CUBO A Mathematical Journal
Vol.11, No¯ 05, (23–38). December 2009

Resonances and SSF Singularities for Magnetic

Schrödinger Operators

Jean-François Bony, Vincent Bruneau

Université Bordeaux I, Institut de Mathématiques de Bordeaux,

UMR CNRS 5251, 351, Cours de la Libération, 33405 Talence, France

emails: bony@math.u-bordeaux1.fr,

vbruneau@math.u-bordeaux1.fr

Philippe Briet

Centre de Physique Théorique,

CNRS-Luminy, Case 907, 13288 Marseille, France

email : briet@cpt.univ-mrs.fr

and

Georgi Raikov

Departamento de Matemáticas, Facultad de Matemáticas,

Pontificia Universidad Católica de Chile,

Vicuña Mackenna 4860, Santiago de Chile

email : graikov@mat.puc.cl

ABSTRACT

The aim of this note is to review recent articles on the spectral properties of magnetic

Schrödinger operators. We consider H0, a 3D Schrödinger operator with constant magnetic

field, and H̃0, a perturbation of H0 by an electric potential which depends only on the variable

along the magnetic field. Let H (resp. H̃) be a short range perturbation of H0 (resp. of H̃0).

In the case of (H, H0), we study the local singularities of the Krein spectral shift function
(SSF) and the distribution of the resonances of H near the Landau levels which play the role

of spectral thresholds. In the case of (H̃, H̃0), we study similar problems near the eigenvalues
of H̃0 of infinite multiplicity.



24 J.-F. Bony, Ph. Briet, V. Bruneau and G. Raikov CUBO
11, 5 (2009)

RESUMEN

El objetivo de esta nota es reseñar artículos recientes sobre las propiedades espectrales de

operadores de Schrödinger magnéticos. Consideramos H0, el operador tridimensional de

Schrödinger con campo magnético constante, y H̃0, perturbación de H0 por un potencial

eléctrico que depende sólo de la variable a lo largo del campo magnético. Sea H, respecti-

vamente H̃, perturbación de corto alcance de H0, respectivamente H̃0. En el caso del par

(H, H0), estudiamos las singularidades locales de la función de corrimiento espectral (SSF) de

Krein y la distribución de las resonancias de H cerca de los niveles de Landau que tienen el

papel de umbrales espectrales. En el caso del par (H̃, H̃0), investigamos problemas similares

cerca de los valores propios de multiplicidad infinita de H̃0.

Key words and phrases: Magnetic Schrödinger operators, resonances, spectral shift function.

Math. Subj. Class.: 35P25, 35J10, 47F05, 81Q10.

1 Introduction

Let

H0 := (−i∇ − A)
2 − b,

be the (shifted) 3D Schrödinger operator with constant magnetic field B = (0, 0,b), b > 0, self-

adjoint in L2(R3), and essentially self-adjoint on C∞0 (R
3
). Here A =

(
− bx2

2
, bx1

2
, 0
)

is a magnetic

potential generating the magnetic field B = curl A. It is well-known that the spectrum of H0 is

absolutely continuous and coincides with the interval [0,∞), i.e.

σ(H0) = σac(H0) = [0,∞).

Moreover, the Landau levels 2bq, q ∈ Z+ := {0, 1, 2, . . .}, play the role of thresholds in σ(H0) (see

[10, 15, 3] and Section 3 below).

For x = (x1,x2,x3) ∈ R
3 we write x = (X⊥,x3) where X⊥ := (x1,x2) ∈ R

2 are the variables on

the plane perpendicular to the magnetic field B.

Further, let V : R3 → R be the electric potential. Throughout the article, we assume that

V ∈ L∞(R3) ∩ C(R3), and V does not vanish identically. Moreover, we will suppose that V

satisfies one of the following decay assumptions:

|V (x)| = O(〈X⊥〉
−m⊥〈x3〉

−m3 ), m⊥ > 2, m3 > 1, x = (X⊥,x3) ∈ R
3, (1.1)

|V (x)| = O(〈x〉−m0 ), m0 > 3, x ∈ R
3, (1.2)

|V (x)| = O(〈X⊥〉
−m⊥ exp (−N|x3|)), m⊥ > 2, ∀N > 0, x = (X⊥,x3) ∈ R

3. (1.3)

Note that each of the conditions (1.2) and (1.3) implies (1.1).

On the domain of H0 define the perturbed operator

H := H0 + V.



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Resonances and SSF Singularities ... 25

We introduce the Krein spectral shift function (SSF) ξ(·; H,H0) for the operator pair (H,H0) (see

Section 2 for its definition), and study its behavior near the Landau levels. In Theorem 4.1 we

show that at least in the case of V of constant sign, the SSF ξ(·; H,H0) has a singularity at each

Landau levels, i.e. it either blows up to +∞ if V ≥ 0, or blows down to −∞ if V ≤ 0, as the

energy approaches the Landau level in an appropriate manner. The singularities of ξ(·; H,H0)

are described in the terms of the spectral characteristics of compact operators of Berezin-Toeplitz

type, studied in Section 3.

One of the possible explanation of the singularities of the SSF ξ(·; H,H0) is the accumulation of

resonances of H to the Landau levels. In Section 5 we define the resonances of H as the poles

of an appropriate meromorphic extension of its resolvent, and in Theorem 5.1 we obtain upper

and lower estimates of the number of the resonances in a vicinity of a given Landau level. The

lower bound confirms our conjecture that the singularities of the SSF at the Landau levels and the

accumulation of the resonances of H to these levels, are intimately related.

Further, in Section 6 we introduce a modified operator pair (H̃,H̃0). Here H̃0 := H0 + v0 where

v0 = v0(x3) decays fast enough at infinity, while H̃ := H̃0 + V where V satisfies (1.1). The

remarkable property of the operator H̃0 is that, under appropriate assumption on v0, it has infinitely

many eigenvalues of infinite multiplicity, most of which are embedded in the continuous spectrum.

We study the asymptotic behavior of the SSF ξ(·; H̃,H̃0) near these eigenvalues of H̃0 of infinite

multiplicity. Finally, in Section 7, under the assumption that the perturbation V is axisymmetric,

we investigate the asymptotic behavior as κ → 0 of the unitary group associated with H̃0 + κV ,

and relate it to certain dynamic resonances for the operator H̃.

2 The Spectral Shift Function

Assume that V satisfies (1.1). Then the resolvent difference (H − i)−1 − (H0 − i)
−1 is a trace-

class operator, and there exists a unique ξ = ξ(·; H,H0) ∈ L
1
(R; (1 + E2)−1dE) 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), and ξ(E; H,H0) = 0 for each E ∈ (−∞, inf σ(H)) (see the original

works [17, 14] or [24, Chapter 8]). The function ξ(·; H,H0) is called the spectral shift function

(SSF) for the operator pair (H,H0).

For almost every E ∈ σac(H) = [0,∞) the SSF ξ(E; H,H0) coincides with the scattering phase for

the operator pair (H,H0) according to the Birman-Krein formula

det S(E; H,H0) = e
−2πiξ(E;H,H0)

where S(E; H,H0) is the scattering matrix for the operator pair (H,H0) (see the original work [4]

or [24, Chapter 8]). On the other hand, for almost every E < 0 the value −ξ(E; H,H0) coincides

with the number of the eigenvalues of H less than E, counted with their multiplicity.



26 J.-F. Bony, Ph. Briet, V. Bruneau and G. Raikov CUBO
11, 5 (2009)

A priori, the SSF ξ(·; H,H0) is defined as an element of L
1
(R; (1 + E2)−1dE). The following

proposition provides more precise information on ξ(·; H,H0):

Proposition 2.1. [7, Proposition 2.5] (i) The SSF ξ(·; H,H0) is bounded on every compact subset

of R \ 2bZ+.

(ii) The SSF ξ(·; H,H0) is continuous on R \ (2bZ+ ∪ σpp(H)) where σpp(H) is the set of the

eigenvalues of H.

3 Auxiliary Toeplitz Operators

We have

H0 = H0,⊥ ⊗ I‖ + I⊥ ⊗ H0,‖

where I⊥ and I‖ are the identities in L
2
(R

2
X⊥

) and L2(Rx3 ) respectively,

H0,⊥ :=

(
−i

∂

∂x1
+
bx2
2

)2
+

(
−i

∂

∂x2
−
bx1
2

)2
− b

is the (shifted) Landau Hamiltonian, self-adjoint in L2(R2X⊥ ), and

H0,‖ := −
d2

dx23

is the 1D free Hamiltonian, self-adjoint in L2(Rx3 ). Note that

H0,⊥ = a
∗a

where

a∗ := −2ieb|z|
2/4 ∂

∂z
e−b|z|

2/4, z = x1 + ix2,

is the creation operator, and

a := −2ie−b|z|
2/4 ∂

∂z̄
eb|z|

2/4, z̄ = x1 − ix2,

is the annihilation operator. Moreover,

[a,a∗] = 2b.

Therefore, σ(H0,⊥) = ∪
∞
q=0{2bq}. Furthermore,

Ker H0,⊥ = Ker a =

{
f ∈ L2(R2)|f = ge−b|z|

2/4,
∂g

∂z̄
= 0

}

is the classical Fock-Segal-Bargmann space (see e.g. [11]), and

Ker (H0,⊥ − 2bq) = (a
∗
)
q
Ker H0,⊥, q ≥ 1.



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Resonances and SSF Singularities ... 27

Evidently,

dim Ker (H0,⊥ − 2bq) = ∞

for each q ∈ Z+.

Let U : R2 → R. Fix q ∈ Z+. Introduce the Toeplitz operator

pqUpq : pqL
2
(R

2
) → pqL

2
(R

2
)

where pq is the orthogonal projection onto Ker (H0,⊥ − 2bq). Obviously, if U ∈ L
∞

(R
2
), then the

operator pqUpq is bounded. Denote by Sr the Schatten-von Neumann class of order r ∈ [1,∞).

Then U ∈ Lr(R2) implies pqUpq ∈ Sr, r ∈ [1,∞) (see [20, Lemma 5.1] or [9, Lemma 3.1]).

Moreover, if U is sufficiently regular, then p0Up0 is unitarily equivalent to a ΨDO with anti-Wick

symbol

ω(y,η) := U(b−1/2η,b−1/2y), (y,η) ∈ T ∗R,

(see [20]), while pqUpq with q ≥ 1 is unitarily equivalent to

p0

(
q∑

s=0

q!

(2b)s(s!)2(q − s)!
∆

sU

)
p0

(see [7, Lemma 9.2]).

For further references, in the following three theorems we describe the eigenvalue asymptotics for

the Toeplitz operator pqUpq, q ∈ Z+, under the assumptions that U admits a power-like decay,

exponential decay, or is compactly supported, respectively.

Theorem 3.1. [20, Theorem 2.6] Let 0 ≤ U ∈ C1(R2), and

U(X⊥) = u0(X⊥/|X⊥|)|X⊥|
−α

(1 + o(1)),

|∇U(X⊥)| = O(|X⊥|
−α−1

),

as |X⊥| → ∞, with α > 0, and 0 < u0 ∈ C(S
1
). Fix q ∈ Z+. Then

Tr 1(s,∞)(pqUpq) = ψα(s)(1 + o(1)), s ↓ 0, (3.1)

where

ψα(s) := s
−2/α b

4π

∫

S1

u0(t)
2/αdt. (3.2)

Theorem 3.2. [22, Theorem 2.1, Proposition 4.1] Let 0 ≤ U ∈ L∞(R2) and

ln U(X⊥) = −µ|X⊥|
2β

(1 + o(1)), |X⊥| → ∞,

with β ∈ (0,∞), µ ∈ (0,∞). Fix q ∈ Z+. Then

Tr 1(s,∞)(pqUpq) = ϕβ (s)(1 + o(1)), s ↓ 0, (3.3)



28 J.-F. Bony, Ph. Briet, V. Bruneau and G. Raikov CUBO
11, 5 (2009)

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). (3.4)

Theorem 3.3. [22, Theorem 2.2, Proposition 4.1] Let 0 ≤ U ∈ L∞(R2). Assume that supp U is

compact, and U ≥ C > 0 on an open subset of R2. Fix q ∈ Z+. Then

Tr 1(s,∞)(pqUpq) = ϕ∞(s)(1 + o(1)), s ↓ 0, (3.5)

where

ϕ∞(s) := (ln | ln s|)
−1| ln s|, s ∈ (0,e−1). (3.6)

Remark: Relations (3.1) and (3.3) with β < 1 are semiclassical in the sense that they are

equivalent to

Tr 1(s,∞)(pqUpq) =
b

2π

∣∣{X⊥ ∈ R2|U(X⊥) > s
}∣∣(1 + o(1)), s ↓ 0.

The asymptotic order in relation (3.3) with β = 1 which corresponds to Gaussian decay of V , is

semiclassical, but the coefficient is not. Finally, even the asymptotic order in (3.3) with β > 1,

and (3.5) is not semiclassical. Moreover the main terms of these asymptotics do not depend on

the Landau levels.

4 Singularities of the SSF at the Landau levels

Let V satisfy (1.1). For X⊥ ∈ R
2, λ ≥ 0, set

W(X⊥) :=

∫

R

|V (X⊥,x3)|dx3, (4.1)

Wλ = Wλ(X⊥) :=

(
w11 w12

w21 w22

)
,

where

w11 :=

∫

R

|V (X⊥,x3)| cos
2
(
√
λx3)dx3,

w12 = w21 :=

∫

R

|V (X⊥,x3)| cos (
√
λx3) sin (

√
λx3)dx3,

w22 :=

∫

R

|V (X⊥,x3)| sin
2
(
√
λx3)dx3.

We have

rank pqWpq = ∞, rank pqWλpq = ∞, λ ≥ 0.



CUBO
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Resonances and SSF Singularities ... 29

Theorem 4.1. [9, Theorems 3.1, 3.2] Let V satisfy (1.2), and V ≥ 0 or V ≤ 0. Fix q ∈ Z+. Then

we have

ξ(2bq − λ; H,H0) = O(1), λ ↓ 0,

if V ≥ 0, and for each ε ∈ (0, 1) we have

−Tr 1((1−ε)2
√

λ,∞) (pqWpq) + O(1) ≤

ξ(2bq − λ; H,H0) ≤

−Tr 1((1+ε)2
√

λ,∞) (pqWpq) + O(1), λ ↓ 0,

if V ≤ 0.

Moreover, for each ε ∈ (0, 1) we have

±
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), λ ↓ 0,

if ±V ≥ 0.

Remark: The proof of Theorem 4.1 is based on a representation of the SSF obtained by A.

Pushnitski in [19].

If we assume that W admits a power-like or exponential decay at infinity, or has a compact

support, we can combine the results of Theorems 3.1 – 3.3 for U = W , with Theorem 4.1 and

obtain explicitly the main asymptotic term of ξ(2bq + λ; H,H0) as λ ↓ 0 or λ ↑ 0:

Corollary 4.1. [9, Corollaries 3.1 – 3.2], [21, Corollary 2.1] Let (1.2) hold with m0 > 3.

(i) Assume that the hypotheses of Theorem 3.1 hold with U = W and α > 2. Then we have

ξ(2bq − λ; H,H0) = −
b

2π

∣∣∣
{
X⊥ ∈ R

2|W(X⊥) > 2
√
λ
}∣∣∣ (1 + o(1)) =

−ψα(2
√
λ) (1 + o(1)), λ ↓ 0, (4.2)

if V ≤ 0, and

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,

if ±V ≥ 0, the function ψα being defined in (3.2).

(ii) Assume that the hypotheses of Theorem 3.2 hold with U = W. Then we have

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



30 J.-F. Bony, Ph. Briet, V. Bruneau and G. Raikov CUBO
11, 5 (2009)

if V ≤ 0, the functions ϕβ being defined in (3.4). 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,∞),

if ±V ≥ 0.

(iii) Assume that the hypotheses of Theorem 3.3 hold with U = W. Then we have

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

if V ≤ 0, the function ϕ∞ being defined in (3.6). 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,

if ±V ≥ 0.

As a corollary we obtain the following result which could be regarded as generalized Levinson

formulae:

Corollary 4.2. Let V satisfy (1.2), and V ≤ 0. Fix q ∈ Z+. Then

lim
λ↓0

ξ(2bq + λ; H,H0)

ξ(2bq − λ; H,H0)
=

1

2 cos
π
α

if W admits a power-like decay with decay rate α > 2 (i.e. if U = W satisfies the hypotheses of

Theorem 3.1), or

lim
λ↓0

ξ(2bq + λ; H,H0)

ξ(2bq − λ; H,H0)
=

1

2

if W decays exponentially or has a compact support (i.e. if U = W satisfies the hypotheses of

Theorems 3.2 – 3.3).

Remark: The classical Levinson formula relates the number of the negative eigenvalues of

−∆ + V and limE↓0 ξ(E; −∆ + V,−∆) (see the original work [16] or the survey article [23]).

5 Resonances Near the Landau Levels

One of the possible explanations of the singularities of the SSF described in Theorem 4.1, is the

accumulation of resonances of H at the Landau levels. In this section we define the resonances

of H as an appropriate extension of the resolvent (H − z)−1 defined a priori for z ∈ C+ := {ζ ∈

C | Im ζ > 0}, following the exposition of [5]. In Theorem 5.1 below we establish upper and lower

estimates on the number of resonances of H in a ring centered at a given Landau level; the lower



CUBO
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Resonances and SSF Singularities ... 31

bounds imply accumulation of the resonances to the Landau level.

For z ∈ C+ we have

(H0 − z)
−1

=

∞∑

q=0

pq ⊗
(
H0,‖ + 2bq − z

)−1
.

For each q ∈ Z+ and N > 0 the operator-valued function

z 7→
(
H0,‖ + 2bq − z

)−1
∈ L(e−N〈x3〉L2(R),eN〈x3〉L2(R)),

admits a holomorphic extension from C \ [2bq,∞) to the 2-sheeted covering

Pq : {ζ ∈ C \ {0}, Im ζ > −N} ∋ k 7→ k
2

+ 2bq ∈ C \ {2bq}.

This extension however depends on q ∈ Z+.

Let π1(C \ 2bZ+) be the fundamental group of C \ 2bZ+, and G be the subgroup of π1(C \ 2bZ+)

generated by {
a21,a2a1a

−1
2 a

−1
1 | a1,a2 ∈ π1(C \ 2bZ+)

}
.

We define PG : M 7→ C \ 2bZ+ as the connected infinite-sheeted covering such that

P∗G(π1(M)) = G.

Fix a base point in M. Let F be the connected component of P−1G (C \ [0,∞)) containing this base

point. By definition, the functions M ∋ z 7→
√
z − 2bq have a positive imaginary part on F. Set

F+ := F ∩ P
−1
G (C+).

For λ0 ∈ C and ε > 0 put

D(λ0,ε) := {λ ∈ C| |λ − λ0| < ε},

D(λ0,ε)
∗

:= {λ ∈ C| 0 < |λ − λ0| < ε}.

Let D∗q ⊂ M be the connected component of P
−1
G (D(2bq, 2b)

∗
) that intersects F+. There exists

an analytic bijection zq:

D(0,
√

2b)∗ ∋ k 7→ zq(k) ∈ D
∗
q,

such that PG(zq(k)) = 2bq + k
2 and z−1q (D

∗
q ∩ F+) is the first quadrant of D(0,

√
2b)∗.

For N > 0 set

MN :=
{
z ∈ M | Im

√
z − 2bq > −N,∀q ∈ Z+

}
.

Evidently, ∪N >0MN = M.

Proposition 5.1. [5, Proposition 1] (i) For each N > 0 the operator-valued function

z 7→ (H0 − z)
−1 ∈ L

(
e−N〈x3〉L2(R3); eN〈x3〉L2(R3)

)

has a holomorphic extension from the open upper half-plane to MN .

(ii) Suppose that V satisfies (1.3) with m⊥ > 0. Then for each N > 0 the operator-valued function

z 7→ (H − z)−1 ∈ L
(
e−N〈x3〉L2(R3); eN〈x3〉L2(R3)

)
,

has a meromorphic extension from the open upper half plane to MN . Moreover, the poles and the

range of the residues of this extension do not depend on N.



32 J.-F. Bony, Ph. Briet, V. Bruneau and G. Raikov CUBO
11, 5 (2009)

We define the resonances of H as the poles of the meromorphic extension of the resolvent

(H − z)−1. The multiplicity of a resonance z0 is defined as

rank
1

2iπ

∫

γ

(H − z)−1dz,

where γ is an appropriate circle centered at z0. In what follows, Resq(H0 + κV ) denotes the set of

the resonances of H0 + κV in D
∗
q .

Theorem 5.1. [5, Theorem 2] Assume that V satisfies (1.3), and V ≥ 0 or V ≤ 0. Fix q ∈ Z+.

Then for any δ > 0 there exist κ0,r0 > 0 such that:

(i) For any 0 < r < r0 and 0 ≤ κ ≤ κ0, we have

#{z = zq(k) ∈ Resq(H0 + κV ) |r < |k| < 2r} = O(Tr 1(r,8r)(κpqWpq)).

(ii) For any 0 ≤ κ ≤ κ0, H0 + κV has no resonances in

{z = zq(k) | 0 < |k| < r0, ∓Im k ≤
1

δ
|Re k|}

if ±V ≥ 0.

(iii) If W satisfies ln W(X⊥) ≤ −C〈X⊥〉
2, then for any 0 ≤ κ ≤ κ0, H0 + κV has an infinite

number of resonances in {z = zq(k) | 0 < |k| < r0, ∓Im k >
1
δ
|Re k|} if ±V ≥ 0. More precisely,

there exists a decreasing sequence (rℓ)ℓ∈N of positive numbers, rℓ ↓ 0 such that,

#{z = zq(k) ∈ Resq(H0 + κV ) | rℓ+1 < |k| < rℓ, ∓ Im k >
1

δ
|Re k|} ≥

Tr 1(2rℓ+1,2rℓ)(κpqWpq).

The setting is summarized by the following figure:

Im k

Re k

V ≥ 0

r0

Re k

Im k

V ≤ 0

r0

Figure 1: Resonances near a Landau level for V of definite sign. They are essentially concentrated
near the semi-axis k = ∓i[0,∞) for ±V ≥ 0. The physical sheet is in white.



CUBO
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Resonances and SSF Singularities ... 33

6 SSF for a Pair of Modified Magnetic Schrödinger Operators

In this section we replace the unperturbed operator H0 by H̃0 = H0 + v0 where v0 is an electric

potential which depends only on the variable x3, and decays fast enough as |x3| → ∞. Under

appropriate assumptions on v0, the operator H̃0 has infinitely many eigenvalues of infinite mul-

tiplicity, most of which are embedded in the continuous spectrum. We introduce the perturbed

operator H̃ := H̃0 + V where V satisfies (1.1), and investigate the asymptotic behavior of the SSF

ξ(·; H̃,H̃0) near the eigenvalues of H̃0 of infinite multiplicity.

Let v0 ∈ C
1
(R; R) satisfy the assumption

|v0(x)| = O(〈x〉
−δ0 ), x ∈ R, δ0 > 1. (6.1)

Set

H‖ := H0,‖ + v0 = −
d2

dx23
+ v0(x3).

We have

σess(H‖) = σac(H‖) = [0,∞),

σpp(H‖) ∩ (0,∞) = ∅, σsc(H‖) = ∅.

For simplicity, assume that inf σ(H‖) > −2b. Then

H̃0 := H0,⊥ ⊗ I‖ + I⊥ ⊗ H‖ = H0 + I⊥ ⊗ v0

is the Hamiltonian of a non-relativistic spinless quantum particle subject to a classical electromag-

netic field (E, B) with B = (0, 0,b) and E = −(0, 0,v′0).

Let Λ ∈ σdisc(H‖). Then 2bq + Λ with q ∈ Z+ is an eigenvalue of H̃0 of infinite multiplicity; if

q ≥ 1, this eigenvalue is embedded in σac(H̃0).

Assume now that V : R3 → R satisfies (1.1) with m⊥ > 2 and m3 > 1, and set

H̃ := H̃0 + V.

Then (H̃ − i)−1 − (H̃0 − i)
−1 is trace-class, and the SSF ξ(·; H̃,H̃0) is well defined.

Set

Z := {E = 2bq + µ | q ∈ Z+, µ ∈ σdisc(H‖) or µ = 0}.

Similarly to Proposition 2.1, we can show that the SSF ξ(·; H̃,H̃0) is bounded on every compact

subset of R \ Z, and is continuous on R \ (Z ∪ σpp(H̃)).

Let Λ ∈ σdisc(H‖), and ψ be the eigenfunction satisfying

H‖ψ = Λψ, ‖ψ‖L2(R) = 1, ψ = ψ̄.

By analogy with (4.1) set

W̃(X⊥) :=

∫

R

|V (X⊥,x3)|ψ(x3)
2dx3.



34 J.-F. Bony, Ph. Briet, V. Bruneau and G. Raikov CUBO
11, 5 (2009)

Theorem 6.1. [2, Theorem 6.1] Let v0 satisfy (6.1), and V satisfy (1.1). Assume inf σ(H‖) >

−2b. Let Λ ∈ σdisc(H‖). Fix q ∈ Z+. Then for each ε ∈ (0, 1) we have

Tr 1((1+ε)λ,∞)(pqW̃pq) + O(1) ≤ ξ(2bq + Λ + λ; H̃,H̃0) ≤ Tr 1((1−ε)λ,∞)(pqW̃pq) + O(1),

ξ(2bq + Λ − λ; H̃,H̃0) = O(1),

as λ ↓ 0, if V ≥ 0, and

−Tr 1((1−ε)λ,∞)(pqW̃pq) + O(1) ≤ ξ(2bq + Λ − λ; H̃,H̃0) ≤ −Tr 1((1+ε)λ,∞)(pqW̃pq) + O(1),

ξ(2bq + Λ + λ; H̃,H̃0) = O(1),

as λ ↓ 0, if V ≤ 0.

Similarly to Corollary 4.2, we can combine the result of Theorem 6.1 with those of Theorems

3.1 – 3.3, and obtain explicit asymptotic formulae describing the singularity of ξ(·; H̃,H̃0) at 2bq+Λ

under explicit assumptions about the decay of W̃ at infinity. We omit here this obvious corollary,

and refer the reader to [2, Corollary 6.1].

7 Dynamical Resonances for Axisymmetric Perturbations V

In this section we assume that the perturbation V is axisymmetric, and investigate the asymptotics

as κ → 0 of the unitary group e−it(H̃0+κV ), t ≥ 0. For m ∈ Z introduce the operator

H
(m)
0,⊥ := −

1

̺

d

d̺
̺
d

d̺
+

(
m

̺
−
b̺

2

)2
− b,

self-adjoint in L2(R+; ̺d̺). We have

σ(H
(m)
0,⊥ ) = ∪

∞
q=m−

{2bq}, m− := max{0,−m},

dim Ker(H
(m)
0,⊥ − 2bq) = 1, ∀q ≥ m−.

For m ∈ Z, q ∈ Z, q ≥ m−, let ϕq,m satisfy

H
(m)
0,⊥ ϕq,m = 2bqϕq,m, ‖ϕq,m‖L2(R+;̺d̺) = 1.

Let the multiplier by v0 be H0,‖-compact. Set

H
(m)
0 := H

(m)
0,⊥ ⊗ I‖ + I⊥ ⊗ H0,‖,

H̃
(m)
0 := H

(m)
0,⊥ ⊗ I‖ + I⊥ ⊗ H‖ = H

(m)
0 + I⊥ ⊗ v0,

where I⊥ is the identity operator in L
2
(R+; ̺d̺);

σ(H
(m)
0 ) = σess(H̃

(m)
0 ) = [2m−b,∞).



CUBO
11, 5 (2009)

Resonances and SSF Singularities ... 35

Let (̺,φ,x3) be the cylindrical coordinates in R
3. The operator H

(m)
0 (resp. H̃

(m)
0 ), m ∈ Z,

is unitarily equivalent to the restriction of H0 (resp., of H̃0) onto Ker (L − m) with L := −i
∂

∂φ
.

Hence, the operator H0 (resp., H̃0) is unitarily equivalent to the orthogonal sum ⊕m∈ZH
(m)
0 (resp.,

⊕m∈ZH̃
(m)
0 ).

Let the multiplier by V : R3 → R be H0-bounded with zero relative bound. Suppose that V is

axisymmetric i.e. ∂V
∂φ

= 0.

Let κ ∈ R. On D(H̃0) introduce the operator

H̃κ := H̃0 + κV

self-adjoint in L2(R3), and on D(H̃
(m)
0 ) introduce the operator

H̃(m)
κ

:= H̃
(m)
0 + κV, m ∈ Z,

self-adjoint in L2(R+ × R; ̺d̺dx3).

Let Λ ∈ σdisc(H‖). For z ∈ C+, m ∈ Z, q ∈ Z+, q ≥ m−, set

Fq,m(z) := 〈(H̃
(m)
0 − z)

−1
(I − Pq,m)V Φq,m,V Φq,m〉

where 〈·, ·〉 is the scalar product in L2(R+ × R; ̺d̺dx3), Φq,m(̺,x3) = ϕq,m(̺)ψ(x3) so that

H̃
(m)
0 Φq,m = (2bq + Λ)Φq,m, and Pq,m := |Φq,m〉〈Φq,m|.

We will say that the Fermi Golden Rule Fq,m,Λ is valid if the limit

Fq,m(2bq + Λ) = lim
δ↓0

Fq,m(2bq + Λ + iδ)

exists and is finite, and

Im Fq,m(2bq + Λ) > 0.

For j ∈ Z+ set vj (x3) := x
j
3

dj v0
dx

j
3

, Vj = x
j
3

∂j V
∂x

j
3

.

We will say that the condition Cν , ν ∈ Z+, holds true if the multipliers by vj , j = 0, 1, are H0,‖-

compact, the multipliers by vj , j ≤ ν, are H0,‖-bounded, the multiplier by V is H0-bounded with

zero relative bound, and the multipliers by Vj , j = 1, . . . ,ν, are H0-bounded.

Theorem 7.1. [2, Theorem 4.1] Let V be axisymmetric. Fix n ∈ Z+, and assume that the

condition Cν holds with ν ≥ n + 5. Suppose that inf σ(H‖) > −2b. Let Λ ∈ σdisc(H‖). Fix m ∈ Z,

q ∈ Z+, q > m−, and suppose that the Fermi Golden Rule Fq,m,Λ is valid.

Then there exists a function g ∈ C∞0 (R; R) such that g = 1 near 2bq + Λ, and

〈e−iH̃
(m)
κ

tg(H̃(m)
κ

)Φq,m, Φq,m〉 = a(κ)e
−iΛq,m (κ)t + b(κ, t), t ≥ 0, (7.1)

where

Λq,m(κ) = 2bq + Λ + κ〈V Φq,m, Φq,m〉 − κ
2Fq,m(2bq + Λ) + oq,m,V (κ

2
), κ → 0. (7.2)

Moreover, a and b satisfy the estimates

|a(κ) − 1| = O(κ2),



36 J.-F. Bony, Ph. Briet, V. Bruneau and G. Raikov CUBO
11, 5 (2009)

|b(κ, t)| = O(κ2| ln |κ||(1 + t)−n),

|b(κ, t)| = O(κ2(1 + t)−n+1),

as κ → 0 uniformly with respect to t ≥ 0.

Remarks: (i) The numbers Λq,m appearing in (7.1) can be interpreted as resonances for the

operator H̃m, and hence of H̃. Another definition of the resonances of H̃ which is in the spirit

of [1] and [12] and includes x3-analyticity assumptions concerning v0 and V , can be found in [2,

Section 3] (see also [13] and [6] where no axial symmetry of V is assumed).

(ii) Note that the numbers 〈V Φq,m, Φq,m〉 appearing in (7.2) are eigenvalues of the Toeplitz oper-

ator pqW̃pq appearing in Theorem 6.1.

(iii) The proof of Theorem 7.1 is based on appropriate Mourre estimates (see [18]), and the ap-

proach developed in [8].

Acknowledgements. G. Raikov thanks the organizers of the Second Symposium on Scatter-

ing and Spectral Theory, Pernambuco, Brazil, August 2008, for having invited him to present there

the results of this paper. J.-F. Bony and V. Bruneau were partially supported by the French ANR

Grant no. JC0546063. V. Bruneau and G. Raikov were partially supported by the Chilean Science

Foundation Fondecyt under Grants 1050716 and 7060245 and by the cooperation program CNRS-

Conicyt RQWUS41561. G. Raikov was partially supported by Núcleo Científico ICM P07-027-F

“Mathematical Theory of Quantum and Classical Magnetic Systems".

Received: December, 2008. Revised: March, 2009.

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