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1 LIFSHITZ TAILS FOR ALLOY TYPE MODELS IN A CONSTANT MAGNETIC FIELD FRÉDÉRIC KLOPP Dedicated to the memory of Pierre Duclos Abstract In this paper, we study Lifshitz tails for a 2D Landau Hamiltonian perturbed by a random alloy-type potential constructed with single site potentials decaying at least at a Gaussian speed We prove that, if the Landau level stays preserved as a band edge for the perturbed Hamiltonian, at the Landau levels, the integrated density of states has a Lifshitz behavior of the type e log2 E 2bq Résumé Dans cette note, nous démontrons qu en dimension 2, la densité d états intégrée d un opérateur de Landau avec un potentiel aléatoire non négatif de type Anderson dont le potentiel de simple site décroît au moins aussi vite qu une fonction gaussienne admet en chaque niveau de Landau, disons, 2bq, si celui-ci est un bord du spectre, une asymptotique de Lifshitz du type e log2 E 2bq Introduction On C R2, consider the Landau Hamiltonian H = H b := i A 2 b where A = bx 2 2, bx 1 2 is the magnetic potential, and b > is the constant scalar magnetic field H is essentially self-adjoint on C R2 It is wellnown that σh, the spectrum of the operator H, consists of the socalled Landau levels {2bq; q N = {,1,2, }}; each Landau level is an eigenvalue of infinite multiplicity of H Consider now the random Z 2 -ergodic alloy-type electric potential V ω x := γ Z 2 ω γ ux γ, x R 2 where we assume that H 1 : Thesingle-sitepotential usatisfies, forsomec > andx R 2, 2 Mathematics Subject Classification 82B44, 47B8, 47N55, 81Q1 Key words and phrases Lifshitz tails, Landau Hamiltonian, continuous Anderson model Part of this wor was done during the conference Spectral analysis of differential operators held at the CIRM, Luminy 7/7-11/7/28; it is a pleasure to than P Briet, F Germinet and G Raiov, the organizers of this event for their invitation to participate to the meeting This wor was partially supported by the grant ANR-8-BLAN

2 2 FRÉDÉRIC KLOPP 1 C 1 {x R 2 ; x x <1/C} ux Ce x 2 /C H 2 : The coupling constants {ω γ } γ Z 2 are non-trivial, almost surely bounded iid random variables These two assumptions guarantee V ω is almost surely bounded On the domain of H, define the operator H ω := H +V ω The integrated density of states IDS of the operator H ω is defined as the non-decreasing leftcontinuous function N : R [, which, almost surely, satisfies ϕedne = lim R R 2 Tr 1 ΛR ϕh1 ΛR, ϕ C R R Here and in the sequel, Λ R := R 2, R 2 2 and 1O denotes the characteristic function of the set O By the Pastur-Shubin formula see eg [9, Section 2], we have ϕedne = ETr 1 Λ1 ϕh1 Λ1, ϕ C R, R where E denotes the mathematical expectation with respect to the random variables ω γ γ Moreover, there exists a set Σ R such that σh ω = Σ almost surely Σ is the support of the positive measure dn The aim of the present article is to study the asymptotic behavior of N near the edges of Σ It is well nown that, for many random models, this behavior is characterized by a very fast decay which goes under the name of Lifshitz tails It was studied extensively see eg [6, 9, 4] and references therein In order to fix the picture of the almost sure spectrum σh ω, we assume: H 3 : the common support of the random variables ω γ γ Z 2 consists of the interval [ω,ω + ] where ω < ω + and ω ω + = H 4 : M + M < 2b where ±M ± := ess-sup ω sup x R 2 ±V ω x Assumptions H 1 H 4 imply that M M + = It also implies that the union [2bq+M,2bq+M + ], which contains Σ, is disjoint Let W be the q= bounded Z 2 -periodic potential defined by Wx := γ Z 2 ux γ, x R 2 On the domain of H, define the operators H ± := H +ω ± W It is easy to see that and Set σh q= [2bq +M,2bq], σh + q= [2bq,2bq +M +], σh [2bq +M,2bq], σh + [2bq,2bq +M + ], q N E q := min σh [2bq+M,2bq], E + q := max σh + [2bq,2bq+M + ]

3 LIFSHITZ TAILS IN A CONSTANT MAGNETIC FIELD 3 The standard characterization of the almost sure spectrum see also [6, Theorem 535] yields Σ = [Eq,E q + ], Eq < E q + q= ie Σ is represented as a disjoint union of compact intervals, and each interval [Eq,E+ q ] contains exactly one Landau level 2bq Actually, one has either Eq = 2bq or E q + = 2bq; more precisely Eq = 2bq if ω = and E q + = 2bq if ω + = In Theorem 21 of [5], the authors describe the behavior of N2bq + E N2bq whene tends to whilein Σ Underthe assumptionthat u does not decayasfastasinassumptionh 1, theycomputethelogarithmicasymptotics oftheidsnear2bq UnderassumptionH 1, theauthorsobtainedtheoptimal logarithmic upper bound and a lower bound that they deemed not to be optimal In our main result, we obtain the optimal lower bound, thus, proving the logarithmic asymptotics Theorem 1 Let b > and assumptions H 1 H 4 hold Assume that, for some C > and κ >, 1 P ω E CE κ, E Then, for any q N, one has ln lnn2bq +E N b 2bq 2 lim = 2 E ln lne E Σ Thus, Theorem 1 states that, at the Landau level 2bq, when it is a spectral edge for H, the IDS decays roughly as e log2 E 2bq This decay is faster than any power of E 2bq This explains why we name this behavior also Lifshitz tails even though it is much slower than the Lifshitz tails obtained when the magnetic field is absent see eg [6] In [5], the upper bound in 2 is proved under less restrictive assumptions; indeed, Theorem 51 of [5] states in particular that, under our assumptions, lim sup E ln lnn2bq +E N b 2bq ln lne 2 So it suffices to prove 3 lim inf E ln lnn2bq +E N b 2bq ln lne 2 The improvement over the results in [5] is obtained through a different analysis that borrows ideas and estimates from [1] The basic idea is to show that, for energies at a distance at most E from 2bq, the single site potential u can be replaced by an effective potential that has a support of size approximately loge 1/2 see section 2 and Lemma 3 therein This can then be used to estimate the probability of the occurrence of such energies

4 4 FRÉDÉRIC KLOPP 1 Periodic approximation AssumethathypothesesH 1 H 4 hold Forthesaeofdefiniteness,fromnow on, we assume that ω = So, for q N, we have Eq = 2bq Moreover, 1 becomes P ω E CE κ for E > small and P ω E = for any E > Up to obvious modifications, the case ω + = is dealt with in the same way We now recall some useful results from [5] Pic a > such that ba2 2π N Set L := 2n+1a/2, n N, and define the random 2LZ 2 -periodic potential 4 V per per L,ω x = VL,ω x := V ω 1 Λ2L x+γ, x R 2 γ 2LZ 2 For q N, let Π q be the orthogonal projection onto the q + 1-st Landau level ie the orthogonal projection onto KerH 2bq Consider the bounded operator Π q V per L,ω Π q It is invariant by the Abelian group of magnetic translations generated by 2LZ 2 see section 2 in [5] Hence, Π q V per L,ω Π q admits an integrated density of states that we denote by ρ q,l,ω E see [5] In [5], we have proved Theorem 2 [5] Assume that hypotheses H 1 H 4 hold and ω = Pic q N and η > Then, there exist ν >, C > 1 and E >, such that for each E,E and L E ν, we have Eρ q,l,ω E/C e E η N2bq +E N2bq Eρ q,l,ω CE+e E η As ρ q,l,ω E is the IDS of the periodic operator Π q V per L,ω Π q at energy E, it vanishes if and only if σπ q V per L,ω Π q,e] Moreover, ρ q,l,ω E is bounded by CL d where the constant C is locally uniform in E see [5] Thus, we get that, for some C >, 5 Eρ q,l,ω E CL d P σπ q V per L,ω Π q,ce] Then, the estimate 3 and, thus, Theorem 1, is a consequence of Theorem 3 For η,1, there exists C η > such that, for E sufficiently small and L 1, one has, for almost all ω, 6 e loge 1 η log loge Π q V per inf γ Λ 2L Z 2 L,ω Π q β γ loge 1 η/2 ω β e loge 1 η /C η Π q The proof of Theorem 3 relies on Lemma 3 which shows that, at the expense of a small error in energy, we can enlarge the support of the single site potential u Lemma 3 is stated and proved in section 2 Let us now use Theorem 3 to complete the proof of 3 and, thus, of Theorem 1 Pic L E ν, ν given by Theorem 2 and fix η,1 arbitrary

5 LIFSHITZ TAILS IN A CONSTANT MAGNETIC FIELD 5 Thus, by Theorem 2, 5 implies that, for E > small, 7 N2bq +E N2bq CL d P σπ q V per L,ω Π q,ce] +e E η Using 6, as the random variables ω γ γ Z 2 are iid, for E > small, we compute P σπ q V per L,ω Π q,ce] P inf ω β e loge 1 η /C η e loge /2 8 γ Λ 2L Z 2 β γ loge 1 η/2 CL d P β loge 1 η/2 ω β 2e loge 1 η/cη Recall that, by 1, as ω =, one has P ω E CE κ and P ω E = for E > small Hence, by a classical standard large deviation result see eg [2], we obtain that P C η e loge 2 2η /C η β loge 1 η/2 ω β 2e loge 1 η/cη Thus, as L E ν, this, 7 and 8 yield, for E > small, N2bq +E N2bq C η e loge 2 2η /C η As this bound holds for any η >, we obtain 3 and, thus, complete the proof of Theorem 1 2 The proof of Theorem 3 Recall that, for q N, Π q is the orthogonal projection on the eigenspace of H corresponding to 2bq, the q +1-st Landau level of H We recall Lemma 1 [7] Pic p > 1 and let V L p R 2 be radially symmetric Let µ q, V N be the eigenvalues of the compact operator Π q VΠ q repeated according to multiplicity Then, for N, one has µ q, V = Vϕ q,,ϕ q, where the functions ϕ q, are given by q! b q+1/2 ϕ q, x := x 1 +ix 2 q L q q b x 2 /2 e b x 2 /4, π! 2 for x = x 1,x 2 R 2, L q q are the generalized Laguerre polynomials given by q ξ L q l q ξ :=, ξ, q N, N, q l l! l=max{,q }

6 6 FRÉDÉRIC KLOPP, denotes the scalar product in L 2 R 2 Finally, for N, a normalized eigenfunctions of Π q VΠ q corresponding to the eigenvalue µ q, V is equal to ϕ q, In particular, the eigenfunctions are independent of V We denote by Dx,R the dis of radius R >, centered at x R 2 We set ν q, R := µ q, 1 D,R where 1 A is the characteristic function of the set A Lemma 2 Fix q N Define = R := br 2 /2 and 9 νq, e q+1+ R = q! ρ 2q 1! Pic β,2 Let f : [1,+ [1,+ be such that 1 2q 1 f 2q +f β + Then, there exists 1 and C > such that, for, ν q, R 2q 1 11 sup 1 R> R C f 2q + R f ν q, f β+1 This lemma is an extension of Corollary 2 in [1] to a larger range of radii R Proof of Lemma 2 By Lemma 1, passing to polar coordinates r,θ in the integral 1 D,R ϕ q,,ϕ q, and changing the variable br 2 /2 = ξ, the eigenvalues ν q, R of the operator Π q 1 D,R Π q are written as For q =, we have 12 ν, R = 1! ν q, R = q!! ξ [ L q q ξ] 2 e ξ dξ ξ e ξ dξ = e +1! 1 e ρt+log1 t dt Now, usingataylor expansion at and the concavity of t ρt+log1 t, write 1 ρ β/2 1 e ρt+log1 t dt = e ρt+log1 t dt+ e ρt+log1 t dt ρ β/2 = ρ β/2 e ρt 1+O ρ β dt = 1 ρ +O ρ β+1 +O This and 12 yields 11 when q = Consider now the case q 1 For some C q >, one has 13 1, sup s q q s! 1 q s C q s {,,q} e ρ1 β/2

7 LIFSHITZ TAILS IN A CONSTANT MAGNETIC FIELD 7 In order to chec 11, we assume that q In this case, using 13, we compute ν q, R = q! q 1 l+m 1 e ξ ξ q+m+l dξ 14! m!l! q l q m where 15 and 16 l,m= = V,q+R,q V,q = 1!q! = 1!q! R,q C q C q q q 1 l+m l l,m= 1!q! 1!q! e ξ ξ q ξ 2q dξ, q l,m= q 2q l m e ξ ξ q+m+l dξ m q q 2q l m e ξ ξ q+m+l dξ l m e ξ ξ q +ξ 2q dξ For ρ f, using 1, one computes 17 R, q V,q C 2q 1 f 2q + On the other hand, as in the case q =, we have where e ξ ξ q ξ 2q dξ = e ρ ρ q+1 ρ 2q I,ρ I,ρ = 1 e ρξ 1 ξ q 1+ ρ The function t ρt+ qlog1 t+2qlog [,1] and its derivative at is 2q ρ ξ dξ 1+ ρ ρ t ρ +q +2qρ/ ρ = ρ 1+O ρ 2 Hence, as in the case q =, we obtain that I,ρ = 1 ρ +O ρ β+1 is concave on Plugging this into 15, using 17 and 16, and replacing in 14, we obtain 11 for q 1 This completes the proof of Lemma 2 We will now use Lemma 2 to derive the enlargement of obstacles lemma for the Landau-Anderson model; we prove Lemma 3 Let q N and fix b > Fix ε > There exists C > and R > 1 such that, for each R R, 18 Π q 1 D,ε Π q e C R 2 logr Π q 1 D,R Π q e R2 /C Π q 1 D,2R Π q

8 8 FRÉDÉRIC KLOPP This lemma is basically Lemma 2 in [1] except that we want to control the behavior of the constants coming up in the inequality in terms of R Proof of Lemma 3 We fix δ,1 Recall Lemma 2, in particular 11 and 9 Pic C > 2b and set = R := CR 2 Let f satisfy 1 Hence, there exists R > such that, for R R and = R, one has f ρr Thus, Lemma 2 implies that, for R [R/2,2R], one has 19 e R R q+1 ρ R 2q 1 1 δ q!! ν q, R e R R q+1 1+δ q! We show that, if R R, then, the operator inequality 2 Π q 1 D,ε Π q C 1 Πq 1 D,R Π q C 2 Π q 1 D,2R Π q holds with the following constants: ν q, ε 21 C 1 := min {,, } ν q, R 1 e 2CR2 logr ; C ρ R 2q 1! the lower bound holds for sufficiently large R and, as = CR 2, is a consequence of 9 and 11 written for ν q, ε; 22 C 2,q := 1+δ 1 δ C 2q q+1 e R+ 2R e R2 /C ; C 2b the upper bound holds for sufficiently large R and follows from ρ2r By Lemma 1, the operators Π q 1 D,ε Π q, Π q 1 D,R Π q, and Π q 1 D,2R Π q, are reducible in the same basis {ϕ q, } N Hence, in order to prove 2, it suffices to chec that, for each N, the following numerical inequality holds 23 ν q, ε C 1 ν q, R C 2,q ν q, 2R If, then 23 holds as ν q, ε C 1 ν q, R by 21 As CR 2 for C > 2bandρ = br 2 /2, or, onehasc ρ2r C 2b ρr Thus, by 19 and 22, we have e R R q+1 ν q, R C 2,q ν q, 2R 1+δ q! 1+δ 1 δ 2 2 q+1 e R+ 2R ρr 2q 1 R! C ρ2r C 2b ρr 2q 1 e 2R 2R q+1 ρr 2q 1 2R 1 δ q!! e R ρr 2q 1 2R q+1 = 1+δ 2 2q q!!

9 LIFSHITZ TAILS IN A CONSTANT MAGNETIC FIELD 9 Hence, we find that ν q, R C 2 ν q, 2R if, which again implies 23 This completes the proof of Lemma 3 We now prove Theorem 3 The magnetic translations for the constant magnetic field problem in twodimensions are defined as follows see eg [8] For any field strength b R, any vector α R 2, the magnetic translation by α, say, Uα b is defined as U b α fx := eib 2 x 1α 2 x 2 α 1 fx+α f C R2 The invariance of H with respect to the group of magnetic translations U b α α Z 2 implies that, for γ Z 2, one has 24 U b γπ q 1 D,ε Π q U b γ = Π q 1 Dγ,ε Π q Hypothesis H 1 on the single-site potential u guarantees that there exists ǫ,1/2 so that V ω γ Z 2 ω γ 1 Dγ,ε Plugging this into 4, we get 25 V per L,ω γ 2LZ 2 β Λ 2L Z 2 ω β 1 Dγ+β,ε Fix η,1 and pic R loge 1 η/2 Lemma 3 and 24 imply that Π q 1 Dγ,ε Π q e C R 2 logr Π q 1 Dγ,R Π q e R2 /C Π q 1 Dγ,2R Π q Hence, as the random variables ω γ γ Z 2 are bounded, this and 25 imply that e C R 2logR Π q V per L,ω Π q e C R 2 logr ω β Π q 1 Dγ+β,ε Π q γ 2LZ 2 β Λ 2L Z 2 ω β Π q 1 Dγ+β,R Π q γ 2LZ 2 β Λ 2L Z 2 Ce R2 /C Π q 1 Dγ+β,2R Π q γ 2LZ 2 β Λ 2L Z 2 Π q 1 x ν 1/2 Π q Ce R2 /C R 2 Π q inf ω β γ 2LZ 2 β Λ 2L Z 2 ν γ β R/2 γ Λ 2L Z 2 β γ R/2 ω β CR 2 e R2 /C Π q Taing into account R loge 1 η/2, this completes the proof of Theorem 3 References [1] Jean-Michel Combes, Peter D Hislop, Frédéric Klopp, and Georgi Raiov Global continuity of the integrated density of states for random Landau Hamiltonians Comm Partial Differential Equations, 297-8: , 24 [2] Amir Dembo and Ofer Zeitouni Large deviations techniques and applications, volume 38 of Applications of Mathematics New Yor Springer-Verlag, New Yor, second edition, 1998 [3] B A Dubrovin and S P Noviov Fundamental states in a periodic field Magnetic Bloch functions and vector bundles Dol Aad Nau SSSR, 2536: , 198

10 1 FRÉDÉRIC KLOPP [4] Werner Kirsch and Bernd Metzger The integrated density of states for random Schrödinger operators In Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon s 6th birthday, volume 76 of Proc Sympos Pure Math, pages Amer Math Soc, Providence, RI, 27 [5] Frédéric Klopp and Georgi Raiov Lifshitz tails in constant magnetic fields Comm Math Phys, 2673:669 71, 26 [6] Leonid Pastur and Alexander Figotin Spectra of random and almost-periodic operators, volume 297 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Springer-Verlag, Berlin, 1992 [7] Georgi D Raiov and Simone Warzel Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing electric potentials Rev Math Phys, 141: , 22 [8] Johannes Sjöstrand Microlocal analysis for periodic magnetic Schrödinger equation and related questions In Microlocal analysis and applications, volume 1495 of Lecture Notes in Mathematics, Berlin, 1991 Springer Verlag [9] Ivan Veselić Existence and regularity properties of the integrated density of states of random Schrödinger operators, volume 1917 of Lecture Notes in Mathematics Springer- Verlag, Berlin, 28 Frédéric Klopp LAGA, UMR 7539 CNRS, Institut Galilée, Université Paris-Nord, 99 Avenue J-B Clément, F-9343 Villetaneuse, France et Institut Universitaire de France address: lopp@mathuniv-paris13fr