On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels

Ann. Henri Poincare 5 (2004) 381 – 403c© Birkhauser Verlag, Basel, 20041424-0637/04/020381-23DOI 10.1007/s00023-004-0173-9 Annales Henri Poincare

On the Singularities of the Magnetic Spectral ShiftFunction at the Landau Levels

Claudio Fernandez and Georgi Raikov

Abstract. We consider the three-dimensional Schrodinger operators H0 and H±where H0 = (i∇+A)2−b, A is a magnetic potential generating a constant magneticfield of strength b > 0, and H± = H0 ± V where V ≥ 0 decays fast enough atinfinity. Then, A. Pushnitski’s representation of the spectral shift function (SSF)for the pair of operators H±, H0 is well defined for energies E = 2qb, q ∈ Z+.We study the behaviour of the associated representative of the equivalence classdetermined by the SSF, in a neighbourhood of the Landau levels 2qb, q ∈ Z+.Reducing our analysis to the study of the eigenvalue asymptotics for a family ofcompact operators of Toeplitz type, we establish a relation between the type of thesingularities of the SSF at the Landau levels and the decay rate of V at infinity.

Resume. On considere les operateurs de Schrodinger tridimensionnels H0 et H±ou H0 = (i∇ + A)2 − b, A est un potentiel magnetique engendrant un champmagnetique constant d’intensite b > 0, et H± = H0±V ou V ≥ 0 decroıt assez vite al’infini. Alors, la representation obtenue par A. Pushnitski de la fonction du decalagespectral pour les operateurs H±, H0 est bien definie pour des energies E = 2qb, q ∈Z+. On etudie le comportement du representant associe de la classe d’equivalencedeterminee par la fonction du decalage spectral, au voisinage des niveaux de Landau2bq, q ∈ Z+. En reduisant l’analyse a l’investigation de l’asymptotique des valeurspropres d’une famille d’operateurs de Toeplitz compacts, on etablit une relationentre le type des singularites de la fonction du decalage spectral aux niveaux deLandau et la vitesse de la decroissance de V a l’infini.

1 Introduction

In this paper we analyze the singularities of the spectral shift function (SSF) for thethree-dimensional Schrodinger operator with constant magnetic field, perturbedby an electric potential which decays fast enough at infinity. Let us recall thedefinition of the abstract SSF for a pair of self-adjoint operators. First, let usconsider two self-adjoint operators T0 and T acting in the same Hilbert space,such that T − T0 ∈ S1 where S1 denotes the space of trace class operators. Then,there exists a unique function ξ(.; T , T0) ∈ L1(R) such that the Lifshits-Kreın traceformula

Tr(φ(T ) − φ(T0)) =∫R

ξ(E; T , T0)φ′(E)dE, φ ∈ C∞0 (R), (1.1)

holds (see, e.g., [17, Theorem 8.3.3]). Let now H0 and H be two lower-bounded self-adjoint operators acting in the same Hilbert space. Assume that for some γ > 0,

382 C. Fernandez and G. Raikov Ann. Henri Poincare

and λ0 ∈ R lying strictly below the infima of the spectra of H0 and H, we havethat

(H− λ0)−γ − (H0 − λ0)−γ ∈ S1. (1.2)

Set

ξ(E;H,H0) :=

−ξ((E − λ0)−γ ; (H− λ0)−γ , (H0 − λ0)−γ) if E > λ0,0 if E ≤ λ0.

Then, similarly to (1.1),

Tr(φ(H) − φ(H0)) =∫R

ξ(E;H,H0)φ′(E)dE, φ ∈ C∞0 (R),

(see [17, Theorem 8.9.1]). The function ξ(.;H,H0) is called the SSF for the pairof the operators H and H0. Evidently, it does not depend on the particular choiceof γ and λ0 in (1.2). If E lies below the infimum of the spectrum of H0, then thespectrum of H below E could be at most discrete, and we have

ξ(E;H,H0) = −N(E;H) (1.3)

where N(E;H) denotes the number of eigenvalues of H in the interval (−∞, E),counted with the multiplicities. On the other hand, for almost every E in the abso-lutely continuous spectrum of H0, the SSF ξ(E;H,H0) is related to the scatteringdeterminant det S(E;H,H0) for the pair (H,H0) by the Birman-Kreın formula

det S(E;H,H0) = e−2πiξ(E;H,H0)

(see [2] or [17, Section 8.4]). A survey of various asymptotic results concerning theSSF for numerous quantum Hamiltonians is contained in [15].In the present paper the role of H0 is played by the operator H0 := (i∇+A)2 − b,which is essentially self-adjoint on C∞

0 (R3). Here the magnetic potential A =(− bx2

2 , bx12 , 0

)generates the constant magnetic field B = curl A = (0, 0, b), b > 0.

It is well known that σ(H0) = σac(H0) = [0,∞) (see [1]), where σ(H0) denotesthe spectrum of H0, and σac(H0) its absolutely continuous spectrum. Moreover,the so-called Landau levels 2bq, q ∈ Z+ := 0, 1, . . ., play the role of thresholdsin σ(H0).For x = (x1, x2, x3) ∈ R3 we denote by X⊥ = (x1, x2) the variables on the planeperpendicular to the magnetic field. We assume that V satisfies

V ≡ 0, V ∈ C(R3), 0 ≤ V (x) ≤ C0〈X⊥〉−m⊥〈x3〉−m3 , x = (X⊥, x3) ∈ R3,(1.4)

with C0 > 0, m⊥ > 2, m3 > 1, and 〈x〉 := (1 + |x|2)1/2, x ∈ Rd, d ≥ 1. Most ofour results will hold under a more restrictive assumption than (1.4), namely

V ≡ 0, V ∈ C(R3), 0 ≤ V (x) ≤ C0〈x〉−m, m > 3, x ∈ R3. (1.5)

Vol. 5, 2004 Singularities of the Magnetic Spectral Shift Function 383

Note that (1.5) implies (1.4) with any m3 ∈ (0,m) and m⊥ = m−m3. In partic-ular, we can choose m3 ∈ (1,m− 2) so that m⊥ > 2.Set H± := H0 ± V so that the electric potential ±V has a definite sign. Ob-viously, inf σ(H+) = 0, inf σ(H−) ≥ −C0. The role of the perturbed opera-tor H is played in this paper by H±. By (1.4) and the diamagnetic inequality(see, e.g., [1]), V 1/2(H0 − λ0)−1 with λ0 < 0 is a Hilbert-Schmidt operator.Therefore, the resolvent identity implies (H± − λ0)−1 − (H0 − λ0)−1 ∈ S1 forλ0 < inf σ(H±) ≤ inf σ(H0), i.e., (1.2) holds with H = H±, H0 = H0, and γ = 1,and, hence, the SSF ξ(.;H±, H0) exists.A priori the SSF ξ(E;H±, H0) is defined only for almost every E ∈ R. In Sec-tion 2 below we introduce a representative ξ(.;H±, H0) of the equivalence classdetermined by ξ(.;H±, H0), which is well defined and uniformly bounded on eachcompact subset of the complement of the Landau levels. Moreover, ξ is continuouson R \ 2bZ+ everywhere except at the eigenvalues, isolated, or embedded in thecontinuous spectrum, of the operator H±.The main goal of the paper is the study of the asymptotic behaviour as λ → 0of ξ(2bq + λ;H±, H0) with fixed q ∈ Z+. Our results establish the asymptoticcoincidence of ξ(2bq + λ;H±, H0) with the traces of certain functions of compactToeplitz operators. Many of the spectral properties of those Toeplitz operatorsare well known, which allows us to describe explicitly the asymptotics as λ → 0of ξ(2bq + λ;H±, H0) in several generic cases. These asymptotic results admit aninterpretation directly in the terms of the SSF, which is independent of the choiceof the representative of the equivalence class. In particular, these results reveal thelink between the type of the singularities of the SSF at the Landau levels, and thedecay rate of V at infinity.The paper is organized as follows. In Section 2 we introduce the representative ξof the SSF. In Section 3 we formulate our main results, summarize some knownspectral properties of compact Toeplitz operators, and obtain as corollaries explicitasymptotic formulas describing the singularities of the SSF at the Landau levels.Section 4 contains preliminary estimates. The proofs of our main results can befound in Section 5. Finally, in Section 6 we prove some of the corollaries of themain results.

2 A. Pushnitski’s representation of the SSF

2.1. In this subsection we introduce some basic notations used throughout thepaper.We denote by S∞ the class of linear compact operators acting in a fixed Hilbertspace.Let T = T ∗ ∈ S∞. Denote by PI(T ) the spectral projection of T associated withthe interval I ⊂ R. For s > 0 set

n±(s;T ) := rank P(s,∞)(±T ).


For an arbitrary (not necessarily self-adjoint) operator T ∈ S∞ put

n∗(s;T ) := n+(s2;T ∗T ), s > 0. (2.1)

If T = T ∗, then evidently

n∗(s;T ) = n+(s, T ) + n−(s;T ), s > 0. (2.2)

Moreover, if Tj = T ∗j ∈ S∞, j = 1, 2, then the Weyl inequalities

n±(s1 + s2, T1 + T2) ≤ n±(s1, T1) + n±(s2, T2) (2.3)

hold for each s1, s2 > 0.Further, we denote by Sp, p ∈ [1,∞), the Schatten-von Neumann class of compact

operators for which the norm ‖T ‖p : =(p∫∞0 sp−1n∗(s;T ) ds

)1/pis finite. If T ∈

Sp, p ∈ [1,∞), then the following elementary inequality

n∗(s;T ) ≤ s−p‖T ‖pp (2.4)

holds for every s > 0. If T = T ∗ ∈ Sp, p ∈ [1,∞), then (2.2) and (2.4) imply

n±(s;T ) ≤ s−p‖T ‖pp, s > 0. (2.5)

Finally, we define the self-adjoint operators Re T := 12 (T + T ∗) and Im T :=

12i (T − T ∗). Evidently,

n±(s; Re T ) ≤ 2n∗(s;T ), n±(s; Im T ) ≤ 2n∗(s;T ). (2.6)

2.2. In this subsection we summarize several results due to A. Pushnitski on therepresentation of the SSF for a pair of lower-bounded self-adjoint operators (see[8]–[10]).Let I ∈ R be a Lebesgue measurable set. Set µ(I) := 1

π

∫I

dt1+t2 . Note that µ(R) = 1.

Lemma 2.1. [8, Lemma 2.1] Let T1 = T ∗1 ∈ S∞ and T2 = T ∗

2 ∈ S1. Then∫R

n±(s1 + s2;T1 + t T2) dµ(t) ≤ n±(s1;T1) +1πs2

‖T2‖1, s1, s2 > 0. (2.7)

Let H± and H0 be two lower-bounded self-adjoint operators acting in thesame Hilbert space. Let λ0 < inf σ(H±) ∪ σ(H0). First of all, assume that (1.2)holds with H = H± for some γ > 0. Further, let

V := ±(H± −H0) ≥ 0, (2.8)

V1/2(H0 − λ0)−1/2 ∈ S∞. (2.9)

Finally, suppose thatV1/2(H0 − λ0)−γ′ ∈ S2 (2.10)

holds for some γ′ > 0. For z ∈ C with Im z > 0 set T (z): = V1/2(H0 − z)−1V1/2.


Lemma 2.2. (see, e.g., [8, Lemma 4.1]) Let (2.8)–(2.10) hold. Then for almostevery E ∈ R the operator-norm limit T (E + i0) := n − limδ↓0 T (E + iδ) exists,and by (2.9) we have T (E + i0) ∈ S∞. Moreover, Im T (E + i0) ∈ S1.

Theorem 2.1. [8, Theorem 1.2] Let (1.2) with H = H±, and (2.8)–(2.10) hold.Then for almost every E ∈ R we have

ξ(E;H±,H0) = ±∫R

n∓(1; Re T (E + i0) + t Im T (E + i0)) dµ(t).

Remark. The representation of the SSF described in the above theorem has beengeneralized to non-sign-definite perturbations in [6] in the case of trace-class per-turbations, and in [10] in the case of relatively trace-class perturbations. Thesegeneralizations are based on the concept of the index of orthogonal projections.We will not use them in the present paper.

Suppose now that V satisfies (1.4). Then relations (1.2) and (2.8)–(2.10)hold with V = V , H0 = H0, and γ = γ′ = 1. For z ∈ C, Im z > 0, set T (z) :=V 1/2(H0 − z)−1V 1/2. By Lemma 2.2, for almost every E ∈ R the operator-normlimit

T (E + i0) := n − limδ↓0

T (E + iδ) (2.11)

exists, andImT (E + i0) ∈ S1. (2.12)

For trivial reasons the limit in (2.11) exists and (2.12) holds for each E < 0. InCorollary 4.3 below we show that this is also true for each E ∈ [0,∞) \ 2bZ+.Hence, by Lemma 2.1, the quantity

∫Rn∓(1; ReT (E + i0) + t ImT (E + i0)) dµ(t)

is well defined for every E ∈ R \ 2bZ+. Set

ξ(E;H±, H0) = ±∫R

n∓(1; ReT (E + i0) + t ImT (E + i0)) dµ(t), E ∈ R \ 2bZ+.

(2.13)By Theorem 2.1 we have

ξ(E;H±, H0) = ξ(E;H±, H0) (2.14)

for almost every E ∈ R.Remark. In [4] it is shown that the function ξ defined on R \ 2bZ+ is continuousaway from the eigenvalues of the operator H±. Note that, in contrast to the caseb = 0, we cannot rule out the possibility of existence of embedded eigenvalues,by imposing short-range assumptions of the type of (1.4) or (1.5): Theorem 5.1of [1] shows that there are axisymmetric potentials V of compact support suchthat below each Landau level 2bq, q ∈ Z+, there exists an infinite sequence ofeigenvalues of H− which converges to 2bq. On the other hand, generically, theonly possible accumulation points of the eigenvalues of the operators H± are theLandau levels (see [1, Theorem 4.7], [5, Theorem 3.5.3 (iii)]). Further informationof the location of these eigenvalues can be found in [4].


3 Main results

3.1. In this subsection we formulate our main results. To this end we need somemore notations. Introduce the Landau Hamiltonian

h(b) :=(i∂

∂x1− bx2

2

)2

+(i∂

∂x2+bx1

2

)2

− b, (3.1)

i.e., the two-dimensional Schrodinger operator with constant scalar magnetic fieldb > 0, essentially self-adjoint on C∞

0 (R2). It is well known that σ(h(b)) = ∪∞q=0

2bq, and each eigenvalue 2bq, q ∈ Z+, has infinite multiplicity (see, e.g., [1]).For x,x′ ∈ R2 denote by Pq,b(x,x′) the integral kernel of the orthogonal projectionpq(b) onto the subspace Ker (h(b) − 2bq), q ∈ Z+. It is well known that

Pq,b(x,x′) =b

2πLq

(b|x− x′|2

2

)exp

(− b

4(|x − x′|2 + 2i(x1x

′2 − x′1x2))

)(3.2)

(see [7] or [12, Subsection 2.3.2]) where

Lq(t) :=1q!etdq(tqe−t)

dtq=

q∑k=0

(q

k

)(−t)kk!

, t ∈ R, q ∈ Z+,

are the Laguerre polynomials. Note that Pq,b(x,x) = b2π for each q ∈ Z+ and

x ∈ R2. Introduce the orthogonal projections Pq : L2(R3) → L2(R3), q ∈ Z+, by

(Pqu)(X⊥, x3) =∫R2

Pq,b(X⊥, X ′⊥)u(X ′

⊥, x3) dX ′⊥, u ∈ L2(R3). (3.3)

Assume that (1.4) holds. Set

W (X⊥) :=∫R

V (X⊥, x3)dx3, X⊥ ∈ R2. (3.4)

If, moreover, V satisfies (1.5), then

0 ≤W (X⊥) ≤ C′0〈X⊥〉−m+1, X⊥ ∈ R2, (3.5)

where C′0 = C0

∫R〈x〉−mdx. For q ∈ Z+ and λ > 0 introduce the operator

ωq(λ) :=1

2√λpqWpq. (3.6)

Evidently, ωq(λ) is self-adjoint and non-negative in L2(R2).

Lemma 3.1. Let U ∈ Lr(R2), r ≥ 1, and q ∈ Z+. Then pqUpq ∈ Sr.


Proof. If U ∈ L∞(R2), then evidently ‖pqUpq‖ ≤ ‖U‖L∞. If U ∈ L1(R2), we writepqUpq = pq|U |1/2ei arg U |U |1/2pq, check that

‖pq|U |1/2‖22 =

b

2π‖U‖L1, ‖ei argU |U |1/2pq‖2

2 =b

2π‖U‖L1,

and conclude that ‖pqUpq‖1 ≤ b2π‖U‖L1. Interpolating, we get ‖pqUpq‖rr ≤

b2π‖U‖rLr which implies the desired result.

Remark. The proof of Lemma 3.1 follows the idea of the proof of [11, Lemma 5.1].We include it here in order to make the exposition self-contained.

If λ > 0, and V satisfies (1.4) with m⊥ > 2 and m3 > 1, then Lemma 3.1with U = W implies ωq(λ) ∈ S1.

Theorem 3.1. Assume that (1.5) is valid. Let q ∈ Z+, b > 0. Then the asymptoticestimates

ξ(2bq − λ;H+, H0) = O(1), (3.7)

and

−n+((1 − ε);ωq(λ)) +O(1) ≤ ξ(2bq − λ;H−, H0) ≤ −n+((1 + ε);ωq(λ)) +O(1),(3.8)

hold as λ ↓ 0 for each ε ∈ (0, 1).

Suppose that the potential V satisfies (1.4). For λ > 0 define the matrix-valued function

Wλ = Wλ(X⊥) :=(w11 w12

w21 w22

), X⊥ ∈ R2, (3.9)

where

w11 :=∫R

V (X⊥, x3) cos2 (√λx3)dx3, w22 :=

∫R

V (X⊥, x3) sin2 (√λx3)dx3,

w12 = w21 :=∫R

V (X⊥, x3) cos (√λx3) sin (

√λx3)dx3.

Introduce the operator

Ωq :=1

2√λpqWλpq. (3.10)

Evidently, Ωq(λ) is self-adjoint and non-negative in L2(R2)2. Moreover, using thefact that ωq(λ) ∈ S1, it is easy to check that Ωq(λ) ∈ S1 as well.

Theorem 3.2. Assume that (1.5) is valid. Let q ∈ Z+, b > 0. Then the asymptoticestimates

± 1π

Tr arctan ((1 ± ε)−1Ωq(λ)) +O(1) ≤ ξ(2bq + λ;H±, H0)

≤ ± 1π

Tr arctan ((1 ∓ ε)−1Ωq(λ)) +O(1) (3.11)



The proofs of Theorems 3.1 and 3.2 can be found in Section 5. In the followingsubsection we will describe explicitly the asymptotics of ξ(2bq − λ;H−, H0) andξ(2bq + λ;H±, H0) as λ ↓ 0 under generic assumptions about the behaviour ofW (X⊥) as |X⊥| → ∞.

3.2. Relations (3.8) and (3.11) allow us to reduce the analysis of the behaviour asλ → 0 of ξ(2bq + λ;H±, H0), to the study of the asymptotic distribution of theeigenvalues of Toeplitz-type operators pqUpq. The following three lemmas concernthe spectral asymptotics of such operators.

Lemma 3.2. [11, Theorem 2.6] Let the function U ∈ C1(R2) satisfy the estimates

0 ≤ U(X⊥) ≤ C1〈X⊥〉−α, |∇U(X⊥)| ≤ C1〈X⊥〉−α−1, X⊥ ∈ R2,

for some α > 0 and C1 > 0. Assume, moreover, that

U(X⊥) = u0(X⊥/|X⊥|)|X⊥|−α(1 + o(1)), |X⊥| → ∞,

where u0 is a continuous function on S1 which is non-negative and does not vanishidentically. Then for each q ∈ Z+ we have

n+(s; pqUpq) =b

2π

∣∣X⊥ ∈ R2|U(X⊥) > s∣∣ (1 + o(1)) =

ψα(s;u0, b) (1 + o(1)), s ↓ 0,

where |.| denotes the Lebesgue measure, and

ψα(s) = ψα(s;u0, b) := s−2/α b

4π

∫S1u0(t)2/αdt, s > 0. (3.12)

Remark. Theorem 2.6 of [11] contains a considerably more general result thanLemma 3.2. For the sake of the simplicity of exposition, here we reproduce onlythe special case of asymptotically homogeneous U .

Lemma 3.3. [13, Theorem 2.1, Proposition 4.1] Let 0 ≤ U ∈ L∞(R2). Assume that

lnU(X⊥) = −µ|X⊥|2β(1 + o(1)), |X⊥| → ∞,

for some β ∈ (0,∞), µ ∈ (0,∞). Then for each q ∈ Z+ we have

n+(s; pqUpq) = ϕβ(s)(1 + o(1)), s ↓ 0,

where

ϕβ(s) = ϕβ(s;µ, b) :=

b2µ1/β | ln s|1/β if 0 < β < 1,

1ln (1+2µ/b) | ln s| if β = 1,β

β−1(ln | ln s|)−1| ln s| if 1 < β <∞,

s ∈ (e,∞).

(3.13)


Lemma 3.4. [13, Theorem 2.2, Proposition 4.1] Let 0 ≤ U ∈ L∞(R2). Assumethat the support of U is compact, and that there exists a constant C > 0 such thatU ≥ C on an open non-empty subset of R2. Then for each q ∈ Z+ we have

n+(s; pqUpq) = ϕ∞(s) (1 + o(1)), s ↓ 0,

whereϕ∞(s) := (ln | ln s|)−1| ln s|, s ∈ (e,∞). (3.14)

Remark. For each β ∈ (0,∞] and c > 0 we have ϕβ(cs) = ϕβ(s)(1 + o(1)) as s ↓ 0.

Employing Lemmas 3.2, 3.3, 3.4, and the above remark, we find that (3.8)immediately entails the following corollary.

Corollary 3.1. Let (1.5) hold with m > 3.i) Assume that the hypotheses of Lemma 3.2 hold with U = W and α = m− 1.

Then we have

ξ(2bq − λ;H−, H0) = − b

2π

∣∣∣X⊥ ∈ R2|W (X⊥) > 2

√λ∣∣∣ (1 + o(1))

= −ψm−1(2√λ;u0, b) (1 + o(1)), λ ↓ 0, (3.15)

the function ψα being defined in (3.12).ii) Assume that the hypotheses of Lemma 3.3 hold with U = W . Then we have

ξ(2bq − λ;H−, H0) = −ϕβ(√λ;µ, b) (1 + o(1)), λ ↓ 0, β ∈ (0,∞),

the functions ϕβ being defined in (3.13).iii) Assume that the hypotheses of Lemma 3.4 hold with U = W . Then we have

ξ(2bq − λ;H−, H0) = −ϕ∞(√λ) (1 + o(1)), λ ↓ 0,

the function ϕ∞ being defined in (3.14).

Remark. In the special case q = 0 when −ξ(−λ;H−, H0) coincides for almostevery λ > 0 with the eigenvalue counting function for the operator H− (see (1.3)),relation (3.15) was established for the first time in [16]. Here we use a differentapproach related to the one developed in [11].

Similarly, the combination of Theorem 3.2 with Lemmas 3.2–3.4 yields thefollowing corollary.

Corollary 3.2.

i) Let (1.5) hold with m > 3. Assume that the hypotheses of Lemma 3.2 arefulfilled for U = W and α = m− 1. Then we have

ξ(2bq + λ;H±, H0) = ± b

2π2

∫R2

arctan ((2√λ)−1W (X⊥))dX⊥ (1 + o(1))

= ± 12 cos (π/(m− 1))

ψm−1(2√λ;u0, b) (1 + o(1)), λ ↓ 0.


ii) Let (1.5) hold with m > 3. Suppose in addition that V satisfies (1.4) for somem⊥ > 2 and m3 > 2. Finally, assume that the hypotheses of Lemma 3.3 arefulfilled for U = W . Then we have

ξ(2bq + λ;H±, H0) = ± 12ϕβ(

√λ;µ, b) (1 + o(1)), λ ↓ 0, β ∈ (0,∞).

iii) Let the assumptions of the previous part be fulfilled, except that the hypothesesof Lemma 3.3 are replaced by those of Lemma 3.4. Then we have

ξ(2bq + λ;H±, H0) = ± 12ϕ∞(

√λ) (1 + o(1)), λ ↓ 0.

The proof of Corollary 3.2 can be found in Section 6.

3.3. In this subsection we present a possible interpretation of our results directlyin the terms of the SSF ξ(.;H±, H0) which is invariant of the choice of the repre-sentative of the equivalence class determined by the SSF. For λ > 0, and q ∈ Z+,introduce the averaged values of the SSF

Ξ±q,<(λ) :=

1λ

∫ 2bq

2bq−λ

ξ(s;H±, H0)ds =1λ

∫ λ

0

ξ(2bq − t;H±, H0)dt,

Ξ±q,>(λ) :=

1λ

∫ 2bq+λ

2bq

ξ(s;H±, H0)ds =1λ

∫ λ

0

ξ(2bq + t;H±, H0)dt.

Since ξ(.;H±, H0) ∈ L1loc(R), the quantities Ξ±

q,<(λ) and Ξ±q,>(λ) are well defined

for every λ > 0. Applying (2.14), we find that the asymptotic bound Ξ+q,<(λ) =

O(1) as λ ↓ 0 follows from (3.7). Further, Corollary 3.1 i) implies

Ξ−q,<(λ) = − m− 1

m− 2ψm−1(2

√λ;u0, b) (1 + o(1)), λ ↓ 0, m > 3,

while Corollary 3.1 ii)–iii) entails

Ξ−q,<(λ) = −ϕβ(

√λ;µ, b) (1 + o(1)), λ ↓ 0, β ∈ (0,∞].

Finally, it follows from Corollary 3.2 i) that

Ξ±q,>(λ) = ± 1

2 cos (π/(m− 1))m− 1m− 2

ψm−1(2√λ;u0, b) (1 + o(1)), λ ↓ 0, m > 3,

while Corollary 3.2 ii)–iii) implies

Ξ±q,>(λ) = ± 1

2ϕβ(

√λ;µ, b) (1 + o(1)), λ ↓ 0, β ∈ (0,∞].


4 Preliminary estimates

For z ∈ C with Im z > 0, define the operator R(z) :=(− d2

dx23− z)−1

bounded in

L2(R), as well as the operators

Tq(z) := V 1/2Pq(H0 − z)−1V 1/2, q ∈ Z+,

bounded in L2(R3) (see (3.3) for the definition of the orthogonal projection Pq).The operator R(z) admits the integral kernel Rz(x3 − x′3) where Rz(x) =iei

√z|x|/(2

√z), x ∈ R, the branch of

√z being chosen so that Im

√z > 0. More-

over, Tq(z) = V 1/2(pq(b) ⊗R(z − 2bq)

)V 1/2.

For λ ∈ R, λ = 0, define R(λ) as the operator with integral kernel Rλ(x3 − x′3)where

Rλ(x) := limδ↓0

Rλ+iδ(x) =

e−√−λ|x|

2√−λ

if λ < 0,iei

√λ|x|

2√λ

if λ > 0,x ∈ R. (4.1)

Evidently, if w ∈ L2(R) and λ = 0, then wR(λ)w ∈ S2. For E ∈ R, E = 2bq,q ∈ Z+, set

Tq(E) := V 1/2(pq(b) ⊗R(E − 2bq)

)V 1/2.

Proposition 4.1. Let E ∈ R, q ∈ Z+, E = 2bq. Let (1.4) hold. Then Tq(E) ∈ S2,and

‖Tq(E)‖22 ≤ C1b/|E − 2bq| (4.2)

with C1 independent of E, b, and q. Moreover,

limδ↓0

‖Tq(E + iδ) − Tq(E)‖2 = 0. (4.3)

Proof. The operator Tq(E) admits the representation

Tq(E) = M (Gq ⊗ JE−2bq) M (4.4)

where M : L2(R3) → L2(R3) is the multiplier by V (X⊥, x3)1/2〈X⊥〉m⊥/2〈x3〉m3/2,Gq : L2(R2

X′⊥

) → L2(R2X⊥) is the operator with integral kernel

〈X⊥〉−m⊥/2Pq,b(X⊥, X ′⊥)〈X ′

⊥〉−m⊥/2, X⊥, X ′⊥ ∈ R2,

(see (3.2) for the definition of Pq,b), while Jλ : L2(Rx′3) → L2(Rx3) is the operator

with integral kernel

〈x3〉−m3/2Rλ(x3 − x′3)〈x′3〉−m3/2, x3, x′3 ∈ R, λ ∈ R \ 0. (4.5)


Evidently, ‖Tq(E)‖22 ≤ ‖M‖4‖Gq‖2

2‖JE−2bq‖22. By (1.4) we have ‖M‖4 ≤ C2

0 .Further,

‖JE−2bq‖22 =

∫R

|RE−2bq(x3 − x′3)|2〈x3〉−m3〈x′3〉−m3 dx3 dx′3

≤ 14|E − 2bq|

(∫R

〈x3〉−m3dx3

)2

.

Finally, since ‖Gq‖ ≤ 1, we have ‖Gq‖22 ≤ ‖Gq‖1 = b

2π

∫R2〈X⊥〉−m⊥dX⊥. Hence,

(4.2) holds with C1 = C20

8π

(∫R〈x3〉−m3dx3

)2 ∫R2〈X⊥〉−m⊥dX⊥ . To prove (4.3), we

write

‖Tq(E + iδ) − Tq(E)‖22 =

∫R2

∫R2

∫R

∫R

V (X⊥, x3)V (X ′⊥, x

′3)|Pq,b(X⊥, X ′

⊥)|2

|RE−2bq+iδ(x3 − x′3) −RE−2bq(x3 − x′3)|2dx3dx′3dX⊥dX ′

⊥,

note that limδ↓0 RE−2bq+iδ(x) = RE−2bq(x) for each x ∈ R, and that the integrandin the above integral is bounded from above for each δ > 0 by the L1(R6)-function

1|E − 2bq|V (X⊥, x3)V (X ′

⊥, x′3)|Pq,b(X⊥, X ′

⊥)|2, (X⊥, x3, X′⊥, x

′3) ∈ R6.

Therefore, the dominated convergence theorem implies (4.3).

Remark. Using more sophisticated tools than those of the proof of Proposition4.1, it is shown in [4] that for E = 2bq we have not only Tq(E) ∈ S2, but alsoTq(E) ∈ S1. We will not use this fact here.

Corollary 4.1. Assume that (1.4) holds. Let E ∈ R, q ∈ Z+, E = 2bq. ThenIm Tq(E) ≥ 0. Moreover, if E < 2bq, then Im Tq(E) = 0.

Proof. The non-negativity of Im Tq(E) follows from the representation

Im Tq(E + iδ) = δV 1/2Pq((H0 − E)2 + δ2)−1PqV1/2, δ > 0,

and the limiting relation Im Tq(E) = n− limδ↓0 Im Tq(E + iδ), E = 2bq, which onits turn is implied by (4.3). Moreover, if E < 2bq, then (4.1) entails Tq(E) = Tq(E)∗

so that Im Tq(E) = 0.

Corollary 4.2. Under the assumptions of Corollary 4.1 we have ImTq(E) ∈ S1.Furthermore, if E > 2bq, then

‖ImTq(E)‖1 = Tr ImTq(E) =b

4π(E − 2bq)−1/2

∫R3V (x)dx. (4.6)


Proof. Bearing in mind the representation (4.4), we find that the inclusionImTq(E) ∈ S1 would be implied by the inclusion Gq ∈ S1 and ImJE−2bq ∈ S1.The first inclusion follows from Lemma 3.1, and the second one from the obviousfact that rank ImJE−2bq ≤ 2.Further, the first equality in (4.6) follows from the non-negativity of the operatorImTq(E) which is guaranteed by Corollary 4.1. Since the operator ImTq(E) withE > 2bq admits the kernel

12√E − 2bq

√V (X⊥, x3) cos

(√E − 2bq(x3 − x′3)

)Pq,b(X⊥, X ′

⊥)√V (X ′

⊥, x′3), (X⊥, x3), (X ′

⊥, x′3) ∈ R3,

the Mercer theorem (see, e.g., the lemma on pp. 65–66 of [14]) implies the secondequality in (4.6).

Proposition 4.2. Let q ∈ Z+, λ ∈ R, |λ| ∈ (0, b], and δ > 0. Assume that V satisfies(1.4). Then the operator series

T+q (2bq + λ+ iδ) :=

∞∑l=q+1

Tl(2bq + λ+ iδ), (4.7)

T+q (2bq + λ) :=

∞∑l=q+1

Tl(2bq + λ) (4.8)

are convergent in S2. Moreover,

‖T+q (2bq + λ)‖2

2 ≤ C0b

8π

∞∑l=q+1

(2b(l − q) − λ)−3/2

∫R3V (x)dx. (4.9)

Finally,limδ↓0

‖T+q (2bq + λ+ iδ) − T+

q (2bq + λ)‖2 = 0. (4.10)

Proof. For each q′, q′′ ∈ Z+ such that q + 1 ≤ q′ < q′′ <∞ we have

‖q′′∑l=q′

T+l (2bq + λ)‖2

2 = ‖q′′∑l=q′

√V pl ⊗

(− d2

dx23

+ 2b(l− q) − λ

)−1 √V ‖2

2

≤ C0

q′′∑l=q′

Tr

(pl ⊗

(− d2

dx23

+ 2b(l− q) − λ

)−2

V

)

=C0b

(2π)2

q′′∑l=q′

∫R

dη

(η2 + 2b(l− q) − λ)2

∫R3V (x)dx

=C0b

8π

q′′∑l=q′

(2b(l− q) − λ)−3/2

∫R3V (x)dx. (4.11)


Since the numerical series∑∞

l=q+1(2b(l − q) − λ)−3/2 is convergent, and S2 is aHilbert (hence, complete) space, we conclude that (4.11) entails the convergence inS2 of the operator series in (4.8), as well as the validity of (4.9). The convergenceof the series in (4.7) is proved in exactly the same manner. Finally, (4.10) followsfrom the estimate

‖T+q (2bq + λ+ iδ) − T+

q (2bq + λ)‖22

≤ δ2C0b

(2π)2

∞∑l=q+1

∫R

dη

(η2 + 2b(l − q) − λ)2((η2 + 2b(l− q) − λ)2 + δ2)

∫R3V (x)dx

≤ δ2C0b

2π2

∞∑l=q+1

(2b(l− q) − λ)−7/2

∫ ∞

0

dη

(η2 + 1)4

∫R3V (x)dx.

For E = 2bq + λ with q ∈ Z+, and λ ∈ R, |λ| ∈ (0, b], set T−q (E) :=

T (E) − Tq(E) − T+q (E) (see (4.8)). Note that if q = 0, then T−

q (E) = 0, and ifq ≥ 1, then T−

q (E) =∑q−1

l=0 Tl(E).

Corollary 4.3. For E = 2bq + λ with q ∈ Z+ and λ ∈ R, |λ| ∈ (0, b] the operator-norm limit (2.11) exists, and

T (E + i0) = T−q (E) + Tq(E) + T+

q (E). (4.12)

Moreover,

Re T (E + i0) = Re T−q (E) + Re Tq(E) + T+

q (E), (4.13)

Im T (E + i0) = Im T−q (E) + Im Tq(E) ∈ S1. (4.14)

Proof. Let δ > 0. Evidently, T (E+ iδ) =∑q

l=0 Tl(E+ iδ)+T+q (E+ iδ) (see (4.7)).

Proposition 4.1 implies that n − limδ↓0∑q

l=0 Tl(E + iδ) = T−q (E) + Tq(E), while

Proposition 4.2 implies that n − limδ↓0 T+q (E) = T+

q (E). Combining the abovetwo relations, we get (4.12). Relation (4.13) follows immediately from (4.12) andT+q (E) = T+

q (E)∗, while (4.14) is implied by (4.12) and Corollaries 4.1 and 4.2.

5 Proof of the main results

5.1. This subsection contains a general estimate which will be used in the proofsof all our main results. Informally speaking, we show that we can replace theoperator T (E + i0) by Tq(E) in the r.h.s of (2.13) when we deal with the firstasymptotic term of ξ(E;H±, H0) as the energy E approaches a given Landau level2bq, q ∈ Z+.


Proposition 5.1. Assume that (1.4) holds. Let E = 2bq+λ with q ∈ Z+, and λ ∈ R,|λ| ∈ (0, b]. Then the asymptotic estimates

∫R

n±(1 + ε; ReTq(E) + t ImTq(E)) dµ(t) +O(1)

≤∫R

n±(1; ReT (E + i0) + t ImT (E + i0)) dµ(t)

≤∫R

n±(1 − ε; ReTq(E) + t ImTq(E)) dµ(t) +O(1) (5.1)

hold as λ→ 0 for each ε ∈ (0, 1).

Proof. Using (4.13) and (4.14), and applying the Weyl inequalities (2.3), we get∫R

n±(1 + ε; ReTq(E) + t ImTq(E)) dµ(t)

−∫R

n∓(ε; ReT−q (E) + T+

q (E) + t ImT−q (E)) dµ(t)

≤∫R

n±(1; ReT (E + i0) + t ImT (E + i0)) dµ(t)

≤∫R

n±(1 − ε; ReTq(E) + t ImTq(E)) dµ(t)

+∫R

n±(ε; ReT−q (E) + T+

q (E) + t ImT−q (E)) dµ(t). (5.2)

In order to conclude that (5.2) implies (5.1), it remains to show that∫R


q (E) + t ImT−q (E)) dµ(t) = O(1), λ→ 0, (5.3)

for each ε > 0. Employing (2.7) and (2.3), we find that∫R


q (E) + t ImT−q (E)) dµ(t)

≤ n±(ε/2; ReT−q (E) + T+

q (E)) +2επ

‖ImT−q (E)‖1

≤ n±(ε/4; ReT−q (E)) + n±(ε/4;T+

q (E)) +2επ

‖ImT−q (E)‖1, ε > 0. (5.4)

If q = 0, and, hence, T−q (E) = 0, we need only to apply (2.5) with p = 2, and

(4.9), in order to get

n±(ε/4;T+q (E)) ≤ 16ε−2‖T+

q (E)‖22 ≤ 2C0b

ε2π

∞∑l=q+1

(2b(l− q) − λ)−3/2

∫R3V (x)dx,

(5.5)


which combined with (5.4) yields (5.3). If q ≥ 1 we should also utilize the estimate

n±(ε/4; ReT−q (E)) ≤ 32ε−2‖T−

q (E)‖22 ≤ 32ε−2qC1b

q−1∑l=0

(2b(q − l) + λ)−1 (5.6)

which follows from (2.6), (2.4) with p = 2, and (4.2), as well as the estimate

‖ImT−q (E)‖1 ≤ b

2π

q−1∑l=0

(2b(q − l) + λ)−1/2

∫R3V (x)dx, (5.7)

which follows from (4.6). Thus, in the case q ≥ 1, estimate (5.3) is implied by fromthe combination of (5.4)–(5.7).

5.2. In this subsection we complete the proof of the first part of Theorem 3.1.Since Im Tq(2bq−λ) = 0 and Re Tq(2bq−λ) = Tq(2bq−λ) ≥ 0 if λ > 0, Proposition5.1 implies immediately the following corollary.

Corollary 5.1. Under the hypotheses of Proposition 5.1 the asymptotic estimates∫R

n−(1; ReT (2bq − λ+ i0) + t ImT (2bq − λ+ i0)) dµ(t) = O(1), (5.8)

andn+(1 + ε;Tq(2bq − λ)) +O(1)

≤∫R

n+(1; ReT (2bq − λ+ i0) + t ImT (2bq − λ+ i0)) dµ(t)

≤ n+(1 − ε;Tq(2bq − λ)) +O(1) (5.9)


Now the combination of (2.13) and (5.8) yields (3.7).

5.3. In this section we complete the proof of the second part of Theorem 3.1. Forq ∈ Z+ and λ > 0 define Oq(λ) : L2(R3) → L2(R3) as the operator with integralkernel

12√λ

√V (X⊥, x3) Pq,b(X⊥, X ′

⊥)√V (X ′

⊥, x′3), (X⊥, x3), (X ′

⊥, x′3) ∈ R3.

Proposition 5.2. Under the hypotheses of Theorem 3.1 the asymptotic estimates

n+((1 + ε)s;Oq(λ)) +O(1) ≤ n+(s;Tq(2bq − λ)) ≤ n+((1 − ε)s;Oq(λ)) +O(1)(5.10)

hold as λ ↓ 0 for each ε ∈ (0, 1) and s > 0.


Proof. Fix s > 0 and ε ∈ (0, 1). Then the Weyl inequalities entail

n+((1 + ε)s;Oq(λ)) − n−(εs;Tq(2bq − λ) −Oq(λ))

≤ n+(s;Tq(2bq − λ))

≤ n+((1 − ε)s;Oq(λ)) + n+(εs;Tq(2bq − λ) −Oq(λ)).

In order to get (5.10), it suffices to show that

n±(t;Tq(2bq − λ) −Oq(λ)) = O(1), λ ↓ 0, (5.11)

for every fixed t > 0. Denote by Tq the operator with integral kernel

−12

√V (X⊥, x3) |x3 − x′3| Pq,b(X⊥, X ′

⊥)√V (X ′

⊥, x′3), (X⊥, x3), (X ′

⊥, x′3) ∈ R3.

(5.12)Pick m′ ∈ (3,m), and write

Tq = Mm,m′Gq,m−m′ ⊗ J(0)m′ Mm,m′

where Mm,m′ is the multiplier by the bounded function√V (X⊥, x3)〈X⊥〉(m−m′)/2

〈x3〉m′/2, (X⊥, x3) ∈ R3, Gq,m−m′ : L2(R2) → L2(R2) is the operator with integral

kernel

〈X⊥〉−(m−m′)/2Pq,b(X⊥, X ′⊥)〈X ′

⊥〉−(m−m′)/2, X⊥, X ′⊥ ∈ R2,

and J (0)m′ : L2(R) → L2(R) is the operator with integral kernel

−12〈x3〉−m′/2|x3 − x′3|〈x′3〉−m′/2, x3, x

′3 ∈ R.

Since m−m′ > 0, Lemma 3.1 implies that the operator Gq,m−m′ is compact, andsince m′ > 3 we have J (0)

m′ ∈ S2. Finally, since Mm,m′ is bounded, we find that theoperator Tq is compact. Further,

Tq(2bq − λ) −Oq(λ) = Mm,m′Gq,m−m′ ⊗ J(λ)m′ Mm,m′

where J (λ)m′ , λ > 0, is the operator with integral kernel

− 1√λ〈x3〉−m′/2e−

12

√λ|x3−x′

3|sinh

(√λ|x3 − x′3|

2

)〈x′3〉−m′/2, x3, x

′3 ∈ R.

Applying the dominated convergence theorem, we easily find that limλ↓0 ‖J (λ)m′ −

J(0)m′ ‖2 = 0. Therefore, Tq = n − limλ↓0(Tq(2bq− λ) −Oq(λ)). Fix t > 0. Choosing


λ > 0 so small that ‖Tq(2bq − λ) − Oq(λ) − Tq‖ < t/2, and applying the Weylinequalities, we get

n±(t;Tq(2bq − λ) −Oq(λ)) ≤n±(t/2;Tq(2bq − λ) −Oq(λ) − Tq) + n±(t/2; Tq) = n±(t/2; Tq). (5.13)

Since the r.h.s. of (5.13) is finite and independent of λ, we conclude that (5.13)entails (5.11).

Proposition 5.3. Assume that (1.4) holds. Then for each q ∈ Z+, λ > 0, and s > 0we have

n+(s;Oq(λ)) = n+(s;ωq(λ)) (5.14)

(see (3.6) for the definition of the operator ωq(λ)).

Proof. Define the operator K : L2(R3) → L2(R2) by

(Ku)(X⊥) :=∫R2

∫R

Pq,b(X⊥, X ′⊥)√V (X ′

⊥, x′3)u(X ′

⊥, x′3) dx′3 dX

′⊥, X⊥ ∈ R2,

where u ∈ L2(R3). The adjoint operator K∗ : L2(R2) → L2(R3) is given by

(K∗v)(X⊥, x3) :=√V (X⊥, x3)

∫R2

Pq,b(X⊥, X ′⊥)v(X ′

⊥) dX ′⊥, (X⊥, x3) ∈ R3,

where v ∈ L2(R2). Obviously,

Oq(λ) =1

2√λK∗K, ωq(λ) =

12√λK K∗.

Since n+(s;K∗K) = n+(s;KK∗) for each s > 0, we get (5.14).

Putting together (2.13), (5.9), (5.10), and (5.14), we get (3.8). Thus, we aredone with the proof of Theorem 3.1.

5.4. In this subsection we complete the proof of Theorem 3.2.

Proposition 5.4. Let q ∈ Z+ and b > 0. Assume that (1.5) holds. Then the asymp-totic estimates

n±(s; ReTq(2bq + λ)) = O(1) (5.15)

are valid as λ ↓ 0 for each s > 0.

Proof. The operator ReTq(2bq + λ+ i0) admits the integral kernel

− 12√λ

√V (X⊥, x3) sin

(√λ|x3 − x′3|

)Pq,b(X⊥, X ′

⊥)√V (X ′

⊥, x′3),

(X⊥, x3), (X ′⊥, x

′3) ∈ R3.

Arguing as in the proof of Proposition 5.2, we find that n− limλ↓0 ReTq(2bq+λ) =Tq (see (5.12)). Fix s > 0. Choosing λ > 0 so small that ‖ReTq(2bq+λ)−Tq‖ < s/2,and applying the Weyl inequalities, we get n±(s; ReTq(2bq + λ)) ≤ n±(s/2; Tq)which implies (5.15).


Taking into account Propositions 5.1 and 5.4, and applying the Weyl inequal-ities and the evident identities∫

R

n±(s; tT )dµ(t) =1π

Tr arctan (s−1T ), s > 0,

where T = T ∗ ≥ 0, T ∈ S1, we obtain the following

Corollary 5.2. Let q ∈ Z+, b > 0. Assume that V satisfies (1.5). Then the asymp-totic estimates

1π

Tr arctan ((1 + ε)−1Im Tq(2bq + λ)) +O(1)

≤∫R

n±(1; Re Tq(2bq + λ) + t Im Tq(2bq + λ))dµ(t)

≤ 1π

Tr arctan ((1 − ε)−1Im Tq(2bq + λ)) +O(1) (5.16)

are valid as λ ↓ 0 for each ε ∈ (0, 1).

Proposition 5.5. Assume that (1.4) holds. Then for each q ∈ Z+, λ > 0, and s > 0,we have

n+(s; Im Tq(2bq + λ)) = n+(s; Ωq(λ)) (5.17)

(see (3.10) for the definition of the operator Ωq(λ)). Consequently,

Tr arctan (s−1Im Tq(2bq + λ)) = Tr arctan (s−1Ωq(λ)) (5.18)

for each q ∈ Z+, λ > 0.

Proof. The proof is quite similar to that of Proposition 5.3. Define the operatorK : L2(R3) → L2(R2)2 by

Ku := v = (v1, v2) ∈ L2(R2)2, u ∈ L2(R3),

where

v1(X⊥) :=∫R2

∫R

Pq,b(X⊥, X ′⊥) cos(

√λx′3)

√V (X ′

⊥, x′3)u(X ′

⊥, x′3) dx′3 dX

′⊥,

v2(X⊥) :=∫R2

∫R

Pq,b(X⊥, X ′⊥) sin(

√λx′3)

√V (X ′

⊥, x′3)u(X ′

⊥, x′3) dx′3 dX

′⊥, X⊥ ∈ R2.

Evidently, the adjoint operator K∗ : L2(R2)2 → L2(R3) is given by

(K∗v)(X⊥, x3) := cos(√λx3)

√V (X⊥, x3)

∫R2

Pq,b(X⊥, X ′⊥)v1(X ′

⊥) dX ′⊥

+ sin(√λx3)

√V (X⊥, x3)

∫R2

Pq,b(X⊥, X ′⊥)v2(X ′

⊥) dX ′⊥, (X⊥, x3) ∈ R3,


where v = (v1, v2) ∈ L2(R2)2. Obviously,

Im Tq(2bq + λ) =1

2√λK∗K, Ωq(λ) =

12√λKK∗.

Since n+(s;K∗K) = n+(s;KK∗) for each s > 0, we get (5.17).

Now the combination of (2.13), (5.1), (5.16), and (5.18) yields (3.11).

6 Proof of Corollary 2.2

Introduce the matrix-valued functions

W(1)(X⊥) :=(W 00 0

),

W(2)(X⊥) = W(2)(X⊥;λ) := Wλ(X⊥) −W(1)(X⊥) =(

−w22 w12

w21 w22

)

(see (3.4) and (3.9) for the definitions of W and Wλ, respectively), as well as theoperators

Ω(j)q (λ) :

12√λpqW(j)pq, λ > 0, q ∈ Z+, j = 1, 2,

compact in L2(R2)2. Evidently, Ω(j)q (λ) ∈ S1, j = 1, 2.

Proposition 6.1.

(i) Let (1.5) hold with m ∈ (3, 4]. Then for each q ∈ Z+, s > 0, and δ > 4−m2 ,

we have

Tr(

arctan (s−1Ωq(λ)) − arctan (s−1Ω(1)q (λ))

)= O(λ−δ), λ ↓ 0, (6.1)

(see (3.10) for the definition of the operator Ωq(λ)).(ii) Let (1.4) hold with m⊥ > 2, m3 > 2. Then for each q ∈ Z+ and s > 0 we

have

Tr(


)= O(1), λ ↓ 0. (6.2)

Proof. Applying the Lifshits-Kreın trace formula (1.1), we easily get

Tr(


)

=∫R

ξ(E; s−1Ωq(λ); s−1Ω(1)q (λ))(1 +E2)−1dE, s > 0. (6.3)


Therefore, ∣∣∣Tr(


)∣∣∣≤∫R

|ξ(E; s−1Ωq(λ); s−1Ω(1)q (λ))|dE ≤ 1

s‖Ωq(λ) − Ω(1)

q (λ)‖1 =1s‖Ω(2)

q (λ)‖1

(6.4)(see [17, Theorem 8.2.1]). Further,

‖Ω(2)q (λ)‖1 ≤ b

2π√λ

∫R2

√w22(X⊥)2 + w12(X⊥)2dX⊥

≤ b

2π√λ

∑j=1,2

∫R2

|wj2(X⊥)|dX⊥ ≤ b

π√λ

∫R2

∫R

V (X⊥, x3)| sin(√λx3)|dx3 dX⊥.

(6.5)Assume now that V satisfies (1.5) with m ∈ (3, 4]. Pick δ > 4−m

2 , and m′ ∈(−2δ + 2,m− 2). Then we have

λ−1/2

∫R2

∫R

V (X⊥, x3)| sin (√λx3)|dx3 dX⊥

≤ λ−δC0

∫R2〈X⊥〉−(m−m′)dX⊥

∫R

〈x3〉−m′ |x3|−2δ+1dx3. (6.6)

Since m −m′ > 2 the integral with respect to X⊥ ∈ R2 is convergent, and sincem′ + 2δ − 1 > 1 the integral with respect to x3 is convergent as well. Now thecombination of (6.3)–(6.6) entails (6.1).Further, suppose that V satisfies (1.4) with m⊥ > 2 and m3 > 2. Then

λ−1/2

∫R2

∫R

V (X⊥, x3)| sin (√λx3)|dx3 dX⊥

≤ C0

∫R2〈X⊥〉−m⊥dX⊥

∫R

〈x3〉−m3 |x3|dx3. (6.7)

Putting together (6.3)–(6.5), and (6.7), we get (6.2).

Now note that if V satisfies (1.5) with m ∈ (3, 4], we can choose 4−m2 < δ <

1m−1 so that in this case Proposition 6.1 entails

Tr(


)= o(λ−1/(m−1)), λ ↓ 0. (6.8)

Moreover, if V satisfies (1.5) with m > 4, then it satisfies (1.4) with m⊥ > 2 andm3 > 2, and, hence, (6.2) is valid. Finally,

Tr arctan (s−1Ω(1)q (λ)) = Tr arctan (s−1ωq(λ))

=∫ ∞

0

n+(st;ωq(λ))1 + t2

dt, s > 0, λ > 0, (6.9)


(see (3.6) for the definition of the operator ωq(λ)). Putting together (6.9), (6.8),and (6.2), we conclude that Corollary 3.2 follows easily from Theorem 3.2 andLemmas 3.2–3.4.

Acknowledgments. The authors are grateful to the anonymous referee whose valu-able remarks contributed to the improvement of the text.Georgi Raikov was partially supported by the Chilean Science Foundation Fonde-cyt under Grant 1020737.

References

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Claudio FernandezDepartamento de MatematicasFacultad de MatematicasPontificia Universidad Catolica de ChileAv. Vicuna Mackenna 4860SantiagoChileemail: [email protected]

Georgi D. RaikovDepartamento de MatematicasFacultad de CienciasUniversidad de ChileLas Palmeras 3425SantiagoChileemail: [email protected]

Communicated by Bernard Helffersubmitted 23/09/03, accepted 15/01/04

On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels

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