We prove an upper bound for the (differentiated) density of states of the
Anderson model at the bottom of the spectrum. The density of states is shown to
exhibit the same Lifshitz tails upper bound as the integrated density of
states

Combes, Jean-Michel

Germinet, François

Klein, Abel

English

arXiv

International audienceWe prove an upper bound for the (differentiated) density of states of the Anderson model at the bottom of the spectrum. The density of states is shown to exhibit the same Lifshitz tails upper bound as the integrated density of states

HAL AMU

Lifshitz tails estimate for the density of states of the Anderson model

COMBES, Jean-Michel

GERMINET, Francois

KLEIN, Abel

Kyoto University Research Information Repository

Title Lifshitz tails estimate for the density of states of the Andersonmodel (Spectra of Random Operators and Related Topics)Author(s)COMBES, Jean-Michel; GERMINET, Francois; KLEIN, AbelCitation数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu(2011), B27: 1-9Issue Date2011-07URL http://hdl.handle.net/2433/187900RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityRIMS Ko^kyu^roku BessatsuB27 (2011), 1{9Lifshitz tails estimate for the density of states of theAnderson modelByJean-Michel Combes, Francois Germinetand Abel KleinAbstractWe prove an upper bound for the (dierentiated) density of states of the Anderson modelat the bottom of the spectrum. The density of states is shown to exhibit the same Lifshitztails upper bound as the integrated density of states.x 1. Introduction and main resultWe consider the Anderson model, the simplest random Schrodinger operator, givenby the random Hamiltonian(1.1) H! :=  + V! on `2(Zd);  is the d-dimensional discrete Laplacian operator (i.e.  u(j) = Pjk jj=1 u(k),where u 2 `2(Zd)) and V! is the random potential given by V!(j) = !j for j 2 Zd,where ! = f!jgj2Zd is a family of independent identically distributed random variableswhose common probability distribution  is non-degenerate, with support bounded frombelow, and has a bounded density . (The requirement inf supp >  1 is equivalentto requiring H! to be bounded from below with probability one.) Note that(1.2) V! =Xj2Zd!jj ;Received August 28, 2010. Revised February 8, 2011.2000 Mathematics Subject Classication(s): Primary 82B44; Secondary 47B80, 60H25Key Words: Lifshitz tail, density of states, Anderson localization, Wegner estimateJM.C. and F.G. were supported by ANR 08 BLAN 0261. A.K was supported in part by NSF GrantDMS-1001509CPT, CNRS, Luminy Case 907, Cedex 9, F-13288 Marseille, France.e-mail: combes@cpt.univ-mrs.frUniversite de Cergy Pontoise, IUF, UMR CNRS 8088, F-95000 Cergy-Pontoise, France.e-mail: francois.germinet@u-cergy.frUniversity of California, Irvine, Department of Mathematics, Irvine, CA 92697-3875, USA.e-mail: aklein@uci.educ 2011 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.2 J.M. Combes, F. Germinet and A. Kleinwhere j is the orthogonal projection onto j , the delta function at site j.The integrated density of states (IDS) of this celebrated model is known to exhibitLifshitz tails behavior at the bottom of its spectrum (e.g., [CaL, K, PF]), a propertythat can be interpreted as a rst signature of localization.In the physics literature there is much interest on the density of the IDS, the densityof states (DOS). There is an implicit assumption that the IDS N(E) is absolutely con-tinuous, and hence has a density n(E) given by its almost everywhere derivative. Veryfew mathematical results are available for the DOS of random Schrodinger operators.For nice enough models, like the Anderson model as above, the existence of the DOSis a consequence of the celebrated Wegner estimate (see (2.4) below), which also showsthe DOS to be bounded. We note that while the absolute continuity of the IDS hasbeen known for a long time for the Anderson model [W, FS], this is a more recent resultfor models in the continuum [CoH, CoHK]. In dimension d = 1 this DOS is knownto be regular [ST, CK, VSW]. For arbitrary dimension d, the DOS of the Andersonmodel described above is known to be regular at high disorder, a result obtained usingsupersymmetric methods by [BoCKP]. Regularity of the DOS is an open question formodels in the continuum.In this article, we show that the DOS exhibits the same Lifshitz tails upper boundas the IDS. To our best knowledge, this is the rst time that such a bound is proved forthe DOS of a random Schrodinger operator.There is another reason to study the DOS n(E). As proved by Minami [M], in thelocalization region the properly rescaled eigenvalues of a discrete Anderson Hamiltonianare distributed as a Poisson point process with intensity n(E). Thus our results (see(2.13)) estimate the intensity of these Poisson point processes.An Anderson Hamiltonian H! is a Zd-ergodic family of random self-adjoint op-erators. It follows from standard results (cf. [KM]) that there exists xed subsets ,pp, ac and sc of R so that the spectrum (H!) of H!, as well as its pure point,absolutely continuous, and singular continuous components, are equal to these xed setswith probability one. This non-random spectrum is given by(1.3)  = ( ) + supp = [0; 4d] + supp:Note inf  = inf supp >  1.If we set !0j = !j   inf supp, the new common probability distribution 0 satisesinf supp0 = 0, and we have H! = H!0 + inf supp. Thus, without loss of generalitywe assume from now on that(1.4) inf  = inf supp = 0:Lifshitz tails and localization are known to hold at the bottom of the spectrum[FMSS, DK, K]. If the support of  is also bounded from above, then so is , andLifshitz tails for the DOS 3Lifshitz tails and localization also hold at the top, i.e., upper edge, of the spectrum. Inthis case the results of this paper also hold at the top of the spectrum.The integrated density of states (IDS), N(E), can be written as (e.g., [K])(1.5) N(E) = Etr 0] 1;E](H!)0	= Etr 0[0;E](H!)0	:The Anderson model is known to satisfy the following Lifshitz tails estimate, whichasserts that the IDS has an exponential fall o as one approaches the edges of . Atthe bottom of the spectrum, i.e., at energy E = 0, the IDS satises (e.g., [K])(1.6) limE#0log jlogN(E)jlogE  d2:Equality is actually known to hold in (1.6). Since N(E) is an increasing function, it hasa derivative n(E) := N 0(E) almost everywhere, which is the density of states.We proveTheorem 1.1. Let H! be an Anderson Hamiltonian. Then there exists a Borelset N  [0; 1] of zero Lebesgue measure, such that(1.7) limE#0; E=2Nlog jlog n(E)jlogE  d2:Remark 1.2. The same Lifshitz tails estimate holds for models in the continuum;see [CoGK3].The proof of Theorem 1.1 takes advantage of a new double averaging procedureintroduced in [CoGK2, Theorem 2.2] to extract better control on the constant in theWegner estimate. Theorem 1.1 will be an immediate consequence of Theorem 2.1.x 2. Finite volume operators, the integrated density of states, and theWegner estimateFinite volume operators will be dened for nite boxes(2.1)  = L(j) := j + L2 ; L2 d ;where j 2 Zd and L 2 2N, L > 1. Given such , we will consider the random Schrodingeroperator H()! on `2() given by the restriction of the Anderson Hamiltonian H! to with periodic boundary condition. To do so, we identify  with the torus Zd=LZd inthe usual way, and dene nite volume operatorsH()! :=  () + V ()! on `2();(2.2)4 J.M. Combes, F. Germinet and A. Kleinwhere () is the Laplacian on  with periodic boundary condition, and the randompotential V()! is the restriction of V!() to , where, given ! = f!igi2Zd , we dene!() =n!()ioi2Zdby!()i = !i if i 2 ;!()i = !()k if k   i 2 LZd:(2.3)The nite volume random operator H()! is covariant with respect to translationsin the torus. If B  R is a Borel set, we write P ()! (B) := BH()!and P!(B) :=B(H!) for the spectral projections.The nite volume operator H()! is a nite dimensional operator, and hence its(!-dependent) spectrum consists of a nite number of isolated eigenvalues with nitemultiplicity. These nite volume operators satisfy a Wegner estimate [W, FS] (see also[CoGK1]), which provides informations on the number of eigenvalues in a given energyinterval: given E0 > 0, there exists a constant KW (E0), independent of , such thatfor all intervals I  [0; E0] we have(2.4) EntrP ()! (I)o KW (E0)kk1 jIj jj :The Wegner estimate is an an immediate consequence of the following spectral averagingproperty (e.g., [CoHK, CoGK1]) : for any k 2 ,(2.5) E!knhk; P ()! (I)kio= E!kntrkP()! (I)ko kk1 jIj ;where E!k denotes integration over the variable !k, and (k)k2Zd stands for the canonicalbasis (so k = jkihkj). It follows that for the Anderson model (2.4) holds with(2.6) KW (E0)  1 for all E0 > 0:We shall prove that at the bottom of the spectrum, in the Lifshitz tails region, theconstant KW (E0) falls o in the same way as the IDS.We set(2.7) N ()! (E) := jj 1 tr ] 1;E]H()!;and recall that (e.g., [CaL, K, PF]) for P-a.e. ! we have, using the fact that N(E) is acontinuous function by (2.4),(2.8) N(E) = limL!1N (L(0))! (E) for all E 2 R;where N(E) is the integrated density of states (IDS) given in (1.5)). Setting(2.9) N ()(E) := EnN ()! (E)o= Entrj] 1;E](H()! )jofor j 2 ;Lifshitz tails for the DOS 5the last equality holding in view of the periodic boundary condition, it follows that(2.10) N(E) = limL!1N (L(0))(E) = Etr 0] 1;E](H!)0	for all E 2 R.Combining (2.4) and (2.10), we conclude that, if N(E) is dierentiable at E, whichis true for a.e. E, we have(2.11) n(E) := N 0(E)  KW (E)kk1:Moreover, to obtain (2.11), it suces to have the Wegner estimate (2.4) for boxes Ln(0)with Ln !1. Thus Theorem 1.1 follows immediately from the following result.Theorem 2.1. Let H! be an Anderson Hamiltonian. Then there is an energyE0 > 0, and for a.e. E 2]0; E0] and all " 2]0; d2 [ there exists a scale L(E; ") < 1,such that given L 2 4N with L  L(E; "), the Wegner estimate (2.4) holds in all boxes = L for all intervals I  [0; E] with a constant KW (E) satisfying(2.12) KW (E)  Cd;" e E  d2+";for some constant Cd;" <1. As a consequence, we have(2.13) n(E)  Cd;" kk1 e E  d2+"for a.e. E 2]0; E0]:x 3. The proof of Theorem 2.1We borrow from [CoGK3] the following observation.Lemma 3.1 ([CoGK3]). Let H = H0 +W , where H;H0 are semi-bounded self-adjoint operators, say H;H0    for some  > 0, such that (H ++ 1) p is a traceclass operator for some p > 0, and W is a bounded self-adjoint operator. Given E0 2 R,let f; g be bounded Borel measurable nonnegative functions such that f = ( 1;E0]f and( 1;E0]  g  1. Then f(H)W and f(H)Wg(H0) are trace class operators, and(3.1) tr f(H)W  tr f(H)Wg(H0):The proof is elementary. It consists in proving that tr f(H)W (1   g(H0))  0,using W = H  H0.Proof of Theorem 2.1. All the operators in this proofs will be nite volume oper-ators on a box  = L(0) as dened in (2.2)-(2.3). We will require L 2 4N.We set !?0 = ! n f!0g and H!?0 = H!   !00. Given E > 0, we x a C1 real-valued non-increasing function fE on R, such that fE(t) = 1 for t  E, fE(t) = 0 for6 J.M. Combes, F. Germinet and A. Kleint  2E, andf (j)E (t)  CE j for all t 2 R and j = 1; 2; : : : ; 2d + 3, where C is aconstant independent of E. We let ~P0 = ~P()!?0 ;E= fE(H()!?0).Given an interval I  [0; E], it follows from Lemma 3.1 thattrP ()! (I)0  trP ()! (I)0 ~P0 =Xk2trP ()! (I)0 ~P0kXk2kP ()! (I)0k2kP ()! (I)kk2k0 ~P0kk(3.2) 12Xk2ntr(0P()! (I)0) + tr(kP()! (I)k)ok0 ~P0kk:Next, for any given k (including k = 0) there exists k0 2 Zd, so that, dening thesublattice  k = k0 + (4Z)d  Zd, we have 0; k 62  k. Since we required L 2 4N, wehave j k \ j = 4 djj. We dene ! k = f!g2 k . We have, using the fast decay ofthe kernel of smooth functions of Schrodinger operators (e.g. [GK]), using the notationhki =q1 + jkj2, and letting r stand for either 0 or k,k0 ~P0kk  k0 ~P0kk 12 kr ~P0k 12  k0 ~P0kk 12 kr ~P0rk 14 CdEd+2 hkid+1trr ~P0r 14(3.3) CdEd+2 hkid+1e2tE trre tH()!?0 r 14 CdEd+2 hkid+1e2tE trre tH()! kr 14;for all t > 0, where we used r 62  k and a positivity preserving argument like [BGKS,Lemma 2.2] to get the last inequality. Hence, again letting r stand for either 0 or k, sor 62  k, using the spectral averaging (2.5) with the bound (2.6) and (3.3) leads toEtrnrP()! (I)rok0 ~P0kk CdEd+2 hkid+1E!?r(e2tE trre tH()! kr 14E!rntrrP()! (I)ro) CdEd+2 hkid+1kk1jIje t2E E! ktrre tH()! kr 14! CdEd+2 hkid+1kk1jIje t2EE! ktrre tH()! kr 14:(3.4)Lifshitz tails for the DOS 7Thanks to (3.2) and (3.4), it now suces to bound the quantityE! ktrre tH()! krwith r = 0; k:(3.5)To alleviate the notation, we write from now on   =  k = k0 + (4Z)d. For all t > 0 wehavetrre tH! r  trr] 1;4E](H()!  )r + e 4tE trr= trr] 1;4E](H()!  )r + e 4tE :(3.6)By the (4Z)d-ergodicity of H()!  , and taking advantage of the periodic boundary condi-tion, we haveEntrr] 1;4E](H()!  )ro=1j  \ jX2(r+(4Z)d)\E! ntr] 1;4E](H()!  )o 4djjE! ntr] 1;4E](H()!  )o:(3.7)We now use the Lifshitz tails estimate for H!  to bound E! ntr] 1;4E](H()!  )o.Note that   is a strict sublattice of Zd, so we lack the so called covering condition. TheLifshitz tails estimate (1.6) is nevertheless valid for H!  (e.g. [K]), and it implies thatfor all " 2]0; d2 [ there is an energy E" > 0 such that (on Zd)(3.8) N (E) = E! trk0] 1;E](H! )k0	  e E  d2 +" for all E  E":Given a box L = L(0), we set, similarly to (2.7) and (2.9),(3.9) N (L)!  (E) = jLj 1 tr] 1;E](H(L)!  ); N (L)  (E) = E! nN (L)!  (E)o:Since H!  is (4Z)d-ergodic, for P-a.e. !  we have [CaL, PF]N (E) = limL!1N (L)!  (E) for a.e. E 2 R;(3.10)so we conclude(3.11) N (E) = limL!1N(L)  (E) for a.e. E 2 R:(Compare with (2.8) and (2.10), where convergence holds for all E. The dierence isthe lack of a Wegner estimate for H!  .) It follows that for a.e. E 2]0; E"] there existsL(E; ") <1 such that for all L  L(E; ") we have(3.12) N(L)  (E)  2N (E)  2e E  d2+":8 J.M. Combes, F. Germinet and A. KleinThus, for a.e. E 2]0; 14E"] and L  L(E; "), combining (3.6), (3.7), and (3.12), wegetEtrre tH! r	  2  4de (4E)  d2 +" + e 4tE :(3.13)Choosing t = tE by(3.14) e 4tE = 2  4de (4E) d2+"; i.e., t = (4E) d2 1+"   (1 + 2d)(log 2)(4E) 1;and letting E0" = maxE  14E"; tE > 0	, we conclude that for a.e. E 2]0; E0"] andL  L(E; ") we haveEtrre tH! r	  4d+1e (4E)  d2 +" = 2e 4tEE :(3.15)Combining (3.2), (3.4), and (3.15), we conclude that, for a.e. E 2]0; E0"] and L  L(E; ")we have, for all intervals I  [0; E],EntrP!(L)(I)0oXk2CdEd+2 hkid+1kk1jIje 12 tEE 2e 4tEE 14(3.16) C 0dE d 2e 12 tEEkk1jIj = C 00dE d 2e 18 (4E)  d2+"kk1jIj:Theorem 2.1 follows.References[BGKS] Bouclet, J.M., Germinet, F., Klein, A., Schenker, J.: Linear response theory formagnetic Schrodinger operators in disordered media. J. Funct. Anal. 226, 301-372(2005)[BoCKP] Bovier, A., Campanino, M., Klein, A., Perez, J.F.: Smoothness of the density ofstates in the Anderson model at high disorder, Commun. Math. Phys. 114, 439-461(1988)[CK] Campanino, M., Klein, A.: A supersymmetric transfer matrix and dierentiabilityof the density of states in the one-dimensional Anderson model. Comm. Math. Phys.104, 227-241 (1986)[CaL] Carmona, R, Lacroix, J.: Spectral theory of random Schrodinger operators. Boston:Birkhauser, 1990[CoGK1] Combes, J.M., Germinet, F., Klein, A.: Generalized eigenvalue-counting estimatesfor the Anderson model, J. Stat. Phys. 135, 201-216 (2009)[CoGK2] Combes, J.M., Germinet, F., Klein, A.: Poisson statistics for eigenvalues of contin-uum random Schrodinger operators, Analysis and PDE 3, 49-80 (2010)[CoGK3] Combes, J.M., Germinet, F., Klein, A.: Local Wegner estimates for randomSchrodinger operators, Minami estimate and Lifshitz behavour of the density ofstates. In preparation.Lifshitz tails for the DOS 9[CoH] Combes, J.M., Hislop, P.D.: Localization for some continuous, random Hamiltoniansin d-dimension. J. Funct. Anal. 124, 149-180 (1994)[CoHK] Combes, J.M., Hislop, P.D., Klopp, F.: Optimal Wegner estimate and its applicationto the global continuity of the integrated density of states for random Schrodingeroperators. Duke Math. J. 140, 469-498 (2007)[DK] von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight bindingmodel. Commun. Math. Phys. 124, 285-299 (1989)[FMSS] Frohlich, J., Martinelli, F., Scoppola, E., Spencer, T.: Constructive proof of lo-calization in the Anderson tight binding model. Commun. Math. Phys. 101, 21-46(1985)[FS] Frohlich, J., Spencer, T.: Absence of diusion with Anderson tight binding modelfor large disorder or low energy. Commun. Math. Phys. 88, 151-184 (1983)[GK] Germinet, F, Klein, A.: Operator kernel estimates for functions of generalizedSchrodinger operators. Proc. Amer. Math. Soc. 131, 911-920 (2003)[K] Kirsch, W.: An invitation to random Schrodinger operator, with an Appendix byFrederic Klopp, Panoramas et syntheses 25 (2008), 1-119[KM] Kirsch, W., Martinelli, F.: On the ergodic properties of the spectrum of generalrandom operators. J. Reine Angew. Math. 334, 141-156 (1982)[M] Minami, N.: Local uctuation of the spectrum of a multidimensional Anderson tightbinding model. Comm. Math. Phys. 177, 709-725 (1996)[PF] Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Heidel-berg: Springer-Verlag, 1992[ST] Simon, B., Taylor, M.: Harmonic analysis on SL(2; R) and smoothness of the densityof states in the one-dimensional Anderson model. Comm. Math. Phys. 101, 1-19(1985)[St] Stollmann, P.: Caught by disorder. Bound States in Random Media. Birkauser, 2001[VSW] Shubin, C. , Vakilian, R. , Wol, T.: Some harmonic analysis questions suggestedby Anderson-Bernoulli models. Geom. Funct. 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Lifshitz tails estimate for the density of states of the Anderson model (Spectra of Random Operators and Related Topics)

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Lifshitz tails estimate for the density of states of the Anderson model

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