A Mathematical Journal
Vol. 7, No 2, (171 - 199). August 2005.

Spectral Shift Function for Schrödinger
Operators in Constant Magnetic Fields

Georgi Raikov 1

Depto Matemática. Facultad de Ciencias. Universidad de Chile
Las Palmeras 3425, Santiago, Chile

graykov@uchile.cl

ABSTRACT
We consider the three-dimensional Schrödinger operator with constant

magnetic field, perturbed by an appropriate short-range electric potential, and
investigate various asymptotic properties of the corresponding spectral shift func-
tion (SSF). First, we analyse the singularities of the SSF at the Landau levels.
Further, we study the strong magnetic field asymptotic behaviour of the SSF;
here we distinguish between the asymptotics far from the Landau levels, and
near a given Landau level. Finally, we obtain a Weyl-type formula describing the
high energy behaviour of the SSF.
This is a survey article on recent published results obtained by the author jointly
with Vincent Bruneau, Claudio Fernández, and Alexander Pushnitski. A shorter
version will appear in the Proceedings of the Conference QMath9, Giens, France,
September 2004.

RESUMEN
Se considera el operador de Schrödinger en tres dimensiones con campo

magnético constante, perturbado por un potencial eléctrico de corto alcance
apropiado, e investigamos una variedad de propiedades asintóticas de la función
de corrimiento espectral (SSF). Analizamos primero las singularidades de la SSF

1I would like to thank my co-authors Vincent Bruneau, Claudio Fernández, and Alexander Push-
nitski for having let me present our joint results in this survey article. The major part of this work
has been done during my visit to the Institute of Mathematics of the Czech Academy of Sciences in
December 2004. I am sincerely grateful to Miroslav Englǐs for his warm hospitality.
The partial support by the Chilean Science Foundation Fondecyt under Grant 1050716 is acknowl-
edged.



172 Georgi Raikov
7, 2(2005)

en los niveles de Landau. Además, estudiamos el comportamiento asintótico
del campo magnético fuerte de la SSF; distinguimos aqúı entre propiedades
asintóticas lejos de los niveles de Landau y cerca de uno determinado. Final-
mente, obtenemos una fórmula de tipo Weyl para describir el comportamiento de
altas enerǵıas de la SSF.
Este es un art́ıculo prospectivo en torno a varios resultados obtenidos por el
autor en conjunto con Vincent Bruneau, Claudio Fernández, y Alexander Push-
nitski. Una versión más compacta aparecerá en los Proceedings of the Conference
QMath9, Giens, France, September 2004.

Key words and phrases: spectral shift function, scattering phase, Schrödinger
operators, constant magnetic fields

Math. Subj. Class.: 81Q10, 35J10, 47N50, 35P05

1 Introduction

In this survey article based on the papers [7], [10], and [8], we consider the 3D
Schrödinger operator with constant magnetic field of scalar intensity b > 0, per-
turbed by an electric potential V which decays fast enough at infinity, and discuss
various asymptotic properties of the corresponding spectral shift function.
More precisely, let H0 = H0(b) := (i∇ + A)2 − b be the unperturbed operator, es-
sentially self-adjoint on C∞0 (R

3). Here A =
(
−bx2

2
, bx1

2
, 0
)

is the magnetic potential
which 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) stands for the spectrum
of H0, and σac(H0) for its absolutely continuous spectrum. Moreover, the so-called
Landau levels 2bq, q ∈ Z+ := {0, 1, . . .}, play the role of thresholds in σ(H0).
For x = (x1,x2,x3) ∈ R3 we denote by X⊥ = (x1,x2) the variables on the plane
perpendicular to the magnetic field. Throughout the paper assume that V satisfies

V �≡ 0, V ∈ C(R3), |V (x)| ≤ C0〈X⊥〉−m⊥〈x3〉−m3, x = (X⊥,x3) ∈ R3, (1.1)

with C0 > 0, m⊥ > 2, m3 > 1, and 〈x〉 := (1 + |x|2)1/2, x ∈ Rd, d ≥ 1. Some of our
results hold under a more restrictive assumption than (1.1), namely

V �≡ 0, V ∈ C(R3), |V (x)| ≤ C0〈x〉−m0, m0 > 3, x ∈ R3. (1.2)

Note that (1.2) implies (1.1) with any m3 ∈ (0,m0) and m⊥ = m0−m3. In particular,
we can choose m3 ∈ (1,m0 − 2) so that m⊥ > 2.
On the domain of H0 define the operator H = H(b) := H0 +V . Obviously, inf σ(H) ≤
inf σ(H0) = 0. Moreover, if (1.1) holds, then for every E < inf σ(H) we have (H −
E)−1 − (H0 − E)−1 ∈ S1 where S1 denotes the trace class. Hence, there exists a
unique function ξ = ξ(·; H,H0) ∈ L1(R; (1 + E2)−1dE) which vanishes identically on



7, 2(2005)
Spectral Shift Function for Magnetic Schrödinger Operators 173

(−∞, inf σ(H)) such that the Lifshits-Krein trace formula

Tr (f(H) −f(H0)) =
∫

R

ξ(E; H,H0)f
′(E)dE

holds for each f ∈ C∞0 (R) (see the original works [22], [20], the survey article [5],
or Chapter 8 of the monograph [45]). The function ξ(·; H,H0) is called the spectral
shift function (SSF) for the operator pair (H,H0). If E < 0 = inf σ(H0), then the
spectrum of H below E could be at most discrete, and for almost every E < 0 we
have

ξ(E; H,H0) = −N(E; H) (1.3)
where N(E; H) denotes the number of eigenvalues of H lying in the interval (−∞,E),
and counted with their multiplicities. On the other hand, for almost every E ∈ [0,∞),
the SSF ξ(E; H,H0) is related to the scattering determinant det S(E; H,H0) for the
pair (H,H0) by the Birman-Krein formula

det S(E; H,H0) = e
−2πiξ(E;H,H0)

(see [4] or [45, Section 8.4]). A survey of various asymptotic results concerning the
SSF for numerous quantum Hamiltonians is contained in [40].
A priori, the SSF ξ(E; H,H0) is defined for almost every E ∈ R. In this article
we will identify this SSF with a representative of its equivalence class which is well-
defined on R \ 2bZ+, bounded on every compact subset of R \ 2bZ+, and continuous
on R \ (2bZ+ ∪ σpp(H)) where σpp(H) denotes the set of the eigenvalues of H. In
the case of perturbations V of definite sign this representative is described explicitly
in Subsection 3.1 below; in the case of general non-sign-definite perturbations its
description can be found in [7, Section 3].
In the present article we investigate the behaviour of the SSF in several asymptotic
regimes:

• First, we analyse the singularities of the SSF at the Landau levels. In other
words, we fix q ∈ Z+, and investigate the behaviour of ξ(2bq + λ; H,H0) as
λ → 0.
• Further, we study the strong-magnetic-field asymptotics of the SSF, i.e. the

behaviour of the SSF as b → ∞. Here we distinguish between the asymptotics
far from the Landau levels, and the asymptotics near a given Landau level.

• Finally, we obtain a Weyl-type formula describing the high-energy asymptotics
of the SSF.

The paper is organised as follows. In Section 2 we formulate our main results, and
discuss briefly on them. More precisely, in Subsection 2.1 we introduce some ba-
sic notations used throughout the paper, Subsection 2.2 contains the results on the
singularities of the SSF at the Landau levels, Subsection 2.3 is devoted to the strong-
magnetic-field asymptotics of the SSF, and Subsection 2.4 to its high-energy behav-
iour. Section 3 contains some auxiliary results. In Subsection 3.1 we describe the



174 Georgi Raikov
7, 2(2005)

representation of the SSF in the case of perturbations of fixed sign, due to A. Push-
nitski (see [29]), while in Subsection 3.2 we establish estimates of some auxiliary
operators of Birman-Schwinger type which are used systematically in the proofs of
the main results. Some of these proofs could be found in Section 4: in Subsection 4.1
we prove the results of Subsection 2.2, and in Subsection 4.2 some of the results of
Subsection 2.3. Since the detailed proofs have already been published in [10] and [7],
the proofs presented here are somewhat sketchy, preference being given to the main
ideas rather than to the technical details.

2 Main Results

2.1 Notations and preliminaries

In this subsection we introduce our basic notations used throughout the paper.
We denote by S∞ the class of linear compact operators acting in a given Hilbert space.
Let T = T ∗ ∈ S∞. Denote by PI (T ) the spectral projection of T associated with the
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+(s
2; 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 ∈ (0,∞), the Schatten-von Neumann class of compact
operators for which the functional ‖T‖p : =

(
p
∫ ∞
0
sp−1n∗(s; T ) ds

)1/p
is finite. If

T ∈ Sp, p ∈ (0,∞), then the following elementary inequality of Chebyshev type
n∗(s; T ) ≤ s−p‖T‖pp (2.4)

holds for every s > 0. If T = T ∗ ∈ Sp, p ∈ (0,∞), then (2.2) and (2.4) imply
n±(s; T ) ≤ s−p‖T‖pp, s > 0. (2.5)

2.2 Singularities of the SSF at the Landau levels

Introduce the Landau Hamiltonian

h(b) :=
(
i
∂

∂x1
− bx2

2

)2
+
(
i
∂

∂x2
+
bx1
2

)2
− b, (2.6)



7, 2(2005)
Spectral Shift Function for Magnetic Schrödinger Operators 175

i.e. the 2D Schrödinger operator with constant scalar magnetic field b > 0, essentially
self-adjoint on C∞0 (R

2). It is well-known that σ(h(b)) = ∪∞q=0 {2bq}, and each eigen-
value 2bq, q ∈ Z+, has infinite multiplicity (see e.g. [1]). Denote by pq = pq(b) the
orthogonal projection onto the eigenspace Ker (h(b) − 2bq), q ∈ Z+.
The estimates of the SSF for energies near the Landau level 2bq, q ∈ Z+, will be given
in the terms of traces of certain functions of Toeplitz-type operators pqUpq where
U : R2 → R decays in a certain sense at infinity.
Lemma 2.1 [31, Lemma 5.1], [10, Lemma 2.1] Let U ∈ Lr(R2), r ≥ 1, and q ∈ Z+.
Then pqUpq ∈ Sr.
Assume that (1.1) holds. Set

W(X⊥) :=
∫

R

|V (X⊥,x3)|dx3, X⊥ ∈ R2.

Since V satisfies (1.1), we have W ∈ L1(R2), and Lemma 2.1 with U = W implies
pqWpq ∈ S1, q ∈ Z+. Evidently, pqWpq ≥ 0, and it follows from V �≡ 0 and
V ∈ C(R2), that rankpqWpq = ∞ for all q ∈ Z+ (see below Lemma 2.4). If,
moreover, V satisfies (1.2), then 0 ≤ W(X⊥) ≤ C′0〈X⊥〉−m0+1, X⊥ ∈ R2, with
C′0 = C0

∫
R
〈x〉−m0dx.

In the following two theorems we assume that V has a definite sign, i.e. that either
V ≤ 0 (then we will write H− instead of H), or V ≥ 0 (then we will write H+ instead
of H).

Theorem 2.1 (cf. [10, Theorem 3.1]) Assume that (1.2) is valid, and ±V ≥ 0. Let
q ∈ Z+, b > 0. Then the asymptotic estimates

ξ(2bq −λ; H+,H0) = O(1), (2.7)
−n+((1−ε)2

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

√
λ; pqWpq)+O(1),

(2.8)
hold as λ ↓ 0 for each ε ∈ (0, 1).
Suppose that the potential V satisfies (1.1). For λ ≥ 0 define the matrix-valued
function

Wλ = Wλ(X⊥) :=
(
w11 w12
w21 w22

)
, X⊥ ∈ R2, (2.9)

where
w11 :=

∫
R

|V (X⊥,x3)|cos2 (
√
λx3)dx3,

w12 = w21 :=
∫

R

|V (X⊥,x3)|cos (
√
λx3) sin (

√
λx3)dx3,

w22 :=
∫

R

|V (X⊥,x3)|sin2 (
√
λx3)dx3.

It is easy to check that for λ ≥ 0 and q ∈ Z+ the operator pqWλpq : L2(R2)2 →
L2(R2)2 satisfies 0 ≤ pqWλpq ∈ S1, and rank pqWλpq = ∞.



176 Georgi Raikov
7, 2(2005)

Theorem 2.2 (cf. [10, Theorem 3.2]) Assume that (1.2) is valid, and ±V ≥ 0. Let
q ∈ Z+, b > 0. Then the asymptotic estimates

±1
π

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

±1
π

Tr arctan (((1 ∓ε)2
√
λ)−1pqWλpq) + O(1) (2.10)

hold as λ ↓ 0 for each ε ∈ (0, 1).
Relations (2.8) and (2.10) allow us to reduce the analysis of the behaviour as λ → 0
of ξ(2bq + λ; H±,H0), to the study of the asymptotic distribution of the eigenvalues
of Toeplitz-type operators pqUpq. The following three lemmas concern the spectral
asymptotics of such operators.

Lemma 2.2 [31, Theorem 2.6] Let the function 0 ≤ U ∈ C1(R2) satisfy the estimates
U(X⊥) = u0(X⊥/|X⊥|)|X⊥|−α(1 + o(1)), |X⊥|→∞,

|∇U(X⊥)| ≤ C1〈X⊥〉−α−1, X⊥ ∈ R2,
where α > 0, and u0 is a continuous function on S1 which is non-negative and does
not vanish identically. Then for each q ∈ Z+ we have

n+(s; pqUpq) =
b

2π

∣∣{X⊥ ∈ R2|U(X⊥) > s}∣∣ (1 + o(1)) = ψα(s) (1 + o(1)), s ↓ 0,
where |.| denotes the Lebesgue measure, and

ψα(s) := s
−2/α b

4π

∫
S1
u0(t)

2/αdt, s > 0. (2.11)

Lemma 2.3 [38, Theorem 2.1, Proposition 4.1] Let 0 ≤ U ∈ L∞(R2). Assume that
ln U(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) :=

⎧⎪⎨
⎪⎩

b
2µ1/β

| ln s|1/β if 0 < β < 1,
1

ln (1+2µ/b)
| ln s| if β = 1,

β
β−1 (ln | ln s|)−1| ln s| if 1 < β < ∞.

s ∈ (0,e−1). (2.12)

Lemma 2.4 [38, Theorem 2.2, Proposition 4.1] Let 0 ≤ U ∈ L∞(R2). Assume that
the support of U is compact, and that there exists a constant C > 0 such that U ≥ 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,
where

ϕ∞(s) := (ln | ln s|)−1| ln s|, s ∈ (0,e−1). (2.13)



7, 2(2005)
Spectral Shift Function for Magnetic Schrödinger Operators 177

Employing Lemmas 2.2, 2.3, 2.4, we easily find that asymptotic estimates (2.8) and
(2.10) entail the following

Corollary 2.1 [10, Corollaries 3.1 – 3.2] Let (1.2) hold with m0 > 3.
i) Assume that the hypotheses of Lemma 2.2 hold with U = W and α > 2. Then we
have

ξ(2bq −λ; H−,H0) = −
b

2π

∣∣∣
{
X⊥ ∈ R2|W(X⊥) > 2

√
λ
}∣∣∣ (1 + o(1)) =

−ψα(2
√
λ) (1 + o(1)), λ ↓ 0, (2.14)

ξ(2bq + λ; H±,H0) = ±
b

2π2

∫
R2

arctan ((2
√
λ)−1W(X⊥))dX⊥ (1 + o(1)) =

± 1
2 cos (π/α)

ψα(2
√
λ) (1 + o(1)), λ ↓ 0,

the function ψα being defined in (2.11).
ii) Assume that the hypotheses of Lemma 2.3 hold with U = W. Then we have

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

the functions ϕβ being defined in (2.12). If, in addition, V satisfies (1.1) for some
m⊥ > 2 and m3 > 2, we have

ξ(2bq + λ; H±,H0) = ±
1
2
ϕβ (2

√
λ) (1 + o(1)), λ ↓ 0, β ∈ (0,∞).

iii) Assume that the hypotheses of Lemma 2.4 hold with U = W. Then we have

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

the function ϕ∞ being defined in (2.13). If, in addition, V satisfies (1.1) for some
m⊥ > 2 and m3 > 2, we have

ξ(2bq + λ; H±,H0) = ±
1
2
ϕ∞(2

√
λ) (1 + o(1)), λ ↓ 0,

the function ϕ∞ being defined in (2.13).

In particular, we find that

lim
λ↓0

ξ(2bq −λ; H−,H0)
ξ(2bq + λ; H−,H0)

=
1

2 cos π
α

(2.15)

if W has a power-like decay at infinity (i.e. if the assumptions of Corollary 2.1 i)
hold), or

lim
λ↓0

ξ(2bq −λ; H−,H0)
ξ(2bq + λ; H−,H0)

=
1
2

(2.16)



178 Georgi Raikov
7, 2(2005)

if W decays exponentially or has a compact support (i.e. if the assumptions of Corol-
lary 2.1 ii) - iii) are fulfilled). Relations (2.15) and (2.16) could be interpreted as
analogues of the classical Levinson formulae (see e.g. [40]).
Remarks: i) Since the ranks of pqWpq and pqWλpq are infinite, the quantities
n+(s2

√
λ; pqWpq) and Tr arctan ((s2

√
λ)−1pqWλpq) tend to infinity as λ ↓ 0 for every

s > 0. Therefore, Theorems 2.1 and 2.2 imply that the SSF ξ(·; H±,H0) has a sin-
gularity at each Landau level. The existence of singularities of the SSF at strictly
positive energies is in sharp contrast with the non-magnetic case b = 0 where the SSF
ξ(E;−∆ + V,−∆) is continuous for E > 0 (see e.g. [40]). The main reason for this
phenomenon is the fact that the Landau levels play the role of thresholds in σ(H0)
while the free Laplacian −∆ has no strictly positive thresholds in its spectrum.
It is conjectured that the singularity of the SSF ξ(·; H±(b),H0(b)), b > 0, at a given
Landau level 2bq, q ∈ Z+, could be related to a possible accumulation of resonances
and/or eigenvalues of H at 2bq. Here it should be recalled that in the case b = 0
the high energy asymptotics (see [27]) and the semi-classical asymptotics (see [28]) of
the derivative of the SSF for appropriate compactly supported perturbations of the
Laplacian, are related by the Breit-Wigner formula to the asymptotic distribution
near the real axis of the resonances defined as poles of the meromorphic continuation
of the resolvent of the perturbed operator.
ii) In the case q = 0, when by (1.3) we have ξ(−λ; H−,H0) = −N(−λ; H−) for λ > 0,
asymptotic relations of the type of (2.14) have been known since long ago (see [43],
[42], [44], [31], [17]). An important characteristic feature of the methods used in [31],
and later in [38], is the systematic use, explicit or implicit, of the connection between
the spectral theory of the Schrödinger operator with constant magnetic field, and the
theory of Toeplitz operators acting in holomorphic spaces of Fock-Segal-Bargmann
type, and the related pseudodifferential operators with generalised anti-Wick sym-
bols (see [12], [3], [41], [15]). Various important aspects of the interaction between
these two theories have been discussed in [37] and [7, Section 9]). The Toeplitz-
operator approach turned out to be especially fruitful in [38] where electric potentials
decaying rapidly at infinity (i.e. decaying exponentially, or having compact support)
were considered (see Lemmas 2.3 - 2.4). It is shown in [11] that the precise spectral
asymptotics for the Landau Hamiltonian perturbed by a compactly supported electric
potential U of fixed sign recovers the logarithmic capacity of the support of U.
iii) Let us mention several other existing extensions of Lemmas 2.2 – 2.4. Lemmas 2.2
and 2.4 have been generalised to the multidimensional case where pq is the orthog-
onal projection onto a given eigenspace of the Schrödinger operator with constant
magnetic field of full rank, acting in L2(R2d), d > 1 (see [31] and [25] respectively).
Moreover, Lemma 2.4 has been generalised in [25] to a relativistic setting where pq
is an eigenprojection of the Dirac operator. Finally, in [36] Lemmas 2.2 – 2.4 have
been extended to the case of the 2D Pauli operator with variable magnetic field from
a certain class including the almost periodic fields with non-zero mean value (in this
case the role of the Landau levels is played by the origin), and electric potentials U
satisfying the assumptions of Lemmas 2.2 – 2.4. In the case of compactly supported
U of definite sign, [11] contains a more precise version of the corresponding result of



7, 2(2005)
Spectral Shift Function for Magnetic Schrödinger Operators 179

[36], involving again the logarithmic capacity of the support of U.
iv) To the author’s best knowledge, the singularities at the Landau levels of the SSF
for the 3D Schrödinger operator in constant magnetic field has been investigated for
the first time in [10]. However, it is appropriate to mention here the article [19] where
an axisymmetric potential V = V (|X⊥|,x3) has been considered. It is well-known
(see e.g. [1]) that in this case the operators H0 and H are unitarily equivalent to the
orthogonal sums

∑
m∈Z ⊕H(m) and

∑
m∈Z ⊕H

(m)
0 respectively, where the operators

H
(m)
0 := −

1
�

∂

∂�
�
∂

∂�
− ∂

2

∂x23
+
(
b�

2
+
m

�

)2
− b, H(m) := H(m)0 + V (�,x3), m ∈ Z,

(2.17)
are self-adjoint in L2(R+ ×R; �d�dx3). For a fixed magnetic quantum number m ∈ Z
the authors of [19] studied the behaviour of the SSF ξ(E; H(m),H(m)0 ) for energies
E near the Landau level 2m if m > 0, and near the origin if m ≤ 0, and deduced
analogues of the classical Levinson formulae for the operator pair

(
H(m),H

(m)
0

)
.

Later, the methods in [19] were developed in [23] and [24]. However, it is not possible
to recover the results of our Theorem 2.1, Theorem 2.2 and/or Corollary 2.1 from the
results of [19], [23], and [24] even in the case of axisymmetric V .
v) Finally, [16] contains general bounds on the SSF for appropriate pairs of magnetic
Schrödinger operators. These bounds are applied in order to deduce Wegner estimates
of the integrated density of states for some random alloy-type models.

2.3 Strong Magnetic Field Asymptotics of the SSF

Our first theorem in this subsection treats the asymptotics as b →∞ of ξ(·; H(b),H0(b))
far from the Landau levels.

Theorem 2.3 (cf. [7, Theorem 2.1]) Let (1.1) hold. Assume that E ∈ (0,∞) \ 2Z+,
and λ ∈ R. Then

ξ(Eb + λ; H(b),H0(b)) =
b1/2

4π2

[E/2]∑
l=0

(E − 2l)−1/2
∫

R3
V (x)dx + O(1), b →∞, (2.18)

where [E/2] denotes the integer part of the real number E/2.
The following two theorems concern the asymptotics of the SSF near a given Landau
level. In order to formulate our next theorem, we introduce the following self-adjoint
operators

χ0 := −d2/dx23, χ = χ(X⊥) := χ0 + V (X⊥, .), X⊥ ∈ R2,
which are defined on the Sobolev space H2(R), and depend on the parameter X⊥ ∈
R2. If (1.1) holds, then (χ(X⊥) − λ0)−1 − (χ0 − λ0)−1 ∈ S1 for each X⊥ ∈ R2
and λ0 < inf σ(χ(X⊥)). Hence, the SSF ξ(.; χ(X⊥),χ0) is well-defined. Set Λ: =
minX⊥∈R2 inf σ(χ(X⊥)). Evidently, Λ ∈ [−C0, 0]. Moreover,

Λ = lim
b→∞

inf σ(H(b)) (2.19)



180 Georgi Raikov
7, 2(2005)

(see [1, Theorem 5.8]).

Proposition 2.1 (cf. [7, Proposition 2.2]) Assume that (1.1) holds.
i) For each λ ∈ R \{0} we have ξ(λ; χ(.),χ0) ∈ L1(R2).
ii) The function (0,∞) � λ �→

∫
R2
ξ(λ; χ(X⊥),χ0)dX⊥ is continuous, while the non-

increasing function

(−∞, 0) � λ �→
∫

R2
ξ(λ; χ(X⊥),χ0)dX⊥ = −

∫
R2
N(λ; χ(X⊥))dX⊥

(see (1.3)), is continuous at the point λ < 0 if and only if

|{X⊥ ∈ R2|λ ∈ σ(χ(X⊥))}| = 0. (2.20)

iii) Assume ±V ≥ 0. If λ > Λ, λ �= 0, then ±
∫

R2
ξ(λ; χ(X⊥),χ0)dX⊥ > 0.

Remark: The third part of Proposition 2.1 was not included in [7, Proposition 2.2].
However, it follows easily from the representation of the SSF described in Subsection
3.1 below, and the hypotheses V �≡ 0 and V ∈ C(R3).
Theorem 2.4 (cf. [7, Theorem 2.3]) Assume that (1.1) holds. Let q ∈ Z+, λ ∈
R \{0}. If λ < 0, suppose also that (2.20) holds. Then we have

lim
b→∞

b−1ξ(2bq + λ; H(b),H0(b)) =
1
2π

∫
R2
ξ(λ; χ(X⊥),χ0) dX⊥. (2.21)

The proofs of Theorems 2.3 and 2.4 are contained in Subsection 4.2. We present
these proofs under the additional assumption that V has a definite sign, and refer the
reader to the original paper [7] for the proofs in the general case.
By Proposition 2.1 iii), if ±V ≥ 0, then the r.h.s. of (2.21) is different from zero if
λ > Λ, λ �= 0. Unfortunately, we cannot prove that the same is true for general non-
sign-definite electric potentials V . On the other hand, it is obvious that for arbitrary
V we have

∫
R2
ξ(λ; χ(X⊥),χ0)dX⊥ = 0 if λ < Λ. The last theorem of this subsection

contains a more precise version of (2.21) for the case λ < Λ.

Theorem 2.5 (cf. [7, Theorem 2.4]) Let (1.1) hold.
i) Let λ < Λ. Then for sufficiently large b > 0 we have ξ(λ; H(b),H0(b)) = 0.
ii) Let q ∈ Z+, q ≥ 1, λ < Λ. Assume that the partial derivatives of 〈x3〉m3V with
respect to the variables X⊥ ∈ R2 exist, and are uniformly bounded on R3. Then we
have

lim
b→∞

b−1/2ξ(2bq + λ; H(b),H0(b)) =
1

4π2

q−1∑
l=0

(2(q − l))−1/2
∫

R3
V (x)dx. (2.22)

The first part of the theorem is trivial, and follows immediately from (2.19). We omit
the proof of Theorem 2.5 ii) and refer the reader to the original work [7].
Remarks: i) Relations (2.18), (2.21), and (2.22) can be unified into a single asymptotic



7, 2(2005)
Spectral Shift Function for Magnetic Schrödinger Operators 181

formula. In order to see this, notice that a general result on the high-energy asymp-
totics of the SSF for 1D Schrödinger operators (see e.g. [40]) implies, in particular,
that

lim
E→∞

E1/2ξ(E; χ(X⊥),χ0) =
1
2π

∫
R

V (X⊥,x3) dx3, X⊥ ∈ R2.

Then relation (2.18) with 0 < E/2 �∈ Z+, or relations (2.21) and (2.22) with E = 2q,
q ∈ Z+, entail

ξ(Eb+λ; H(b),H0(b)) =
b

2π

[E/2]∑
l=0

∫
R2
ξ(b(E−2l)+λ; χ(X⊥),χ0)dX⊥ (1+o(1)), b →∞.

(2.23)
On its turn, (2.23) can be re-written as

ξ(Eb+λ; H(b),H0(b)) =
∫

R

∫
R2
ξ(Eb+λ−s; χ(X⊥),χ0)dX⊥dνb(s) (1 +o(1)), b →∞,

where νb(s) := b2π
∑∞

l=0 Θ(s − 2bl), s ∈ R, and Θ(s) :=
{

0 if s ≤ 0,
1 if s > 0,

is the

Heaviside function. It is well-known that ν is the integrated density of states for the
2D Landau Hamiltonian (see (2.6)).
ii) By (1.3) for λ < 0 we have ξ(λ; H(b),H0) = −N(λ; H(b)). The asymptotics as
b → ∞ of the counting function N(λ; H0(b)) with λ < 0 fixed, has been investigated
in [32] under considerably less restrictive assumptions on V than in Theorems 2.3 –
2.5. The asymptotic properties as λ ↑ 0, and as λ ↓ Λ if Λ < 0, of the asymptotic
coefficient − 1

2π

∫
R2
N(λ; χ(X⊥)dX⊥ which appears at the r.h.s. of (2.21) in the case

of a negative perturbation, have been studied in [33]. The asymptotic distribution
of the discrete spectrum for the 3D magnetic Pauli and Dirac operators in strong
magnetic fields has been considered in [35] and [34] respectively. The main purpose in
[32], [34], and [35] was to obtain the main asymptotic term (without any remainder
estimates) of the corresponding counting function of the discrete spectrum under
assumptions close to the minimal ones which guarantee that the Hamiltonians are
self-adjoint, and the asymptotic coefficient is well-defined. Other results which again
describe the asymptotic distribution of the discrete spectrum of the Schrödinger and
Dirac operator in strong magnetic fields, but contain also sharp remainder estimates,
have been obtained [17], [9], and [18] under assumptions on V which, naturally, are
considerably more restrictive than those in [32], [34], and [35].
iii) Generalisations of asymptotic relation (2.18) in several directions can be found
in [26]. In particular, [26, Theorem 4] implies that if V ∈ S(R3), then the SSF
ξ(Eb + λ; H(b),H0(b)), E ∈ (0,∞) \ 2Z+, λ ∈ R, admits an asymptotic expansion of
the form

ξ(Eb + λ; H(b),H0(b)) ∼
∞∑

j=0

cjb
1−2j

2 , b →∞.

iv) Together with the pointwise asymptotics as b → ∞ of the SSF for the pair
(H0(b),H(b)) (see (2.18), (2.21), or (2.22)), it also is possible to consider its weak



182 Georgi Raikov
7, 2(2005)

asymptotics, i.e. the asymptotics of the convolution of the SSF with an arbitrary
ϕ ∈ C∞0 (R). Results of this type are contained in [6].

2.4 High energy asymptotics of the SSF

Theorem 2.6 [8, Theorem 2.1] Assume that V satisfies (1.1). Then we have

lim
E→∞,E∈Or

E−1/2ξ(E; H,H0) =
1

4π2

∫
R3
V (x)dx, r ∈ (0,b), (2.24)

where Or := {E ∈ (0,∞)|dist(E, 2bZ+)}.
We omit the proof of Theorem 2.6 which is quite similar to that of Theorem 2.3, and
refer the reader to the original paper [8].
Remarks: i) It is essential to avoid the Landau levels in (2.24), i.e. to suppose that
E ∈Or, r ∈ (0,b), as E →∞, since by Theorems 2.1 - 2.2, the SSF has singularities
at the Landau levels, at least in the case ±V ≥ 0.
ii) For E ∈ R set

ξcl(E) :=
∫

T ∗R3

(
Θ(E −|p + A(x)|2) − Θ(E −|p + A(x)|2 −V (x))

)
dxdp =

4π
3

∫
R3

(
E

3/2
+ − (E −V (x))3/2+

)
dx

where Θ, as above, is the Heaviside function. Note that ξcl(E) is independent of
the magnetic field b ≥ 0. Evidently, under the assumptions of Theorem 2.6 we have
limE→∞ E−1/2ξcl(E) = 2π

∫
R3
V (x)dx. Hence, if

∫
R3
V (x)dx �= 0, then (2.24) is

equivalent to

ξ(E; H,H0) = (2π)
−3ξcl(E)(1 + o(1)), E →∞, E ∈Or, r ∈ (0,b).

iii) As far as the author is informed, the high-energy asymptotics of the SSF for 3D
Schrödinger operators in constant magnetic fields was investigated for the first time
in [8]. Nonetheless, in [19] the asymptotic behaviour as E →∞, E ∈Or, of the SSF
ξ(E; H(m),H(m)0 ) for the operator pair (H

(m),H
(m)
0 ) (see (2.17)) with fixed m ∈ Z has

been been investigated. It does not seem possible to deduce (2.24) from the results
of [19] even in the case of axial symmetry of V .

3 Auxiliary Results

3.1 A. Pushnitski’s representation of the SSF

In the first part of this subsection we summarise several results due to A. Pushnitski
on the representation of the SSF for a pair of lower-bounded self-adjoint operators
(see [29]).
Let I ∈ R be a Lebesgue measurable set. Set µ(I) := 1

π

∫
I

dt
1+t2

. Note that µ(R) = 1.



7, 2(2005)
Spectral Shift Function for Magnetic Schrödinger Operators 183

Lemma 3.1 [29, Lemma 2.1] Let T1 = T ∗1 ∈ S∞ and T2 = T ∗2 ∈ S1. Then∫
R

n±(s1 + s2; T1 + tT2) dµ(t) ≤ n±(s1; T1) +
1
πs2
‖T2‖1, s1,s2 > 0. (3.1)

Let H± and H0 be two lower-bounded self-adjoint operators. Assume that

V := ±(H± −H0) ≥ 0. (3.2)

Let λ0 < inf σ(H±) ∪σ(H0). Suppose that

(H0 −λ0)−γ − (H0 −λ0)−γ ∈ S2, γ > 0, (3.3)

V1/2(H0 −λ0)−1/2 ∈ S∞, (3.4)
V1/2(H0 −λ0)−γ

′ ∈ S2, γ′ > 0. (3.5)
For z ∈ C with Im z > 0 set T (z): = V1/2(H0 −z)−1V1/2.
Lemma 3.2 [29, Lemma 4.1] Let (3.3) – (3.5) hold. Then for almost every E ∈ R
the operator-norm limit T (E + i0) := n − limδ↓0 T (E + iδ) exists, and by (3.4) we
have T (E + i0) ∈ S∞. Moreover, 0 ≤ ImT (E + i0) ∈ S1.

Theorem 3.1 [29, Theorem 1.2] Let (3.2) – (3.5) hold. Then the SSF ξ(·;H±,H0)
for the operator pair (H±,H0) is well-defined, and for almost every E ∈ R we have

ξ(E;H±,H0) = ±
∫

R

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

Remark: The representation of the SSF described in the above theorem was gener-
alised to non-sign-definite perturbations in [14] in the case of trace-class perturbations,
and in [30] in the case of relatively trace-class perturbations. These generalisations
are based on the concept of index of orthogonal projections (see [2]).
Suppose now that V satisfies (1.1), and ±V ≥ 0. Then relations (3.2) – (3.5)
hold with V = |V |, H0 = H0, and γ = γ′ = 1. For z ∈ C, Im z > 0, set
T (z) := |V |1/2(H0−z)−1|V |1/2. By Lemma 3.2, for almost every E ∈ R the operator-
norm limit

T (E + i0) := n − lim
δ↓0

T (E + iδ) (3.6)

exists, and
0 ≤ Im T (E + i0) ∈ S1. (3.7)

The following proposition contains a more precise version of the above statement, and
provides estimates of the norm of T (E +i0), and the trace-class norm of Im T (E +i0).

Proposition 3.1 [7, Lemma 4.2] Assume that (1.1) holds, and E ∈ R \2bZ+. Then
the operator limit (3.6) exists, and we have

‖T (E + i0)‖≤ C1 (dist (E, 2bZ+))−1/2 (3.8)



184 Georgi Raikov
7, 2(2005)

with C1 independent of E and b.
Moreover, (3.7) holds, and if E < 0 then Im T (E+i0) = 0, while for E ∈ (0,∞)\2bZ+
we have

‖Im T (E + i0)‖1 = Tr Im T (E + i0) =
b

4π

[ E2b ]∑
l=0

(E − 2bl)−1/2
∫

R3
|V (x)|dx. (3.9)

By Lemma 3.1 and Proposition 3.1, the quantity

ξ̃(E; H±,H0) = ±
∫

R

n∓(1; Re T (E + i0) + t Im T (E + i0)) dµ(t), E ∈ R \ 2bZ+,
(3.10)

is well-defined for every E ∈ R \ 2bZ+, and bounded on every compact subset of
R \ 2bZ+. Moreover, by [7, Proposition 2.5], ξ̃(·; H±,H0) is continuous on R \
{2bZ+ ∪σpp(H±)}. On the other hand, by Theorem 3.1 we have

ξ̃(E; H±,H0) = ξ(E; H±,H0) (3.11)

for almost every E ∈ R. As explained in the introduction, in the case of sign-definite
perturbations we will identify the SSF ξ(E; H±,H0) with ξ̃(E; H±,H0), while in the
case of non-sign-definite perturbations, we will identify it with the generalisation of
ξ̃(E; H±,H0) described in [7, Section 3] on the basis of the general results of [14] and
[30].
Here it should be underlined that in contrast to the case b = 0, we cannot rule
out the possibility that the operator H has infinite discrete spectrum, or eigenvalues
embedded in the continuous spectrum by imposing conditions about the fast decay
of the potential V at infinity. First, it is well-known that if V satisfies

V (x) ≤−Cχ(x), x ∈ R3, (3.12)

where C > 0, and χ is the characteristic function of a non-empty open subset of R3,
then the discrete spectrum of H is infinite (see [1, Theorem 5.1], [38, Theorem 2.4]).
Further, if V is axisymmetric and satisfies (3.12), then the operator H(q) defined in
(2.17) with q ≥ 0 has at least one eigenvalue in the interval (2bq −‖V‖L∞(R3), 2bq),
and hence the operator H has infinitely many eigenvalues embedded in its continuous
spectrum (see [1, Theorem 5.1]). Assume now that V is axisymmetric and satisfies
the estimate

V (X⊥,x3) ≤−Cχ⊥(X⊥)〈x3〉−m3, (X⊥,x3) ∈ R3, (3.13)

where C > 0, χ⊥ is the characteristic function of a non-empty open subset of R2, and
m3 ∈ (0, 2) which is compatible with (1.1) if m3 ∈ (1, 2). Then, using the argument
of the proof of [1, Theorem 5.1] and the variational principle, we can easily check
that for each q ≥ 0 the operator H(q) has infinitely many discrete eigenvalues which
accumulate to the infimum 2bq of its essential spectrum. Hence, if V is axisymmetric
and satisfies (3.13), then below each Landau level 2bq, q ∈ Z+, there exists an infinite



7, 2(2005)
Spectral Shift Function for Magnetic Schrödinger Operators 185

sequence of finite-multiplicity eigenvalues of H, which converges to 2bq. Note however
that the claims in [10, p. 385] and [8, p. 3457] that [1, Theorem 5.1]) implies the
same phenomenon for axisymmetric non-positive potentials compactly supported in
R3, are not justified. The challenging and interesting problem about the accumula-
tion at a given Landau level of embedded eigenvalues and/or resonances of H will be
considered in a future work.
Finally, we note that generically the only possible accumulation points of the eigen-
values of H are the Landau levels (see [1, Theorem 4.7], [13, Theorem 3.5.3 (iii)]).
Further information on the location of the eigenvalues of H can be found in [7, Propo-
sition 2.6].

3.2 Estimates for Birman-Schwinger operators

For x, x′ ∈ R2 denote by Pq,b(x, x′) the integral kernel of the orthogonal projection
pq(b) onto the subspace Ker (h(b)−2bq), q ∈ Z+, the Landau Hamiltonian h(b) being
defined in (2.6). It is well-known that

Pq,b(x, x′) =
b

2π
Lq

(
b|x − x′|2

2

)
exp

(
−b

4
(|x − x′|2 + 2i(x1x′2 −x′1x2))

)
(3.14)

(see [21] or [37, Subsection 2.3.2]) where Lq(t) := 1q!e
t d

q(tq e−t)
dtq

=
∑q

k=0

(
q
k

)
(−t)k

k!
,

t ∈ R, q ∈ Z+, are the Laguerre polynomials. Note that

Pq,b(x, x) =
b

2π
, q ∈ Z+, x ∈ R2. (3.15)

Define the orthogonal projections Pq : L2(R3) → L2(R3), q ∈ Z+, by Pq := pq ⊗ I3
where I3 is the identity operator in L2(Rx3 ).

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

dx23
−z
)−1

bounded in

L2(R). Note that 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.

Define that the operators

Tq(z) := |V |1/2Pq(H0 −z)−1|V |1/2, q ∈ Z+,
bounded in L2(R3). We have 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) := lim
δ↓0
Rλ+iδ (x) =

⎧⎨
⎩

e−
√−λ|x|

2
√
−λ if λ < 0,

iei
√

λ|x|

2
√

λ
if λ > 0,

x ∈ R. (3.16)

Evidently, if w1,w2 ∈ L2(R) and λ �= 0, then w1R(λ)w2 ∈ S2. For E ∈ R, E �= 2bq,
q ∈ Z+, set

Tq(E) := |V |1/2
(
pq(b) ⊗R(E − 2bq)

)
|V |1/2.

Then limδ↓0 ‖Tq(E + iδ) −Tq(E)‖2 = 0 (see [10, Proposition 4.1]).



186 Georgi Raikov
7, 2(2005)

Proposition 3.2 Let E ∈ R, q ∈ Z+, E �= 2bq. Let (1.1) hold. Then
‖Tq(E)‖≤ C2|E − 2bq|−1/2, (3.17)
‖Tq(E)‖22 ≤ C2b|E − 2bq|−1, (3.18)

with C2 independent of E, b, and q.

Proof. We have
Tq(E) = MGq,m⊥ ⊗ t(E − 2bq)M (3.19)

where M is the multiplier by the bounded function |V (X⊥,x3)|1/2〈X⊥〉m⊥/2〈x3〉m3/2,
(X⊥,x3) ∈ R3, Gq,m⊥ : L2(R2) → L2(R2) is the operator with integral kernel

〈X⊥〉−m⊥/2Pb,q(X⊥,X′⊥)〈X′⊥〉−m⊥/2, X⊥,X′⊥ ∈ R2,
and t(λ) : L2(R) → L2(R), λ ∈ R \{0}, is the operator with integral kernel

〈x3〉−m3/2Rλ(x3 −x′3)〈x′3〉−m3/2, x3,x′3 ∈ R.
Then we have

‖Tq(E)‖≤‖M‖2∞‖Gq,m⊥‖‖t(E − 2bq)‖≤‖M‖2∞‖Gq,m⊥‖‖t(E − 2bq)‖2, (3.20)
‖Tq(E)‖2 ≤‖M‖2∞‖Gq,m⊥‖2 ‖t(E − 2bq)‖2, (3.21)

where ‖M‖∞ := ‖M‖L∞(R2). Evidently,
‖Gq,m⊥‖≤ 1, (3.22)

‖Gq,m⊥‖22 ≤ Tr pq〈X⊥〉−m⊥pq =
b

2π

∫
R2
〈X⊥〉−m⊥dX⊥ (3.23)

(see (3.15)), and

‖t(E − 2bq)‖22 ≤
1

4|E − 2bq|
∫

R

〈x3〉−m3dx3 (3.24)

(see (3.16)). Now the combination of (3.20), (3.22), and (3.24) yields (3.17), while
the combination of (3.21), (3.23), and (3.24) yields (3.18).
Remark: Using more sophisticated tools than those of the proof of Proposition 3.2, it
is shown in [7] that for E �= 2bq we have not only Tq(E) ∈ S2, but also Tq(E) ∈ S1.
We will not use this fact here.

Proposition 3.3 Assume that V satisfies (1.1). Let E ∈ R \ 2bZ+, q ∈ Z+. Then
we have 0 ≤ Im Tq(E) ∈ Sp with any p > 2/m⊥. If E < 2bq, then Im Tq(E) = 0. If
E > 2bq, then the estimate

n+(s; Im Tq(E)) ≤ C3
(
1 + b (E − 2bq)−1/m⊥s−2/m⊥

)
(3.25)

holds for each s > 0 with C3 independent of s, b, and E. Moreover, if E > 2bq, then
we have

‖Im Tq(E)‖1 = Tr Im Tq(E) =
b

4π
(E − 2bq)−1/2

∫
R3
|V (x)|dx. (3.26)



7, 2(2005)
Spectral Shift Function for Magnetic Schrödinger Operators 187

Proof. By (3.19), we have

Im Tq(E) = MGp,m⊥ ⊗ Im t(E − 2bq)M.

If E < 2bq, then Im t(E−2bq) = 0. If E > 2bq, then Im t(E−2bq) admits the integral
kernel

1
2
√
E − 2bq〈x3〉

−m3/2 cos (
√
E − 2bq (x3 −x′3))〈x′3〉−m3/2, x3,x′3 ∈ R.

Since the function 〈X⊥〉−m⊥/2 is radially symmetric, the eigenvalues νk, k ∈ N,
of the operator Gp,m⊥ ≥ 0 can be computed explicitly, and for k ≥ k0 we have
νk ≤ C′3bm⊥/2k−m⊥/2 with k0 ∈ N and C′3 independent of b and E (see the proof
of [7, Lemma 9.4]). Further, if E > 2bq, we have rank Im t(E − 2bq) = 2, and the
eigenvalues of Im t(E−2bq) are upper-bounded by 1

2
√

E−2bq
∫

R
〈x3〉−m3dx3. Therefore,

n+(s; Im Tq(E)) ≤ k0 + 2
(
C′3‖M‖2∞bm⊥/2s−1

2
√
E − 2bq

∫
R

〈x3〉−m3dx3
)2/m⊥

, s > 0,

which entails immediately (3.25). Finally, if we write the trace of the operator
Im Tq(E) as the integral of the diagonal value of its kernel, and take into account
(3.15) and (3.16), we get (3.26).

Proposition 3.4 [10, Proposition 4.2] Let q ∈ Z+, λ ∈ R, |λ| ∈ (0,b), and δ >
0. Assume that V satisfies (1.1). Then the operator series T +q (2bq + λ + iδ) :=∑∞

l=q+1 Tl(2bq + λ + iδ), and

T +q (2bq + λ) :=
∞∑

l=q+1

Tl(2bq + λ) (3.27)

converge in S2. Moreover,

‖T +q (2bq + λ)‖22 ≤
C0b

8π

∞∑
l=q+1

(2b(l−q) −λ)−3/2
∫

R3
V (x)dx. (3.28)

Finally, limδ↓0 ‖T +q (2bq + λ + iδ) −T +q (2bq + λ)‖2 = 0.

4 Proofs of the Main Results

4.1 Proofs of the results on the singularities of the SSF at the
Landau levels

The first step in the proofs of both Theorems 2.1 and 2.2 is to show that we can
replace the operator T (E + i0) by Tq(E) in the r.h.s of (3.10) when we deal with the
first asymptotic term of ξ̃(E; H±,H0) as the energy E approaches a given Landau



188 Georgi Raikov
7, 2(2005)

level 2bq, q ∈ Z+. More precisely, we pick q ∈ Z+, λ ∈ R with |λ| ∈ (0,b), and
set T −q (2bq + λ) :=

∑q−1
l=0 Tl(2bq + λ); if q = 0 the sum should be set equal to zero.

Evidently,

T (2bq + λ + i0) = T −q (2bq + λ) + Tq(2bq + λ) + T
+
q (2bq + λ),

Re T (2bq + λ + i0) = Re T −q (2bq + λ) + Re Tq(2bq + λ) + T
+
q (2bq + λ),

Im T (2bq + λ + i0) = Im T −q (2bq + λ) + Im Tq(2bq + λ),

the operator T +q (2bq + λ) being defined in (3.27). Combining the Weyl inequalities
(2.3), Lemma 3.1, (3.26), the Chebyshev-type estimates (2.5) with p = 2, (3.18), and
(3.28), we easily find that the asymptotic estimates

∫
R

n±(1 + ε; Re Tq(2bq + λ) + t Im Tq(2bq + λ)) dµ(t) + O(1) ≤
∫

R

n±(1; Re T (2bq + λ + i0) + t Im T (E + i0)) dµ(t) ≤
∫

R

n±(1 −ε; Re Tq(2bq + λ) + t Im Tq(2bq + λ)) dµ(t) + O(1) (4.1)

hold as λ → 0 for each ε ∈ (0, 1) (see [10, Proposition 5.1] for details).
If λ > 0, then Tq(2bq −λ) is a self-adjoint operator with integral kernel

1
2π

√
|V (X⊥,x3)| Pq,b(X⊥,X′⊥)

∫
R

eip(x3−x
′
3)

p2 + λ
dp
√
|V (X′⊥,x′3)| =

1
2
√
λ

√
|V (X⊥,x3)| Pq,b(X⊥,X′⊥)e−

√
λ|x3−x′3|

√
|V (X′⊥,x′3)|, (X⊥,x3), (X′⊥,x′3) ∈ R3.

In particular, Im Tq(2bq −λ) = 0, and Re Tq(2bq−λ) = Tq(2bq −λ) ≥ 0. Therefore,∫
R

n±(s; Re Tq(2bq−λ) + t Im Tq(2bq−λ)) dµ(t) = n±(s; Tq(2bq−λ)), s > 0, λ > 0.
(4.2)

Since Tq(2bq −λ) ≥ 0, we have n−(s; Tq(2bq −λ)) = 0 for all s > 0 and λ > 0, which
combined with (3.10), (4.1), and (4.2), implies (2.7). In order to prove (2.8), we write

Tq(2bq −λ) = Oq(λ) + T̃q(λ)
where Oq(λ) is an operator with integral kernel

1
2
√
λ

√
|V (X⊥,x3)| Pq,b(X⊥,X′⊥)

√
|V (X′⊥,x′3)|, (X⊥,x3), (X′⊥,x′3) ∈ R3,

and T̃q(λ) := Tq(2bq −λ) −Oq(λ). By (1.2) we have n − limλ↓0 T̃q(λ) = T̃q(0) where
T̃q(0) is a compact operator with integral kernel

−1
2

√
|V (X⊥,x3)| Pq,b(X⊥,X′⊥)|x3 −x′3|

√
|V (X′⊥,x′3)|, (X⊥,x3), (X′⊥,x′3) ∈ R3.



7, 2(2005)
Spectral Shift Function for Magnetic Schrödinger Operators 189

Hence, the Weyl inequalities easily imply that the asymptotic estimates

n+(s
′;Oq(λ)) + O(1) ≤ n+(s; Tq(2bq −λ)) ≤ n+(s′′;Oq(λ)) + O(1) (4.3)

hold for every 0 < s′ < s < s′′ as λ ↓ 0. Further, 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
√

λ
K∗K, pqWpq = K K∗. Therefore,

n+(s;Oq(λ)) = n+(s2
√
λ; pqWpq), s > 0, λ > 0. (4.4)

Now the combination of (3.10) with (4.1) – (4.4) entails (2.8). Thus, we are done
with the proof of Theorem 2.1.
In order to complete the proof of Theorem 2.2, we recall that if λ > 0, then the
operator Re Tq(2bq + λ) admits the integral kernel

− 1
2
√
λ

√
|V (X⊥,x3)|sin (

√
λ|x3 −x′3|)Pq,b(X⊥,X′⊥)

√
|V (X′⊥,x′3)|, (X⊥,x3),

(X′⊥,x
′
3) ∈ R3, and hence n − limλ↓0 Re Tq(2bq + λ) = T̃q(0). Applying the Weyl

inequalities and the evident identities
∫

R

n±(s; tT )dµ(t) =
1
π

Tr arctan (s−1T ), s > 0,

where T = T ∗ ≥ 0, T ∈ S1, we find that asymptotic estimates
1
π
Tr arctan (((1 + ε)s)−1Im Tq(2bq + λ)) + O(1)

≤
∫

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

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

(4.5)

are valid as λ ↓ 0 for each s > 0 and ε ∈ (0, 1). Define the operator K : 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.



190 Georgi Raikov
7, 2(2005)

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
√
λ
K∗K, pqWλpq = KK∗,

n+(s; Im Tq(2bq + λ)) = n+(s2
√
λ; pqWλpq), s > 0, λ > 0,

and, therefore,

Tr arctan (s−1Im Tq(2bq + λ)) = Tr arctan ((s2
√
λ)−1pqWλpq), s > 0, λ > 0. (4.6)

Now the combination of (3.10), (4.1), (4.5), and (4.6) yields (2.10).

4.2 Proofs of the results on the strong-magnetic-field asymp-
totics of the SSF

In this subsection we prove Theorems 2.3 and 2.4 under the additional assumption
that ±V ≥ 0. As before if V ≥ 0 (or if V ≤ 0), we will write H+ and χ+(X⊥),
X⊥ ∈ R2, (or H− and χ−(X⊥)) instead of H and χ(X⊥) respectively.
First, we prove Theorem 2.3. For brevity set

A = A(b) = Re T (Eb + λ), B = B(b) = Im T (Eb + λ).

Note that if E ∈ (0,∞) \ 2Z+, and λ ∈ R, then (3.8) and (3.9) imply

‖A(b)‖ = O(b−1/2), ‖B(b)‖ = O(b−1/2), ‖B(b)‖1 = O(b1/2), b →∞. (4.7)

Assume that b is so large that ‖A(b)‖ < 1. Then the operator I − A is boundedly
invertible, and limb→∞ ‖(I − A(b))−1‖ = 1. By the Birman-Schwinger principle we
have ∫

R

n±(1; A + tB)dµ(t) =
∫

R

n±(1; tB
1/2(I ∓A)−1B1/2)dµ(t) =

∫ ∞
0

n+(s; B
1/2(I ∓A)−1B1/2)dµ(s) = 1

π
Tr arctan

(
B1/2(I ∓A)−1B1/2

)
. (4.8)

Further,
Tr arctan

(
B1/2(I ±A)−1B1/2

)

≤ Tr
(
B1/2(I ±A)−1B1/2

)
= Tr B ∓ Tr

(
(I ±A)−1AB

)
, (4.9)



7, 2(2005)
Spectral Shift Function for Magnetic Schrödinger Operators 191

Tr arctan
(
B1/2(I ±A)−1B1/2

)
≥

Tr
(
B1/2(I ±A)−1B1/2

)
− 1

3
‖B1/2(I ±A)−1B1/2‖33 =

Tr B ∓ Tr
(
(I ±A)−1AB

)
− 1

3
‖B1/2(I ±A)−1B1/2‖33. (4.10)

By (4.7) we have

|Tr
(
(I ±A)−1AB

)
| ≤ ‖(I ±A)−1A‖‖B‖1 = O(1), b →∞, (4.11)

‖B1/2(I ±A)−1B1/2‖33 ≤
≤‖B1/2(I ±A)−1B1/2‖2 ‖B1/2(I ±A)−1B1/2‖1 = O(b−1/2), b →∞.

(4.12)
Putting together (4.8) – (4.12), and bearing in mind (3.10), we get

ξ(Eb + λ; H±(b),H0(b)) = ±
1
π

Tr B(b) + O(1), b →∞. (4.13)

Recalling (3.9), we find that the asymptotic estimate

Tr B(b) =
b1/2

4π

[E/2]∑
l=0

(E − 2l)−1/2
∫

R3
|V (x)|dx + O(b−1/2) (4.14)

holds as b →∞. Now the combination of (4.13) and (4.14) yields (2.18).
Next, we pass to the proof of Theorem 2.4 under the additional assumption that
±V ≥ 0. To this end we establish some auxiliary results. Introduce the operator

τ(X⊥; z) := |V (X⊥, .)|1/2(χ0 −z)−1|V (X⊥, .)|1/2,
defined on L2(R), and depending on the parameters X⊥ ∈ R2 and z ∈ C with
Im z > 0. The operator τ(X⊥; z) admits the integral kernel

|V (X⊥,x3)|1/2Rz(x3 −x′3)|V (X⊥,x′3)|1/2, x3,x′3 ∈ R.
Evidently, τ(X⊥; z) ∈ S2. For X⊥ ∈ R2, λ ∈ R\{0}, define the operator τ(X⊥; λ+i0) :
L2(R) → L2(R) as the operator with integral kernel

|V (X⊥,x3)|1/2Rλ(x3 −x′3)|V (X⊥,x′3)|1/2, x3,x′3 ∈ R,
the function Rλ(x), x ∈ R, being defined in (3.16). Some explicit simple calculations
with the kernel of the operator τ(X⊥; λ + i0) yield the following

Proposition 4.1 Let X⊥ ∈ R2, λ ∈ R \{0}. Assume that (1.1) holds.
i) We have τ(X⊥; λ + i0) ∈ S2,

‖τ(X⊥; λ + i0)‖22 ≤
1

4|λ|
(∫

R

|V (X⊥,x3)|dx3
)2

, (4.15)



192 Georgi Raikov
7, 2(2005)

and τ(X⊥; λ+iδ) → τ(X⊥; λ+i0) in S2 as δ ↓ 0, uniformly with respect to X⊥ ∈ R2.
ii) We have Im τ(X⊥; λ + i0) ≥ 0, and Im τ(X⊥; λ + i0) = 0 if λ < 0. If λ > 0, then
rank Im τ(X⊥; λ + i0) = 2, and

n+(s; Im τ(X⊥; λ + i0)) ≤ 2Θ
(

1
2
√
λ

∫
R

|V (X⊥,x3)|dx3 −s
)
, s > 0. (4.16)

For X⊥ ∈ R2, λ ∈ R \{0}, s > 0, set

Ξ±λ,s(X⊥) :=
∫

R

n±(s; Re τ(X⊥; λ + i0) + t Im τ(X⊥; λ + i0) dµ(t). (4.17)

Corollary 4.1 Let (1.1) hold. Fix X⊥ ∈ R2, λ ∈ R \{0}, s > 0. Then we have
ξ(λ; χ±(X⊥),χ0) = ± Ξ∓λ,1(X⊥) (4.18)

where ξ(·; χ±(X⊥),χ0) is the representative of SSF for the operator pair (χ±(X⊥),χ0)
which is monotonous and left-continuous for λ < 0, and continuous for λ > 0.

Proof. It suffices to apply Theorem 3.1 with H± = χ±(X⊥) and H0 = χ0.
Corollary 4.2 Under the assumptions of Corollary 4.1 we have

Ξ±λ,s(·) ∈ L1(R2). (4.19)
Proof. Combine Lemma 3.1 for T1 = Re τ(X⊥; λ + i0) and T2 = Im τ(X⊥; λ + i0),
with Proposition 4.1.

Proposition 4.2 Let λ > 0. Assume that (1.1) holds. Then the function∫
R2

Ξ±λ,s(X⊥)dX⊥ is continuous with respect to s > 0.

Proof. Fix s > 0. First of all we will show that for almost every (X⊥, t) ∈ R2 × R
the functions

s′ �→ n±(s′; Re τ(X⊥; λ + i0) + t Im τ(X⊥; λ + i0))
are continuous at the point s′ = s. Evidently, this is equivalent to

±s �∈ σ(Re τ(X⊥; λ + i0) + t Im τ(X⊥; λ + i0)). (4.20)
In order to prove (4.20), we will use an argument quite close to the one of the proof of
[29, Lemma 4.1]. Note that the compact operator Re τ(X⊥; λ+i0)+t Im τ(X⊥; λ+i0)
depends linearly on t. By the Fredholm alternative the sets

Ω±(s,X⊥,λ) := {z ∈ C |±s ∈ σ(Re τ(X⊥; λ + i0) + z Im τ(X⊥; λ + i0))}
either coincide with C, or are discrete. However, i ∈ Ω±(s,X⊥,λ) is equivalent to
dim Ker (χ0 ∓ s−1|V (X⊥, .)| − λ) ≥ 1. On the other hand, it is well-known that
the operators χ0 ∓s−1|V (X⊥, .)| have no positive eigenvalues (see e.g. [39, Theorem



7, 2(2005)
Spectral Shift Function for Magnetic Schrödinger Operators 193

XIII.58]) since (1.1) implies lim|x3|→∞ |x3|V (X⊥,x3) = 0.
Therefore, dim Ker (χ0 ∓ s−1|V (X⊥, .)| − λ) = 0, i �∈ Ω±(s,X⊥,λ), and the sets
Ω±(s,λ,X⊥) are discrete. In particular, |R ∩ Ω±(s,X⊥,λ)| = 0. Put

Ω̃±(s,λ) := {(X⊥, t) ∈ R2 × R|±s ∈ σ(Re τ(X⊥; λ + i0) + t Im τ(X⊥; λ + i0))}.
The eigenvalues of the compact operator Re τ(X⊥; λ + i0) + t Im τ(X⊥; λ + i0) are
continuous, and hence measurable with respect to (X⊥, t) ∈ R2 × R. Therefore, the
sets Ω̃±(s,λ) are measurable, and by the Fubini-Tonelli theorem

|Ω̃±(s,λ)| =
∫

R2

∫
R

1Ω̃±(s,λ)(X⊥, t)dtdX⊥ =
∫

R2
|R ∩ Ω±(s,X⊥,λ)|dX⊥ = 0

where 1Ω̃±(s,λ) denotes the characteristic function of Ω
±(s,λ). On the other hand,

lim
s′→s

n±(s
′; Re τ(X⊥; λ + i0) + t Im τ(X⊥; λ + i0)) =

= n±(s; Re τ(X⊥; λ + i0) + t Im τ(X⊥; λ + i0)) (4.21)

if (X⊥, t) �∈ Ω̃±(s,λ). The Weyl inequalities (2.3) and estimates (4.15) – (4.16) imply
n±(s

′; Re τ(X⊥; λ + i0) + t Im τ(X⊥; λ + i0)) ≤
1
s′2λ

(∫
R

|V (X⊥,x3)|dx3
)2

+ 2Θ
( |t|√

λ

∫
R

|V (X⊥,x3)|dx3 −s′
)
. (4.22)

Note that the r.h.s. is in L1(R2 × R; dX⊥ dµ(t)) for each s′ > 0, and is a sum
of two monotonous functions of s′ > 0. Bearing in mind (4.21) – (4.22),
we apply the dominated convergence theorem, and get lims′→s

∫
R2

Ξ±λ,s′ (X⊥)dX⊥ =∫
R2

Ξ±λ,s(X⊥)dX⊥.

Set

Φ±λ,s(t) :=
∫

R2
n±(s; Re τ(X⊥; λ + i0) + t Im τ(X⊥; λ + i0)) dX⊥, t ∈ R.

Corollary 4.3 Assume that (1.1) holds. Let λ > 0, s > 0. Then lims′→s Φ
±
λ,s′ (t) =

Φ±λ,s(t) for almost every t ∈ R.
Proof.

Since the functions Φ±λ,s(t) are non-increasing with respect to s > 0, the
one-sided limits Φ±λ,s−0(t) ≥ Φ±λ,s+0(t) exist. Next, Proposition 4.2 implies∫

R2
Ξ±λ,s−0(X⊥)dX⊥ =

∫
R2

Ξ±λ,s+0(X⊥)dX⊥. By the Fubini theorem
∫

R2
Ξ±λ,s(X⊥)dX⊥

=
∫

R
Φ±λ,s(t)dµ(t). Hence,

∫
R

(
Φ±λ,s−0(t) − Φ±λ,s+0(t)

)
dµ(t) = 0. Since the functions

Φ±λ,s−0(t) − Φ±λ,s+0(t) are non-negative, we conclude that
∣∣∣
{
t ∈ R

∣∣∣Φ±λ,s−0(t) > Φ±λ,s+0(t)
}∣∣∣ = 0.

The following proposition contains key limiting relations used in the proof of The-
orem 2.4.



194 Georgi Raikov
7, 2(2005)

Proposition 4.3 (cf. [7, Proposition 7.1]) Let (1.1) hold. Then we have

lim
b→∞

b−1Tr (Re Tq(2bq + λ) + t Im Tq(2bq + λ))
p =

1
2π

∫
R2

Tr (Re τ(X⊥; λ + i0) + t Im τ(X⊥; λ + i0))
p
dX⊥ (4.23)

for every t ∈ R and each integer p ≥ 2.

Proof. Let t ∈ R, x ∈ R. If λ > 0, set R̃λ,t(x) := −sin(
√

λ|x|)
2
√

λ
+ t cos(

√
λ x)

2
√

λ
. If λ < 0,

then R̃λ,t(x) = Rλ(x) =
e−

√
−λ|x|

2
√
−λ

. We have

Tr (Re Tq(2bq + λ) + t Im Tq(2bq + λ))
p =

�
R2p

�
Rp

Π
p
j=1|V (X⊥,j , x3,j )|Π′

p
j=1Pq,b(X⊥,j , X⊥,j+1)R̃λ,t(x3,j − x3,j+1)Πpj=1dX⊥,j dx3,j

where the notation Π′pj=1 means that in the product of p factors the variables X⊥,p+1
and x3,p+1 should be set equal respectively to X⊥,1 and x3,1. Change the variables

X⊥,1 = X
′
⊥,1, X⊥,j = X

′
⊥,1 + b

−1/2X′⊥,j, j = 2, . . . ,p. (4.24)

Thus we obtain
Tr (Re Tq(2bq + λ) + t Im Tq(2bq + λ))

p =

b

∫
R2p

∫
Rp

|V (X′⊥,1,x3,1)|Πpj=2|V (X′⊥,1 + b−1/2X′⊥,j,x3,j )|Pq,1(0,X′⊥,2)×

Πp−1j=2Pq,1(X⊥′,j,X⊥′,j+1)Pq,1(X′⊥,p, 0) Π′
p
j=1R̃λ,t(x3,j −x3,j+1)Πpj=1dX′⊥,jdx3,j.

(4.25)
Here and in the sequel, if p = 2, then the product Πp−1j=2Pq,b(X⊥′,j,X⊥′,j+1) should
be set equal to one. Bearing in mind (1.1) and (3.14), and applying the dominated
convergence theorem, we easily find that (4.25) entails

lim
b→∞

b−1Tr (Re Tq(2bq + λ + i0) + t Im Tq(2bq + λ + i0))
p =

∫
R2

∫
Rp

Πpj=1|V (X⊥,1,x3,j )|Π′
p
j=1R̃λ,t(x3,j −x3,j+1)dX⊥,1Πpj=1dx3,j ×

∫
R2(p−1)

Pq,1(0,X⊥,2)Πp−1j=2Pq,1(X⊥,j,X⊥,j+1)Pq,1(X⊥,p, 0)Πpj=2dX⊥,j =
∫

R2
Tr
(
Re τ(X⊥,1; λ + i0) + t Im τ(X⊥,1; λ + i0)

)p
dX⊥,1 ×

∫
R2(p−1)

Pq,1(0,X⊥,2)Πp−1j=2Pq,1(X⊥,j,X⊥,j+1)Pq,1(X⊥,p, 0)Πpj=2dX⊥,j.



7, 2(2005)
Spectral Shift Function for Magnetic Schrödinger Operators 195

In order to conclude that the above limiting relation is equivalent to (4.23), it remains
to recall (3.15), and note that
∫

R2(p−1)
Pq,1(0,X⊥,2)Πp−1j=2Pq,1(X⊥,j,X⊥,j+1)Pq,1(X⊥,p, 0)Πpj=2dX⊥,j = Pq,1(0, 0) =

1
2π
.

Corollary 4.4 Assume that the assumptions of Theorem 2.4 hold. Then we have

lim
b→∞

b−1n±(s; Re Tq(2bq + λ) + t Im Tq(2bq + λ)) =

1
2π

∫
R2
n±(s; Re τ(X⊥; λ + i0) + t Im τ(X⊥; λ + i0))dX⊥,

for each t ∈ R, provided that s > 0 is a continuity point of the r.h.s.
Proof. It suffices to notice that the norm of the operator Tq(2bq + λ) is uniformly
bounded with respect to b, and to apply a suitable version of the Kac-Murdock-
Szegö theorem (see e.g. [32, Lemma 3.1]) which tells us that under appropriate
hypotheses the convergence of the moments of a given measure implies the convergence
of the measure itself, and to take into account Proposition 4.3.
Now we are in position to prove Theorem 2.4. By A. Pushnitski’s representation of
the SSF (see (3.10) and (4.18)), in order to check the validity of (2.21), it suffices to
show that

lim
b→∞

b−1
∫

R

n±(1; Re T (2bq + λ + i0) + tIm T (2bq + λ + i0))dµ(t) =

1
2π

∫
R

∫
R2
n±(1; Re τ(X⊥; λ + i0) + tIm τ(X⊥; λ + i0))dµ(t)dX⊥. (4.26)

Arguing as in the derivation of (4.1), we easily find that the asymptotic estimates
∫

R

n±(1 + ε; Re Tq(2bq + λ) + t Im Tq(2bq + λ)) dµ(t) + o(b) ≤
∫

R

n±(1; Re T (2bq + λ + i0) + t Im T (2bq + λ + i0)) dµ(t) ≤
∫

R

n±(1 −ε; Re Tq(2bq + λ) + tIm Tq(2bq + λ)) dµ(t) + o(b), (4.27)

hold as b → ∞ for each ε ∈ (0, 1). Assume λ > 0. Corollary 4.3 and Corollary 4.4
imply

lim
b→∞

b−1n±(s; Re Tq(2bq + λ) + t Im Tq(2bq + λ)) =

1
2π

∫
R2
n±(s; Re τ(X⊥; λ + i0) + tIm τ(X⊥; λ + i0)) dX⊥ (4.28)



196 Georgi Raikov
7, 2(2005)

for any fixed s > 0, and almost every t ∈ R. Further, by (2.3), (2.5) with p = 2,
Proposition 3.2, and Proposition 3.3 we have

b−1n±(s; Re Tq(2bq+λ+i0)+t Im Tq(2bq+λ+i0)) ≤ C4(1+|t|2/m⊥ ), t ∈ R, (4.29)

with C4 which may depend on s > 0, λ ∈ R \{0}, q and m⊥ but is independent of
b ≥ 1 and t. Note that the function on the r.h.s of (4.29) is in L1(R; dµ). By (4.28)
– (4.29), the dominated convergence theorem and the Fubini Theorem imply

lim
b→∞

b−1
∫

R

n±(s; Re Tq(2bq + λ + i0) + t Im Tq(2bq + λ + i0) dµ(t) =

1
2π

∫
R2

∫
R

n±(s; Re τ(X⊥; λ + i0) + t Im τ(X⊥; λ + i0)) dµ(t) dX⊥, s > 0. (4.30)

Putting together (4.27) and (4.30), we find that the following estimates
∫

R2
Ξ±λ,1+ε(X⊥)dX⊥ ≤ lim inf

b→∞
b−1

∫
R

n±(1; Re T (2bq+λ+i0)+tImT (2bq+λ+i0)dµ(t) ≤

lim sup
b→∞

b−1
∫

R

n±(1; Re T (2bq+λ+i0)+t Im T (2bq+λ+i0) dµ(t) ≤
∫

R2
Ξ±λ,1−ε(X⊥)dX⊥

are valid for each ε ∈ (0, 1). Letting ε ↓ 0, and taking into account Proposition
4.2, we obtain (4.26), and hence (2.21), in the case λ > 0. The modifications of the
argument for λ < 0 are quite obvious; in this case we essentially use assumption (2.20)
guaranteeing that λ is a continuity point of the r.h.s of (2.21).

Received: April 2005. Revised: May 2005.

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