwillincreasetoO( n). n √ Theorem2.6 Let P(n) be defined as in (1.5) and let α,β ∈ R+ and θ ∈ [0,2π). Set γ = πα β α,β,θ andassumethat 4 ≤ γ < 1. n2 (1)Form = 0,1,...,N,whereN = O(1),thereholdsforallx = −n,..., n −1 2 2 (cid:12) (cid:18) (cid:19)(cid:12) (cid:12)(cid:12)(P(n) ϕ )(x)−λ ϕ x+ θ (cid:12)(cid:12) ≤ O(γ2), (2.4) (cid:12) α,β,θ m,n m m,n 2πα (cid:12) where λ = 2β +2e−γ −4mγe−γ. (2.5) m (2)Form = −n+1,−n+2,...,−N,whereN = O(1),thereholdsforallx = −n,..., n −1 2 2 (cid:12) (cid:18) (cid:19)(cid:12) (cid:12)(cid:12)(P(n) ϕ )(x)−λ (−1)xϕ x+ θ − 1 (cid:12)(cid:12) ≤ O(γ2), (2.6) (cid:12) α,β,θ m,n m m,n 2πα 2α (cid:12) where λ = −(2β +2e−γ −4mγe−γ). (2.7) m Proof We prove the result for θ = 0 and for even m; the proofs for θ (cid:54)= 0 and for odd m are similarandlefttothereader. Furthermore,forsimplicityofnotation,throughouttheproofwewill writeH insteadofP(n) andϕ insteadofϕ . α,β m m,n Wecomputeforx = −n,..., n −1(recallthatϕ (n −1) = ϕ(−n)duetoourassumptionof 2 2 m 2 2 periodicboundaryconditions) (cid:1) (Hϕ )(x) = ϕ (x+1)+ϕ (x−1)+2βcos(2παx) ϕ (x) (2.8) m m m m (cid:16) (cid:88) (cid:88) (cid:17) = e−γx2e−γ e−2γx · c γl(x+1)2l +e2γx · c γl(x−1)2l + (2.9) m,l m,l l l (cid:88) +e−γx22βcos(2παx) c γlx2l. (2.10) m,l l Weexpandeachoftheterms(x±1)2l intoitsbinomialseries,i.e., 2l 2l (cid:18) (cid:19) (cid:88) (cid:88) 2l (x+1)2l = b x2l−k1k and (x−1)2l = b x2l−k(−1)k, where b = , (2.11) 2l,k 2l,k 2l,k k k=0 k=0 7 andobtainaftersomesimplecalculations m/2 (cid:104)(cid:88) (cid:16) (cid:17)(cid:105) (Hϕ )(x) = e−γx2 c γlx2l e−γb (e−2γx +e2γx)+2βcos(2παx) + (2.12) m m,l 2l,0 l=0 m/2 2l (cid:104)(cid:88) (cid:88) (cid:105) +e−γx2 c γl b x2l−ke−γ(e−2γx +(−1)ke2γx) (2.13) m,l 2l,k l=0 k=1 = (I) +(II). (2.14) Wewillnowshowthat(I) = ϕ (x)·(2β+2e−γ)+O(γ2)and(II) = −ϕ (x)·4mγe−γ+O(γ2), m m fromwhich(2.4)and(2.5)willfollow. Wefirstconsidertheterm(I).Werewrite(I)as m/2 (cid:16) (cid:17)(cid:88) (I) = e−γx2 e−γ(e−2γx +e2γx)+2βcos(2παx) c γlx2l. m,l l=0 Using Taylor approximations for e−2γx,e2γx, and cos(2παx) respectively, we obtain after some rearrangements(whicharejustifiedduetotheabsolutesummabilityofeachoftheinvolvedinfinite series) (cid:16) (cid:17) e−γx2 e−γ(e−2γx +e2γx)+2βcos(2παx) = (cid:104) (−2γx)2 (−2γx)3 = e−γx2e−γ (cid:0)1+(−2γx)+ + +R (x)(cid:1) (2.15) 1 2! 3! (2γx)2 (2γx)3 (cid:105) (cid:0) (cid:1) + 1+(2γx)+ + +R (x) 2 2! 3! (2παx)2 +e−γx22β(cid:0)1+ +R (x)(cid:1) 3 2! (cid:16) (2γx)2 (2παx)2 (cid:17) = e−γx2 2e−γ +2e−γ +2β −2β +e−γ(cid:0)R (x)+R (x)(cid:1)+2βR (x) , (2.16) 1 2 3 2! 2! whereR ,R ,andR aretheremaindertermsoftheTaylorexpansionfore−2γx,e2γx,andcos(2παx) 1 2 3 (inthisorder)respectively,givenby (−2γ)4e−2γξ1 (2γ)4e2γξ2 (2πα)4cos(ξ ) R (x) = x4, R (x) = x4, R (x) = 3 x4, 1 2 3 4! 4! 4! withrealnumbersξ ,ξ ,ξ between0andx. Weuseasecond-orderTaylorapproximationfore−γ 1 2 3 withcorrespondingremaindertermR (γ) = e−ξ4γ3 (forsomeξ ∈ (0,γ))in(2.16). 4 3! 4 Hence(2.16)becomes (cid:18) (cid:18) γ2 (cid:19) (cid:19) e−γx2 2e−γ +(2γx)2 + −γ + +R (γ) (2γx)2 +2β −β(2παx)2 + (2.17) 4 2! (cid:18)(cid:18) γ2 (cid:19) (cid:19) +e−γx2 1−γ + +R (γ) (cid:0)R (x)+R (x)(cid:1)+2βR (x) . (2.18) 4 1 2 3 2! 8 √ Since γ = πα β, the terms (2γx)2 and −β(2παx)2 in (2.17) cancel. Clearly, |R (x)| ≤ (2γx)4, 1 4! |R (x)| ≤ e2γx(2γx)4,and|R (γ)| ≤ (γ)3. Itisconvenienttosubstituteα = √γ inR (x),inwhich 2 4! 4 3! π β 3 caseweget|R (x)| ≤ (2γx)4. Thus,wecanboundtheexpressionin(2.18)fromaboveby 3 β24! (cid:12) (cid:16) γ2 (cid:17)(cid:12) (cid:12)e−γx2 (cid:0)−γ + +R (γ)(cid:1)(cid:0)R (x)+R (x)(cid:1)+2βR (x) (cid:12) (2.19) (cid:12) 2! 4 1 2 3 (cid:12) (cid:16) γ2 (γ)3 (2γx)4 e2γx(2γx)4 (2γx)4 (cid:17) ≤ e−γx2 (cid:0)γ + + (cid:1)(cid:0) + +2β (cid:1) . (2.20) 2! 3! 4! 4! β24! Assumenowthat|x| ≤ √1 thenwecanfurtherboundtheexpressionin(2.20)fromaboveby γ (cid:16) γ2 γ3 (2γx)4 e2γx(2γx)4 (2γx)4 (cid:17) e−γx2 (cid:0)γ + + (cid:1)(cid:0) + +2β (cid:1) (2.21) 2! 3! 4! 4! β24! √ (cid:16) γ2 γ3 2γ2 2e2 γγ2 4γ2 (cid:17) (cid:0) (cid:1)(cid:0) (cid:1) ≤ γ + + + + ≤ O(γ3)+O(γ3/β). (2.22) 2! 3! 3 3 3β Moreover,if|x| ≤ √1γ,wecanboundtheterm(cid:0)−γ + γ22! +R4(γ)(cid:1)(2γ2x!)2 in(2.17)by (cid:12) γ2 (2γx)2(cid:12) γ4 (cid:0) (cid:1) (cid:12) −γ + +R (γ) (cid:12) ≤ 2γ2 +γ3 + ≤ O(γ2). (cid:12) 2! 4 2! (cid:12) 3 (cid:113) (cid:16)(cid:113) (cid:17)c Nowsuppose|x| ≥ 1. Weset|x| = 1 forsomecwith γ γ 2log(n/2) 1 < c ≤ . (2.23) log(1/γ) The upper bound in (2.23) ensures that |x| ≤ n and the assumption 4 ≤ γ < 1 implies that 2 n2 condition(2.23)isnotempty. Then (cid:16) γ2 (γ)3 (2γx)4 e2γx(2γx)4 (2γx)4 (cid:17) e−γx2 (cid:0)γ + + (cid:1)(cid:0) + +2β (cid:1) = (2.24) 2! 3! 4! 4! β24! (cid:16) γ2 (γ)3 (2γ)4−2c e2γ1−c(2γ)4−2c 4γ4−2c (cid:17) = e−γ1−c (cid:0)γ + + (cid:1)(cid:0) + + (cid:1) ≤ O(γ2), (2.25) 2! 3! 4! 4! 3β where the last inequality follows from basic inequalities like e−γ1−cγ4−2c ≤ γ2 (which in turn followsfromey ≥ y2 forally ≥ 0). Thuswehaveshownthat (cid:16) (cid:17) e−γx2 e−γ(e−2γx +e2γx)+2βcos(2παx) = e−γx2(cid:0)2e−γ +2β(cid:1)+O(γ2). (2.26) Returningtotheterm(I)in(2.12),weobtain,using(2.26), m/2 (cid:88) (I) = ϕ (x)(2e−γ +2β)+C c γlx2l (2.27) m γ m,l l=0 9 whereC = O(γ2). γ Let us analyze the error term C (cid:80)m/2c γlx2l. Using Stirling’s Formula ( [1, Page 257]) we γ l=0 m,l notethatc growsatleastasfastas(cid:0)m/e(cid:1)m/2,butnotfasterthan(m/e)m. Hence,theerrorterm m,l willremainofsizeO(γ2),aslongasweensurethatmdoesnotexceedO(1). (cid:0) (cid:1) We now proceed to showing that (II) = −ϕ (x)· 4mγe−γ +O(γ2). The key to this part of m the proof is the observation that H acts “locally” on the powers x2l that appear in the definition of ϕ . Recallthattheterm(II)hastheform m m/2 2l (cid:104)(cid:88) (cid:88) (cid:105) e−γx2 c γl b x2l−ke−γ(e−2γx +(−1)ke2γx) . (2.28) m,l 2l,k l=0 k=1 Analogoustothecalculationsleadingupto(2.26)wecanshowthatforoddk thereholds e−γx2e−γ(e−2γx +(−1)ke2γx) = (−4γx)e−γx2e−γ +O(γ2), (2.29) andforevenk e−γx2e−γ(e−2γx +(−1)ke2γx) = 2e−γx2e−γ +O(γ2). (2.30) Using(2.29)and(2.30)wecanexpress(2.28)as m/2 (cid:0)e−γx2e−γ +O(γ2)(cid:1)(cid:104)(cid:88)c γl(cid:0)(cid:88)(−4γx)b x2l−k + (cid:88) 2b x2l−k(cid:1)(cid:105). (2.31) m,l 2l,k 2l,k l=0 oddk evenk Furthermore, using estimates similar to the ones used in deriving the bounds for (I), one easily verifiesthat e−γx2e−γγl(−4γx)x2l−k ≤ O(γ2), fork ≥ 3, and e−γx2e−γc γlx2l−2 ≤ O(γ2), fork ≥ 4. m,l Therefore, m/2 e−γx2e−γ(cid:104)(cid:88)c γl(cid:0)(cid:88)(−4γx)b x2l−k + (cid:88) 2b x2l−k(cid:1)(cid:105) = (2.32) m,l 2l,k 2l,k l=0 oddk evenk m/2 (cid:0)e−γx2e−γ +O(γ2)(cid:1)(cid:104)(cid:88)c γl(cid:0)(−4γx)b x2l−1 +2b x2l−2(cid:1)(cid:105). (2.33) m,l 2l,1 2l,2 l=0 TheexpressionaboveimpliesthatH acts“locally”onthepowersx2l ofϕ . m Moreover, m/2 e−γx2e−γ(cid:104)(cid:88)c γl(cid:0)(−4γx)b x2l−1 +2b x2l−2(cid:1)(cid:105) (2.34) m,l 2l,1 2l,2 l=0 (cid:104) = e−γx2e−γ (−4γm)c γm/2xm +2c γm/2b xm−2 (2.35) m,m m,m m,2 2 2 m/2−1 (cid:88) (cid:1)(cid:105) + (−4γ)c γlb x2l +2c γlb x2l−2 , (2.36) m,l 2l,1 m,l 2l,2 l=0 10