Classical behavior of the integrated density of states for the uniform magnetic field and a randomly perturbed

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1 Classical behavior of the integrated density of states for the uniform magnetic field and a randomly perturbed lattice * Naomasa Ueki 1 1 Graduate School of Human and Environmental Studies, Kyoto University Kyoto , Japan. ueki@math.h.kyoto-u.ac.jp Abstract. For the Schrödinger operators on L R ) and L R 3 ) with the uniform magnetic field and the scalar potentials located at all sites of a randomly perturbed lattice, the asymptotic behavior of the integrated density of states at the infimum of the spectrum is investigated. The randomly perturbed lattice is the model considered by Fukushima and this describes an intermediate situation between the ordered lattice and the Poisson random field. In this paper the scalar potentials are assumed to decay slowly and its effect to the leading term of the asymptotics are determined explicitly. As the perturbed lattice tends to the Poisson model, the determined leading term tends to that for the Poisson model. Keywords: Lifshitz tails, magnetic Schrödinger operator AMS Subject Classification: 34L40, 8B44 1. Introduction Let H = A A = i Bx ) + i + Bx ) 1 B 1.1) x 1 x be the smaller component of the Pauli Hamiltonian R with the uniform magnetic field B > 0, i = 1 and Let A = i Bx ) + i i + Bx ) 1. 1.) x 1 x V ξ x) = q Z ux q ξ q ) 1.3) be a random potential on R, ξ = ξ q ) q Z is a collection of independently and identically distributed R -valued random variables with the distribution P θ ξ q dx) = exp x θ )dx/zθ), 1.4) * This work was partially supported by KAKENHI ) 1

2 Naomasa Ueki x = x 1 + x, θ 0, ), Zθ) is the normalizing constant, and u is a nonnegative function belonging to the Kato class K cf. [] p-53). Let H ξ,λr be the self-adjoint restriction of the random Schrödinger operator H ξ = H + V ξ, 1.5) to the L space on the cube Λ R := R/, R/) with the Dirichlet boundary condition and N ξ,r λ) be its number of eigenvalues not exceeding λ. It is well-known that there exists a deterministic increasing function N such that R N ξ,r λ) Nλ) as R 1.6) for any point of continuity of N and almost all ξ under the condition sup x α ux) < for some α > cf. [], [7], [10]). This function Nλ)) λ 0 is called as the integrated density of states for the random Schrödinger operator 1.5). For this function, we show the following: Theorem 1. i) If holds for any R 1 and essinf ux) > 0 1.7) x R ux) = x α 1 + o1)) 1.8) as x for some 0, ) and α, ), then we have { } λ κ κ κ κ+1 log Nλ) = λ 0 κ + 1) κ+1 dq inf R y R q + y α + y θ), 1.9) κ = + θ)/α ). ii) If 1.7) holds for any R 1 and x α ) ux) = exp 1 + o1)) 1.10) as x for some 0, ) and α 0, ), then we have log1/λ)) +θ)/α log Nλ) = πc+θ)/α 0 λ 0 θ + 1)θ + ). 1.11) We also consider the 3-dimensional problem in the following formulation referring to [8] and [17]. Let H = i Bx ) + i + Bx ) 1 B x 1 x x 3 1.1)

3 magnetic Lifshitz tails 3 be the smaller component of the Pauli Hamiltonian on R 3 with the constant magnetic field B parallel to the x 3 -axis. We write any element x of R 3 as x, x 3 ) R R and set x θ, x 3 θ3 p = x θ p + x 3 θ3p ) 1/p if p [1, ), x θ p := x θ x 3 θ3 if p =, 1.13) for arbitrarily fixed θ = θ, θ 3 ) 0, ) and p [1, ]. Let ξ = ξ q ) q Z 3 be a collection of independently and identically distributed R 3 -valued random variables with the distribution P θ ξ q dx) = exp x θ p)dx/zθ, p), 1.14) Zθ, p) is the normalizing constant. Let u be a nonnegative function belonging to the Kato class K 3 cf. [] p-53) and satisfying ux) = x α p 1 + o1)) 1.15) as x for some 0, ), p [1, ] and α = α, α 3 ) 0, ) such that α + 1 α 3 < ) Since dq q R 3 : q 1 q α p under the condition 1.16), we can show < 1.17) V ξ x) = q Z 3 ux q ξ q ) 1.18) belong to the local Kato class K 3,loc by the same method in Lemma 7.1 in [7]. As in the -dimensional case we will consider the integrated density of states Nλ)) λ 0 of the random Schrödinger operators H ξ = H + V ξ : 1.19) it is a deterministic increasing function satisfying R 3 N ξ,r λ) Nλ) as R 1.0) for any point of continuity of N and almost all ξ, N ξ,r λ) is the number of eigenvalues not exceeding λ of the self-adjoint operator H ξ,λr on the L space on the cube Λ R := R/, R/) 3 with the Dirichlet boundary condition cf. [], [7], [10]). Our result is the following:

4 4 Naomasa Ueki Theorem. We assume holds for any R 1 and 1.15) as x for some 0, ) and essinf ux) > 0 1.1) x R α + 3 α 3 > 1. 1.) We set and κα, θ) = θ α θ 3 α 3 + α + 1 α 3 1 α 1 α 3, 1.3) Cα, θ, ) = dq inf R 3 cf. 1.17) ). Then we have y=y,y 3) R 3 q + y α p ) 1.4) + 1 θ α θ 3 y θ, 1 θ α 3 α θ 3 y 3 θ3 p α 3 λ κα,θ) κα, θ) κα,θ) log Nλ) = λ κα, θ)) Cα, θ, ) 1+κα,θ). 1.5) 1+κα,θ) These results are extensions of the results in [6] and [7], the same problem is considered in the case without magnetic fields. As is discussed in [6] and [7], our model describes an intermediate situation between a completely ordered situation and a completely disordered situation since the point process {q + ξ q } q Z converges weakly to the Poisson point process with the intensity 1 as θ 0 and converges weakly to the complete lattice Z as θ by slightly modifying the definition as P θ ξ q dx) = exp 1 + x ) θ )dx/zθ), 1.6) which brings no essential changes for our results. The results in [6] and [7] show that the leading term of the integrated density of states for each case also tends to that for the corresponding Poisson case as θ 0 and decays as θ which reflects that the infimum of the spectrum is strictly positive if the perturbations ξ q of sites are all bounded. In the case with uniform magnetic fields the asymptotics of the integrated density of states has been investigated mainly for the Poisson case. For this topic and the relation with other topics, refer to a recent survey by Kirsch and Metzger [11]. The first result was given by Broderix, Hundertmark, Kirsch and Leschke [1]: they determined the leading term for the case corresponding to Theorem 1 i) in this paper the point process {q + ξ q } q Z is replaced by the Poisson point process. As in 1.9), this leading term was determined mainly by the classical effect of the

5 magnetic Lifshitz tails 5 scalar potential as in Pastur s case [14] without magnetic fields. In fact these asymptotics coincide with those of the corresponding classical integrated density of states defined by N c λ) = E θ [ {x, p) Λ R R : H ξ,c x, p) λ} ]πr) 1.7) for any R N, is the 4-dimensional Lebesgue measure and H ξ,c x, p) { = p 1 Bx + V ξ x) = p 1 Bx ) + i ) + p + Bx 1 p + Bx 1 )} { ) + Vξ x) p 1 Bx ) + i p + Bx 1 )} 1.8) is the classical Hamiltonian cf. [9]). Therefore we may say that only the classical effect from the scalar potential determines the leading term. Then Erdös [4] treated the case the single site potential u is replaced by a function with a compact support and he determined the corresponding leading term of the integrated density of states, which depends only on the magnetic field and the intensity of the point process and is independent of the precise informations on the single site potential as in Nakao s case [13] without magnetic field referring to Donsker and Varadhan s result [3]. This behavior is different from that of the classical integrated density of states. Thus we may say that the quantum effect appears in this behavior. The borderline between the classical and quantum behaviors was determined by Hupfer, Leschke and Warzel [9]. The borderline corresponds to the case of α = in Theorem 1 ii) in this paper. They also determined the leading term for the case of Theorem 1 ii) in this paper for the Poisson case. The leading term for the borderline case was determined by Erdös [5]. The leading term for the case that only the classical effect appears was determined also in the 3-dimensional case by Hundertmark, Kirsch and Warzel [8]. Results appearing the quantum effect in 3-dimensional cases were obtained by Warzel [17], general bounds and the leading order for special cases were obtained. Now our problem is to consider the same problem in our setting. As is discussed in [16] we conjecture that the borderline between the classical and quantum behaviors is the case of α = in Theorem 1 ii) and the case of /α + 3/α 3 = 1 in Theorem as in the Poisson case. In this paper we treat only the classical case and we will treat the quantum case in [16]. Now Theorems 1 and show that our leading terms coincide with those of the corresponding classical integrated densities of states. They also tend to the corresponding leading term for the Poisson case as θ 0.

6 6 Naomasa Ueki The leading term we obtained coincides with that for the classical case without magnetic fields obtained in [7]. The classical case without magnetic fields is that of 1.8) with < α < 4 if the dimension is. Theorem 1 shows that this condition for only the classical effect to appear is weakened by the magnetic field, since the effect of the Landau Hamiltonian to the asymptotics of the integrated density of states is weaker than that of the usual Laplacian. The 3-dimensional classical case without magnetic fields is that of 1.15) with 4/α + 1/α 3 > 1 and 1.16). This result was obtained for the Poisson case by Kirsch and Warzel [1], the results for the single site potential with an anisotropic decay are summarized. Theorem shows that this condition for only the classical effect to appear is weakened in the direction perpendicular to the magnetic field and is strengthen in the direction parallel to the magnetic field. The organization of this paper is as follows. We first prove Theorem 1 in Sections and 3: we give the upper estimate in Section and the lower estimate in Section 3. We next prove Theorem in Sections 4 and 5: we give the upper estimate in Section 4 and the lower estimate in Section 5. In Section 6 we give the leading terms of the low energy asymptotics of the classical integrated densities of states defined in a general setting.. -dimensional upper estimate In this section we prove upper estimates necessary to prove Theorem 1. We denote the Laplace-Stieltjes transform of the integrated density of states Nλ) by Ñt): Then the results are the following: : Ñt) = 0 e tλ dnλ). Proposition.1. i) Under the conditions of Theorem 1 i) we have t log Ñt) t dq inf +θ)/α+θ) R ii) Under the conditions of Theorem 1 ii) we have t log Ñt) log t) y R πc+θ)/α 0 +θ)/α q + y α + y θ)..1) θ + 1)θ + )..) i) is proven by the following proposition and the same proof of Proposition 4 in [7]. The following proposition is an extension of the basic inequality 3.6) in [1] for a R -stationary random potential to a Z - stationary random potential, which is proven by applying the basic inequality in [1] to the R -stationary random potential V ξ,ζ x) = q Z ux q ξ q ζ q ),.3)

7 magnetic Lifshitz tails 7 ζ = ζ q ) q Z is an independent family of random vectors uniformly distributed on Λ 1 : Proposition.. Ñt) B π1 e Bt )Ñ1t), Ñ 1 t) = dxe θ [exp tv ξ x))]. Λ 1.4) Proof of Proposition.1 ii). By replacing the summation by the integration, we have [ )] log Ñ1t) dq log E θ exp t inf ux q ξ 0 ). R x Λ We restrict the integration to q L for some finite L. For any ε 1 > 0, there exists R 1 such that ux) exp 1 + ε 1 ) x α / ) for any x R 1, x = x 1 x. Thus the right hand side is dominated by { dy x q ) y α dq log exp t inf exp 1 + ε 1 ) y θ) Zθ) x Λ q+y R 1+1 q L + exp t inf Λ R1 +4 u) }. By changing the variables, this equals C0 log t ) /α 1 + ε 1 q L L = L1 + ε 1 )/ log t)) 1/α, C0 log t ) /α Ñ t, q) = 1 + ε 1 and exp st, q, y) = dq log {Ñ t, q) + exp t 1 st,q,y) t inf Λ R1 +4 q+y R 1+1)1+ε 1)/ log t)) 1/α C0 log t 1 + ε 1 ) θ/α y θ ), sup x q y α. x Λ 1+ε1 )/ log t)) 1/α )} u, dy Zθ) We take L as an arbitrary positive constant less than 1 and independent of t. Taking ε 0, 1) arbitrarily, we dominate Ñt, q) by exp Ñ3t, q))ε /θ, { Ñ 3 t, q) = inf t 1 st,q,y) C0 log t ) θ/α + 1 ε ) y θ : y R }. 1 + ε 1 We use the polar coordinate to express as { C0 log t ) θ/α } Ñ 3 t, q) = inf t 1 st,q)+r)α + 1 ε ) r θ : r > 0,.5) 1 + ε 1

8 8 Naomasa Ueki st, q) = sup x Λ 1+ε1 )/ log t)) 1/α x q. We take small ε 3 > 0 and estimate the infimum in.5) as { C0 log t ) θ/α } Ñ 3 t, q) inf 1 ε ) r θ : r > 1 st, q) ε ε 1 } inf {t 1 st,q)+r)α : 0 < r 1 st, q) ε 3 The right hand side is bounded from below by { C0 log t ) θ/α = 1 ε ) 1 st, q) ε3 ) θ} t 1 1 ε3) α 1 + ε 1 C0 log t ) θ/α 1 ε ) 1 q ε3 ) θ 1 + ε 1 for sufficiently large t. By taking L as 1 3ε 3 and using also the positivity assumption 1.7), we obtain t Since ε 1, ε, ε 3 are arbitrary, we have log Ñ1t) log t) 1 ε C0 ) +θ)/α ) 1 q ε +θ)/α 3 ) θ dq. 1 + ε 1 q 1 3ε 3 t 3. -dimensional lower estimates log Ñ1t) C+θ)/α log t) +θ)/α 0 1 q ) θ dq. q 1 In this section we prove the following lower estimates necessary to prove Theorem 1: Proposition 3.1. i) If 1.8) holds, then we have t log Ñt) t dq inf +θ)/α+θ) R ii) If 1.10) holds with some α 0, ), then we have t log Ñt) log t) y R πc+θ)/α 0 +θ)/α q + y α + y θ). 3.1) θ + 1)θ + ). 3.) Proof. We use the bound Ñt) R) exp tλ 1 H BR) ))Ñ1t), 3.3) for any R N, [ Ñ 1 t) = E θ exp t dx ψ R x) V ξ x)) ],

9 magnetic Lifshitz tails 9 and ψ R is a normalized eigenfunction for the lowest eigenvalue λ 1 H BR) ) of the operator H restricted to the disk BR) = {x R : x < R} by the Dirichlet boundary condition. This is proven by the same method as for the corresponding bound in Theorem 9.6) in [15] for the R -stationary random field. λ 1 H BR) ) exp B ) R 1 ε 1 ) 3.4) is proven by Erdös [4] for sufficiently large R for an arbitrarily taken ε 1 > 0. If 1.8) holds, then we can show 3.1) by the same proof of Proposition 5 in [7], R is taken as ε 0 t 1/α+θ) with an arbitrarily small ε 0 > 0. We now assume 1.10). log Ñ1t) = Ñ t, q), q Z [ Ñ t, q) = log E θ exp t )] dx ψ R x) ux q ξ 0 ). If q is sufficiently large and ξ 0 is less than q /, then we have x q ξ 0 q / R and Ñ t, q) t exp 1 q ) α1 ) R ε ) + log P θ ξ 0 q /), ε is an arbitrarily small positive constant. By a simple estimate using log1 X) X for 0 X 1/, we have q Z \BR 1+ε 3 ) Ñ t, q) c 1 t exp c R 1+ε3)α ) c 3 exp c 4 R 1+ε3)θ ) 3.5) for large enough R, ε 3 is an arbitrarily small positive constant, and c 1,..., c 4 are positive finite constants. We dominate the other part as q Z BR 1+ε 3 ) q Z BR 1+ε 3 ) exp t Ñ t, q) log sup z R y ε 4 dy Zθ) dx ψ R x) ux q y z) y + z θ) 3.6) c 5 R 1+ε3) log πε 4 Zθ) inf sup z R q Z BR 1+ε y ε 3 ) 4 t dx ψ R x) ux q y z) + y + z θ),

10 10 Naomasa Ueki c 5 is a positive finite constant and ε 4 is an arbitrarily small positive constant. We take z = 0 for q Z satisfying q 1 + s)r + ε 4 and z = s)r + ε 4 )/ q )q for other q Z, s is a finite positive number specified later. Then the second term of the right hand side of 3.6) is bounded from below by q Z BR 1+ε 3 )\B1+s)R+ε 4)) q Z B1+s)R+ε 4) c s) R t exp { t exp 1 ε q R ε 4 ) α) } + ε θ 4 { t exp 1 ε sr) α) s)r + ε 4 q ) θ} 1 ε sr) α) c 7 ε θ C 4R 1+ε3) ε 5 ) 1 + s)r + ε 4 q ) θ dq q R : q 1+s)R+ε 4 for large enough R, ε 5 is an arbitrarily small positive constant, and c 6 and c 7 are positive finite constants. By changing the variables, the last term is bounded from below by 1 + ε 5 )1 + s)r + ε 4 ) +θ π θ + 1)θ + ) for large enough R. The above estimate is summarized as follows: log Ñt) t exp B ) R 1 ε 1 ) c s) R t exp 1 ε sr) α) logr) c 1 t exp c R 1+ε3)α ) c 3 exp c 4 R 1+ε3)θ ) 3.7) + c 5 R 1+ε3) log πε 4 Zθ) c 7ε θ 4R 1+ε3) 1 + ε 5 )1 + s)r + ε 4 ) +θ π θ + 1)θ + ) for large enough R. The first term and the second term change the role according to the value of α. When α <, we take 1/s as an arbitrarily small positive constant and set R = sr. Then the right hand side of 3.7) is bounded from below by c 6 t exp 1 ε ) Rα 1 + ε 5 ) C R +θ π 0 θ + 1)θ + ) for large enough R. We now take R = C0 log t 1 ε ) 1/α.

11 magnetic Lifshitz tails 11 Then we obtain log Ñt) c C0 log t ) +θ)/α π ε 5 ) 1 ε θ + 1)θ + ) for large enough t, from which we obtain 3.) dimensional upper estimate In this section we prove an upper estimates necessary to prove Theorem. We denote the Laplace- Stieltjes transform of the integrated density of states Nλ) by Ñt): Then the result is the following: Ñt) = 0 e tλ dnλ). Proposition 4.1. Under the conditions of Theorem, we have log Ñt) Cα, θ, C t t κα,θ) 0 ), 4.1) θ θ α κα, θ) = α 3 α α 3 θ θ. 4.) α α 3 To prove this proposition, we use the following simple extension of Proposition.: Proposition 4.. B Ñt) π1 e Bt ) Ñ 1 t), 4πt Ñ 1 t) = dxe θ [exp tv ξ x))]. 4.3) Λ 1 Proof of Proposition 4.1. By replacing the summation by the integration, we have [ )] log Ñ1t) dq log E θ exp t inf ux q ξ 0 ). R 3 x Λ We restrict the integration to q L and q 3 L 3 for some finite L and L 3. For any ε 1 > 0, there exists R 1 such that ux) 1 ε 1 )/ x α p for any x R 1, x = max i x i. Thus the right hand side is dominated by dq log q L, q 3 L 3 + exp t inf Λ R1 +6 { dy Zθ, p) q+y R 1+ u) }. 1 ε 1 ) exp t inf x Λ x q y α p y θ p )

12 1 Naomasa Ueki By changing the variables q, q 3 ) to t η q, t η3 q 3 ) and y, y 3 ) to t η y, t η3 y 3 ) with η, η 3 0, ) satisfying η α = η 3 α 3, we see that this equals Ñ t, q) t η +η 3 =t η +η 3 q L dy Zθ, p) q + y R 1 + )t η t η α y θ, q 3 + y 3 R 1 + )t η 3 y θ t η θ η 3θ 3 dq log {Ñ t, q) + exp ), p exp t 1 η α t inf Λ R1 +6 inf x t η x 3 t η 3 )} u, 4.4) 1 ε 1 ) x q y α p we take L as an arbitrary constant independent of t by taking L, L 3 ) appropriately, and we take and as elements of {, 3} such that η θ = η θ ) η 3 θ 3 ) and. Moreover we take η and η 3 as η = 1/θ +α ) and η = α /{α θ +α )} so that 1 η α = η θ. Then, taking ε, ε 3 > 0 sufficiently small and using the positivity assumption, we can dominate Ñt, q) by exp t η θ Ñ 3t, q))ε /θ 1/θ 3 for large enough t, Therefore we obtain Ñ 4 q) = Ñ 3 t, q) = inf x Λ ε3,y R 3 { C0 1 ε 1 ) x q y α p t log Ñt) t η +η 3+η θ { C0 1 ε 1 ) inf x Λ ε3,y R 3 x q y α p + 1 ε ) y θ, Ñ 4 q)dq, q L Since ε 1, ε, ε 3 and L are arbitrary, we can complete the proof. y θ t η θ η 3θ 3 }. p θ + 1 ε ) y θ, 1 θ η =θ 3η 3 y θ p }. 4.5) 5. 3-dimensional lower estimate In this section we prove the following lower estimates necessary to prove Theorem : Proposition 5.1. If 1.15) holds with 1.16) and 1.), then we have with 4.) and 1.4). log Ñt) Cα, θ, C t t κα,θ) 0 ) 5.1)

13 magnetic Lifshitz tails 13 Proof. We use the bound Ñt) { R ) R 3 ) 1 exp tλ 1 H BR )) 5.) d tλ 1 )}Ñ1 dx ) R t), 3,R 3) R, R 3 N, H R is the same symbol used in 3.3), d /dx )) R3,R 3) is the restriction to the interval R 3, R 3 ) of the 1-dimensional Laplacian by the Dirichlet boundary condition, [ Ñ 1 t) = E exp t dxψ R x ) φ R3 x 3 ) V ξ x)) ], and ψ R and φ R3 are the normalized ground states of H BR ) and d /dx )) R3,R 3), respectively. This can be proven by the same method as for the corresponding bound in 3.3) and Theorem 9.6) in [15] for R d -stationary random fields. We have Erdös s bound λ 1 H R ) exp B ) R 1 ε 1 ) 5.3) as in 3.4) and d ) π ), λ 1 dx ) R = 5.4) 3,R 3) R 3 ε 1 is an arbitrarily fixed positive constant. By replacing the summation by integration, we have log Ñ1t) R 3 Ñ t, q)dq, [ Ñ t, q) = log E exp t )] dxψ R x ) φ R3 x 3 ) sup ux q z ξ 0 ). z Λ 1 For any ε > 0, there exists R such that ux) 1 + ε )/ x α p for any x R by the assumption 1.15). To use this bound in the above right hand side, we need inf{ x q z ξ 0 : x < R, x 3 < R 3, z Λ 1 } R. A sufficient condition of this is q R + R + ) with ξ 0, q / or q 3 R + R 3 + 1) with ξ 0,3 q 3 /. We fix β, β 3 > 0 and take t large enough so that t β R + R + ) and t β3 R + R 3 + 1). Then, by using also log1 X) X for 0 X 1/

14 14 Naomasa Ueki we obtain dq Ñ t, q) q t β, q 3 1/β 3 q 1/β dq q t β, q 3 1/β 3 q 1/β ) + log P ξ 0, q /) tc0 1 + ε ) α q R + )) α 5.5) c 1 t 1+β3 β α ) c exp c 3 t β θ ) if α > + β 3 /β, 5.6) and dq Ñ t, q) q 3 t β 3, q 1/β q 3 1/β 3 dq q 3 t β 3, q 1/β q 3 1/β 3 ) + log P ξ 0,3 q 3 /) tc0 1 + ε ) α3 q 3 R 3 + 1)) α 5.7) c 4 t 1+β β 3α 3 1) c 5 exp c 6 t β3θ3 ) if α 3 > 1 + β /β ) The other part is estimated as dq Ñ t, q) q t β, q 3 t β 3 dq q t β, q 3 t β 3 exp dy log Zθ, p) q +y R +R + or q 3+y 3 R +R 3+1 t 1 + ε ) inf x <R, x 3 <R 3,z Λ 1 { x q z y α p } y θ p ). 5.9) By the same change of variables as in 4.4) and 4.5), we find that the right hand side equals t η +η 3 dq q t β η, q 3 t β 3 η 3 dyt η +η 3 log Zθ, p) q + y R + R + )/t η or q 3 + y 3 R + R 3 + 1)/t η3 exp t η θ Ñ 3t, y, q)),

15 magnetic Lifshitz tails 15 Ñ 3 t, q, y) = 1 + ε )/ inf{ x q z y α p : x R /t η, x 3 R 3 /t η3, + y θ, y θ t η θ η 3θ 3. p z 1/t η ), z 3 1/t η3 )} 5.10) Taking γ, γ 3 0, ) and w R 3, we restrict the integration to Bw, t γ ) w 3 t γ3, w 3 + t γ3 ). Then we can bound the integrand with respect to q from below by Ñ 4 t, q) log πtη γ )+η 3 γ 3) Zθ, p) { = inf sup Ñ 3 t, q, y) y Bw,t γ ) w 3 t γ 3,w 3+t γ 3 ) t η θ Ñ 4t, q), 5.11) : w 3 R, w R, } dbw, t γ ), q ) R + R + )t η { inf sup Ñ 3 t, q, y) y Bw,t γ ) w 3 t γ 3,w 3+t γ 3 ) 5.1) : w R, w 3 R, dw 3 t γ3, w 3 + t γ3 ), q 3 ) R + R 3 + 1)t η3 }. We now specify R and R 3 as the integer parts of ε 3 t η and ε 3 t η3, respectively, ε 3 is an arbitrarily fixed positive number. We take β and β 3 as β = η + ζε 4 and β 3 = η 3 + ε 4, respectively, ε 4 is also an arbitrarily fixed positive number and ζ is a number satisfying 1 α < ζ < α 3 1. By taking ε 4 small, we have 1 η 3, 1 + β 3 β α ), 1 + β β 3 α 3 1), β + β 3 < η + η 3 + η θ and 5.6) and 5.8). Thus we obtain t log Ñt) t η +η 3+η θ t q t β η, q 3 t β 3 η 3 Ñ 4 t, q)dq. 5.13)

16 16 Naomasa Ueki When q t β η and q 3 t β3 η3, we have t 1 q 1/β η ) and t 1 q 3 1/β3 η3). Thus, for large q, by taking w as 0, we can dominate Ñ4t, q) by q 1) α + q 3 1) α3 + + q γ θ /β η ) q 3 γ3θ3/β3 η3). This is integrable if we take γ and γ 3 large enough so that γ θ /β η ) > 3 and γ 3 θ 3 /β 3 η 3 ) > 3. Thus, by the Lebesgue convergence theorem, we have Ñ 4 t, q)dq t q t β η, q 3 t β 3 η 3 { = dq inf R ε ) inf x, x 3 ε 3 x q y α p : y + q y 3 + q 3 ε 3 }. + y θ, 1 η3θ 3=η θ y θ p Since ε and ε 3 are arbitrary, this completes the proof. 6. Classical integrated density of states In this section we summarize the results on the leading terms of the low energy asymptotics of the classical integrated densities of states in a general setting. The -dimensional results are the following: Theorem 3. i) If 1.7) holds for any R 1 and 1.8) as x for some 0, ) and α, ), then the classical integrated density of states defined by 1.7) has the same leading term with 1.9): { } λ κ κ κ κ+1 log N c λ) = λ 0 κ + 1) κ+1 dq inf R y R q + y α + y θ). 6.1) ii) If 1.7) holds for any R 1 and 1.10) as x for some 0, ) and α 0, + θ), then the classical integrated density of states defined by 1.7) has the same leading term with 1.11): log1/λ)) +θ)/α log N c λ) = πc+θ)/α 0 λ 0 θ + 1)θ + ). 6.) iii) If 1.7) holds for any R 1 and 1.10) as x for some 0, ) and α + θ, ), then we have have λ 0 log1/λ)) 1 log N c λ) = ) iv) If 1.7) holds for any R 1 and 1.10) as x for some 0, ) and α = + θ, then we log1/λ)) 1 π log N c λ) = 1 λ 0 θ + 1)θ + ). 6.4)

17 magnetic Lifshitz tails 17 v) If supp u is compact, then the classical integrated density of states defined by 1.7) satisfies the following: as λ 0, N c λ) = K cλ 1 + o1)) 6.5) 4π K c = dx P θ x q ξ q supp u). 6.6) Λ 1 q Z i)-iv) are proven by the same methods in Sections and 3, and v) is proven by the Lebesgue convergence theorem. Similarly by the methods in Sections 4 and 5, and the Lebesgue convergence theorem, we have the following 3-dimensional results: Theorem 4. Let N c λ) be the classical integrated density of states of our 3-dimensional setting defined similarly as in 1.7). i) If 1.1) holds for any R 1 and 1.15) as x for some 0, ), p [1, ] and α = α, α 3 ) 0, ) satisfying 1.16), then N c λ) has the same leading term with 1.5): λ κα,θ) κα, θ) κα,θ) log N c λ) = λ κα, θ)) Cα, θ, ) 1+κα,θ). 6.7) 1+κα,θ) ii) If 1.1) holds for any R 1 and x ) α p ux) = exp 1 + o1)) 6.8) as x for some 0, ), p [1, ] and α = α, α 3 ) 0, ) satisfying χα, θ) > 1, 6.9) then N c λ) satisfies log1/λ)) χα,θ) log N c λ) = C χα,θ) 0 Dα, θ, p, p), 6.10) λ 0 and Dα, θ, p, p) := q α p 1 dq χα, θ) := + 1 θ + θ ) 3 α α 3 α α 3 1 inf θ y: q+y α p 1 α θ 3 α 3 y θ, 1 θ α θ 3 α 3 y 3 θ3 p. 6.11) 6.1)

18 18 Naomasa Ueki iii) If 1.1) holds for any R 1 and 6.8) as x for some 0, ), p [1, ] and α = α, α 3 ) 0, ) satisfying then N c λ) satisfies χα, θ) < 1, 6.13) λ 0 log1/λ)) χα,θ) log N c λ) = ) iv) If 1.1) holds for any R 1 and 6.8) as x for some 0, ), p [1, ] and α = α, α 3 ) 0, ) satisfying then N c λ) satisfies χα, θ) = 1, 6.15) log1/λ)) 1 log N c λ) = 3 λ 0 Dα, θ, p, p). 6.16) v) If supp u is compact, then N c λ) satisfies the following: N c λ) = K cλ 3/ 6π 1 + o1)) 6.17) as λ 0, K c = dx P θ x q ξ q supp u). 6.18) Λ 1 q Z 3 References [1] K. Broderix, D. Hundertmark, W. Kirsch and H. Leschke 1995) The fate of Lifshits tails in magnetic fields, J. Statist. Phys., 80, 1. [] R. Carmona and J. Lacroix 1990) Spectral theory of random Schrödinger operators, Birkhäuser, Boston. [3] M. D. Donsker and S. R. S. Varadhan 1975) Asymptotics for the Wiener sausage Comm. Pure Appl. Math., 8, [4] L. Erdős 1998) Lifschitz tail in a magnetic field: the nonclassical regime, Probab. Theory Related Fields, 11, [5] L. Erdős 001) Lifschitz tail in a magnetic field: coexistence of classical and quantum behavior in the borderline case Probab. Theory Related Fields, 11, [6] R. Fukushima 009) Brownian survival and Lifshitz tail in perturbed lattice disorder J. Funct. Anal., 56, [7] R. Fukushima and N. Ueki 010) Classical and quantum behavior of the integrated density of states for a randomly perturbed lattice, Ann. Henri Poincaré, 11, [8] D. Hundertmark, W. Kirsch and S. Warzel 003) Classical magnetic Lifshits tails in three space dimensions: impurity potentials with slow anisotropic decay, Markov Process. Related Fields, 9, [9] T. Hupfer, H. Leschke and S. Warzel 1999) Poissonian obstacles with Gaussian walls discriminate between classical and quantum Lifshits tailing in magnetic fields, J. Statist. Phys., 97,

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