Resonances and Spectral Shift Function for a Magnetic SCHRÖdinger Operator

0.32 MB·English

by  Abdallah Khochman

RESONANCES AND SPECTRAL SHIFT FUNCTION FOR A
MAGNETIC SCHRÖDINGER OPERATOR
9 ABDALLAHKHOCHMAN
0
0
2

n H
0
a Abstra
t. We
onsider the3vD S
hrödinger operator with
onstantmagneti
(cid:28)eldaxnd
0 3
J subje
ttoanele
tri
potential dependingonlyonthevariablealongthemagneti
(cid:28)eld .
H
0
4 Theoperator hasin(cid:28)nitelyHm0anyeigenvaluesofin(cid:28)nitemuVltip=li
Oit(yh(exm1,bxe2d)di−edδ⊥inhxi3tis−
δokn)-
1 tinuousspe
trum. Weperturb bysmooths
alarpotentials ,
δ⊥ >2, δk >1. We assume also that V and v0 have an analyti

ontinuation, in the mag-

] neti
(cid:28)elddire
tion,in a
omplexse
toroutside a
ompa
tset. We de(cid:28)netheresonan
es of
P H =H0+V H
.S distqor∈tioNns ofaRsxt3h.eWeiegesntuvadlyuetsh2eobifrqt+dhiesλtnriobnu-tsieolnfadnjeoairnta(rno,yp2er(cid:28)r)xaetodrroeabltaeiingeednvfarloumeofHby0,a2nbaqly+tiλ
h for . Inaring
enteredat withradiuses ,weestablishanupperbound,as
r 0 V
t tendsto , of the numberof resonan
es. This upper bounddepends on thede
ay of at
a (x1,x2)
m in(cid:28)nityonly in the dire
tions . Finally, we dedu
Hea,Hrepresentation of the derivative
0
ofthespe
tralshiftfun
tion(SSF)fortheoperatorpair( )intermsofresonan
es. This
[
representation justi(cid:28)es theBreit-Wigner approximationand implies a lo
al tra
e formula.

1
Mathemati
s
lassi(cid:28)
ation: 35P25, 35J10, 47F05, 81Q10.
v
0 Keywords: Ele
tromagneti
S
hrödinger operator, Resonan
es, Embedded eigenvalue,
8
Spe
tral shift fun
tion, Breit-Wigner approximation, Tra
e formula.
9
1
. 1. Introdu
tion
1
0 The resonan
e theory for non-relativisti
parti
les satisfying the S
hrödinger equation has
9
been developed following several approa
hes. Among them we
an mention the analyti
dila-
0
: tion (see Aguilar-Combes [1℄) or the analyti
distortion (see Hunziker [10℄) and meromorphi
v

ontinuation of the resolvent or of the s
attering matrix (see Lax-Philips [14℄ and Vainberg
i
X
[20℄). For S
hrödinger operators with
onstant magneti
(cid:28)eld, the resonan
es
an be de-
r (cid:28)ned by analyti
dilation (only) with respe
t to the variable along the magneti
(cid:28)eld (see
a
Avron-Herbst-Simon [3℄, Wang [21℄, Astaburuaga-Briet-Bruneau-Fernández-Raikov [2℄) and
by meromorphi

ontinuation of the resolvent (see J.F.Bony-Bruneau-Raikov [4℄).
The link between the resonan
es and the spe
tral shift fun
tion (SSF) by the so-
alled
Breit-Wigner approximation has been developed in di(cid:27)erent situations. Su
h a representa-
tion of the derivative of the spe
tral shift fun
tion related to the resonan
es, implies tra
e
formulas. In the semi-
lassi
al regime we
an mention Sjöstrand [18℄, [19℄, Petkov-Zworski
[15℄, J.F.Bony-Sjöstrand [5℄, Bruneau-Petkov [6℄ and Dimassi-Zerzeri [8℄ for the S
hrödinger
operator and [12℄ for the Dira
operator. In [4℄, J.-F.Bony, Bruneau and Raikov obtain a
3
Breit-Wigner approximation of the spe
tral shift fun
tion near a Landau level for the D
S
hrödinger operator with
onstant magneti
(cid:28)eld. For the last operator, under more general
assumptions, Fernández-Raikov [9℄ studied the singularities of the spe
tral shift fun
tion at a
Date: January 14, 2009.
1
2 ABDALLAHKHOCHMAN
Landau level. These singularities has been analysed at eigenvalues of in(cid:28)nite multipli
ity by
Astaburuaga, Briet, Bruneau, Fernández and Raikov [2℄ for a magneti
S
hrödinger operator
having ele
tri
potential depending (only) on the variable along the magneti
(cid:28)eld.
H
0
In this paper we
onsider the magneti
S
hrödinger operator with an ele
tromagneti
(cid:28)eld introdu
ed in [2℄. We suppose that the magneti
(cid:28)eld is
onstant and that the ele
tri
v x
0 3
potential depends only on the variable and is analyti
outside a
ompa
t set. This
operator isremarkablebe
auseofthegeneri
presen
eofin(cid:28)nitely many eigenvaluesofin(cid:28)nite
H H
0 0
multipli
ity, embedded in the
ontinuous spe
trum of . We perturb the operator by a
V x
3
smooth s
alar potential analyti
outside a
ompa
t set with respe
t to the variable .
The purpose of this work is to de(cid:28)ne the resonan
es of the ele
tromagneti
S
hrödinger
H = H +V
0
operator for analyti
perturbation outside a
ompa
t set in the third dire
tion
x H
3
. We de(cid:28)ne the resonan
es for as the dis
rete eigenvalues of the non-selfadjoint operator
H
θ
Robtained from the magneti
S
hrödinger operator by agenHeral
lass of
omplex distortions
of x3. In Se
tion 3, we prove that the dis
rete eigenvalues of θ are thezeros of aregularized
det (·)
2
determinant whi
h is independent of the distortion. This justi(cid:28)es the de(cid:28)nition of the
resonan
es. We
al
ulate the essential spe
trum of the distorted operator to determine the
se
tor where we
an de(cid:28)ne the resonan
es. In Se
tion 4, we establish an upper bound for the
H r → 0 H
0
number of resonan
es of in a domain of size near an embedded eigenvalue of .
The se
ond goal of this work is to obtain a Breit-Wigner approximation for the derivative of
ξ(λ) H
the spe
tral shift fun
tion related to the resonan
es of the operator , as well as a lo
al
tra
e formula (see Se
tion 5).
2. Assumptions and results
H
0
In this se
tion, we summariBze=so(m0,e0s,pb)e
tbra>l p0roperties of the 3D S
hrödinger operator
with
onstant magneti
(cid:28)eld , and subje
t to a non-
onstant ele
tri
(cid:28)eld
E = −(0,0,v (x )) x
0′ 3 depending onlyonthevariable 3 (see[2℄). Wealsostatethemainresults.
Let
H = H ⊗I +I ⊗H ,
0 0, 0,
(2.1) ⊥ k ⊥ k
I I L2(R ) L2(R2 )
where k and ⊥ are the identity operators in x3 and x1,x2 respe
tively,
2 2
∂ bx ∂ bx
H := i − 2 + i + 1 −b, (x ,x ) ∈R2,
0, 1 2
(2.2) ⊥ (cid:18) ∂x1 2 (cid:19) (cid:18) ∂x2 2 (cid:19)
b L2(R2)
is the Landau Hamiltonian shifted by the
onstant , self-adjoint in , and
d2
H := − +v , x ∈ R.
(2.3) 0,k dx23 0 3
v v (x )
0 0 3
The operator isvth∈eLmu(ltRip)li
ation operator by an one dimensional s
alar potential .
0 ∞
We suppose that and satis(cid:28)es
|v | = O(hx i δ0),
0 3 −
(2.4)
hxi = (1+|x|2)21 δ0 > 1
with and . Then using Weyl theorem, we have
d2
σ (H )= σ (− ) =[0,+∞[.
(2.5) ess 0,k ess dx2
3
MAGNETIC SCHRÖDINGER RESONANCES AND SSF 3
H 2bq q ∈
0,
INt:i=s w{e0l,l1k,n2o.w..n}that the spe
trum of the operator ⊥
2oqnbsists of the Landau levels ,
, aHndthemultipli
ity2obfqe+a
hλeigenvaqlu∈e N isinλ(cid:28)nite(see[3℄). Consequently,
0
the eigenvalues of have the forHm = − d2w+hvere(x ) and is an eigenvalue of the one
dimensional S
hrödinger operator 0,k dx23 0 3 . For simpli
ity, throughout thearti
le
we suppose also that
σ(H ) > −2b.
0,
(2.6) inf k
v > −2b H 2bq+λ, q ∈ N
0 0 ∗
Note that, (2.6) holds true if . The eigenvalues of , , are embedded
[0,+∞[= ∪ [2bq,∞[
∞q=0
in its
ontinuous spe
trum and are of in(cid:28)nite multipli
ity.
H = H + V V
0
Now, we introdu
e theVp(exr)turbed operatorV ∈L (R3) where is the multipli
ation
∞
operator by the potential . Assume that and satis(cid:28)es
|V(x)| = O(hX i−δ⊥hx3i−δk), X = (x1,x2),
(2.7) ⊥ ⊥
δ > 2 δ > 1 V v
0
with ⊥ and k .xWe suppose also that and have holomorphi
extensions in the
3
magneti
(cid:28)eld dire
tion in the se
tor
C := {z ∈ C;| (z)| ≤ ǫ| (z)|, | (z)| ≥ R > 0}, 0 < ǫ < 1,
ǫ,0 0
(2.8) Im Re Re for
x ∈ C
3 ǫ,0
and satisfy respe
tively (2.7) and (2.4) for .
θ ∈ D ∩R D := {θ ∈ C; |θ|≤ r := ǫ }
For ǫ with ǫ ǫ √1+ǫ2 , we denote
H := (I ⊗U )H(I ⊗U 1)= H +V ,
θ θ θ− 0,θ θ
⊥ ⊥
where
H := (I ⊗U )H (I ⊗U 1) = H ⊗I +I ⊗H (θ)
(2.9) 0,θ ⊥ θ 0 ⊥ θ− 0,⊥ k ⊥ 0,k
H (θ) = U H U 1 U
and 0,k θ 0,k θ−H(see (3.2) for the de(cid:28)nition ofθ ∈θ).DWe will prouve in the next
θ ǫ
se
tion that the operator has an analyti
extension for .
θ D+ := D ∩{θ ∈ C; (θ)≥ 0} q ∈ N r ∈ R
0 ǫ ǫ
For (cid:28)xed in Im , and , we de(cid:28)ne
Γ := 2br+(1+θ ) 2[0,+∞[
(2.10) r,θ0 0 −
and
S := Γ .
(2.11) q,θ0 r,θ0
q<r[<q+1
H
The spe
trum of 0,θ0 is purely essential and we have
σ(H ) = σ (H ) = 2bq+σ(H (θ ))
(2.12) 0,θ0 ess 0,θ0 q[N(cid:0) 0,k 0 (cid:1)
∈
= Γ ∪ 2bq+σ (H (θ )) ,
q,θ0 disc 0, 0
q[N(cid:0) (cid:0) k (cid:1)(cid:1)
∈
σ (H (θ )) = σ (H )∪{z ,z ,...} σ (H )
disc 0, 0 disc 0, 1 2 disc 0,
where k k , k denotes the dis
rete spe
trum of
H z ,z ,... H (θ )
0, 1 2 0, 0
k and are the
omplex eigenvalues of k . In the following, we assume that
σ (H )= {λ} λ
disc 0,
k . Note that is ne
essarily simple.
H H
θ 0,θ
The essential spe
trum of
oin
ides with that of . We prove also that the dis
rete
H S = S θ D+ θ , θ
spe
trum of θ in θH Sq∈NHq,θ is independSent∩ofS in ǫ (i.e. for two values 1 2, the
dis
rete spe
trum of θ1 and θ2
oin
ide on θ1 θ2), (see Proposition 3.2). This justi(cid:28)es
the following de(cid:28)nition.
4 ABDALLAHKHOCHMAN
H S H
De(cid:28)nition 2.1. The resonan
es of in θ0 are the dis
rete eigenvalues of θ0. The multi-
z
0
pli
ity of a resonan
e is de(cid:28)ned by
1
(z ) := (z−H ) 1dz,
(2.13) mult 0 rank2iπ Z θ0 −
Γ0
Γ z (H)
0 0
where is a small positively oriented
ir
le
entered at . We will denote Res the set of
resonan
es.
H {z ∈ C; (z) < 0}
Remark 2.1. The resonan
es of in Re are the real dis
rete eigenvalues
H
of .
PSfrag repla
ements
(z)
Im
Ω
q
H
Resonan
es of
H
e.v. of θ0
λ 0 2b+λ 2b 2bq+λ 2bq (z)
◦ ◦ ◦ ◦ Re
⋆ ⋆⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆◦⋆⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆⋆ ⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆⋆⋆ ⋆⋆Γ⋆⋆⋆⋆◦⋆⋆⋆ ⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆ ⋆
Γ q,θ0
1,θ0
Γ
0,θ0
S
θ0
S
Fig.1. The set θ0
r → 0 H Ω
q
Now, we state(ra,n2ru)pper bound as 2bq+oλn tqh∈e nNumber of resonan
es of in a ring in
∗
with radiuses and
entered at , (cid:28)xed.
V v
0
Theorem 2.1. [Upper bound℄ Suppose that and satisfy the above hypotheses. Then
r > 0 ν > 0 0< r < r
0 0
there exist and , su
h that, for any ,
#{z ∈ (H)∩Ω ;r < |z−2bq−λ|< 2r} = O(n (r,νp Wp )|lnr|),
q + q q
(2.14) Res
whereW = supx3∈Cǫ,0|hx3iδkV|, pq istheorthogonalproje
tionontoHq := ker(H0,⊥−2bq)and
n (r,p Wp ) r
+ q q
is the
ounting fun
tion of the eigenvalues larger than of the Toeplitz operator
pqWpq n+(r,pqWpq) = O(r−2/δ⊥)
. In parti
ular, under our assumption we have always .
n (r,p Wp ):= 1 (p Wp )
+ q q (r,+ ) q q
The
ounting fun
tion W rank ∞ satis(cid:28)es asymptoti
relations
depending on the de
ay of at in(cid:28)nity. The following three lemmas give an upper bound
n (r,p Wp ) W
+ q q
of in the
ase power-like de
ay, exponential de
ay, or
ompa
t support of ,
respe
tively. For more pre
ise results
on
erning the asymptoti
properties, we refer to the

ited theorem.
MAGNETIC SCHRÖDINGER RESONANCES AND SSF 5
U ∈ L (R2)
∞
Lemma 2.1. (Theorem 2.6 of [16℄) Let the fun
tion satisfy the estimate
U(X )≤ ChX i α, X ∈ R2,
−
⊥ ⊥ ⊥
α> 0 q ∈ N
for some . Then for ea
h , we have
n (r,p Up )= O(r 2/α).
+ q q −
U ∈L (R2)
∞
Lemma 2.2. (Theorem 2.1 of [17℄) Let . Assume that
lnU(X )
limsup ⊥ < 0, X ∈ R2,
|X⊥|→∞ |X⊥|2β ⊥
β > 0 ln(u) = −∞ u≤ 0 q ∈N
for some (with the
onvention if ). Then for ea
h , we have
n (r,p Up ) = O(ϕ (r))
+ q q β
0 < r < e 1
−
where, for ,
1
|lnr|β 0< β < 1,
if
ϕβ(r):=  |lnr| β = 1,
(2.15)  (ln|lnr|) 1|lnr| if β > 1.
−
if
 U ∈ L (R2) U
∞
Lemma 2.3. (Theoremq2∈.4Nof [17℄) Let . Assume that the support of is

ompa
t. Then for ea
h , we have
n (r,p Up )= O(ϕ (r)),
+ q q
∞
0 < r < e 1
−
where, for , ϕ (r):= (ln|lnr|) 1|lnr|.
−
∞
(H,H ) ξ(λ)
0
Now, we study the spe
tralHsh,iHft fun
tion (SSF) for thDe p(aRi)r . The SSF for a
0 ′
pair of self-adjoint operators ( ) is a distribution in whose derivative is
ξ :f ∈ C (R) 7−→ − (f(H)−f(H )).
′ 0∞ 0
(2.16) tr
|V|21(H0+i)−1
Iξn(λo)ur
ase, L1 (R)is in the Hilbert-S
hmidt
lass, (2.16) is well de(cid:28)ned and the SSF
is a fun
tion in loc .
H S
Wewillseefurtherthattheresonan
esof in θ0 arethezerosoftheholomorphi
extension
of z ∈ {z ∈ C, z > 0} 7−→ D(z) = ((H −z)(H −z) 1)
2 0 −
Im det
S det
into θ0 (see (3.1) for the de(cid:28)nition of 2). Thus in order to obtain a link between the SSF
and the resonan
es, it will be
onvenient to introdu
e the regularized spe
tral shift fun
tion
1
ξ (ν)= lim arg (H −ν −iε)(H −ν −iε) 1 ,
2 2 0 −
(2.17) π ε 0+ det
→ (cid:0) (cid:1)
whose derivative is the following distribution (see [4℄)
d
ξ : f ∈ C (R)7−→ − f(H)−f(H )− f(H +εV)| .
(2.18) 2′ 0∞ tr(cid:18) 0 dε 0 ε=0(cid:19)
We will dedu
e the properties of the SSF from those of the regularized SSF using the relation
1
ξ = ξ + V (H −z) 2 .
(2.19) ′ 2′ πIm tr θ 0,θ −
(cid:0) (cid:1)
2bq +λ
We represent now, the derivative of the spe
tral shift fun
tion near as a sum of a
harmoni
measurerelatedtotheresonan
esandtheimaginary partofaholomorphi
fun
tion.
6 ABDALLAHKHOCHMAN
Ω ⊂⊂Ω C\{0}
Let be open relatively
ompa
t subsets of . We assume that these sets are
r Ω
indepenedent of and that is simply
onne
ted. Also assume that the interse
tions between
Ω R I
and is a non-empty inteerval .
e V v
0
Theorem 2.2. [Breit-Wigner approximation℄ We suppose and satisfy the above
Ω ⊂⊂Ω I g Ω
hypothesis. For and as above, there exists a fun
tion holomorphi
in , su
h that
µ ∈ 2bq+λ+rI
for e , we have
1 µ−2bq−λ − w
ξ′(µ) = g′( ,r)− Im − δ(µ−w)
πrIm r π|µ−w|2
X X
w∈Res(H)∩2bq+λ+rΩ w∈Res(H)∩2bq+λ+rI
w=0
Im 6
g(z,r)
where satis(cid:28)es the estimate
2
g(z,r) = O(n+(r,νpqWpq)|lnr|+n1(r/ν)+n2(r/ν)) = O(|lnr|r−δ⊥), ν > 0,
(2.20)
0< r < r z ∈eΩ ne, p = 1, 2,
0 p
uniformly with respe
t to and , with de(cid:28)ned by
n (r) := pqWpq1 e(p Wpe) p, r > 0.
(2.21) p (cid:13) r [0,r] q q (cid:13)
(cid:13) (cid:13)p
e (cid:13) (cid:13)
(cid:13) (cid:13)
k·k (p = 1) (p = 2)
p
Here, stands for the tra
e-
lass norms and Hilbert-S
hmidt norms .
W
Using [4, Corollary1℄, for de(cid:28)ned above satisfying the assumption of Lemma 2.1 with
α ≥ 2
, we have
2
np(r) = O(r−α), p = 1,2.
(2.22)
e W = U
Finally, if the assumption of Lemma 2.2 or 2.3 hold for , we have
n (r) = o(ϕ (r)) r ց 0,
p β
(2.23)
ϕβ(r) e
the fun
tion being de(cid:28)ned in Lemma 2.2 or 2.3.
As in [15℄, [6℄ or [4℄ and repeating the arguments used in the proof of [4, Corollary 3℄, we
dedu
e from Theorem 2.1-2.2 the following theorem
Ω ⊂⊂ Ω f
Theorem 2.3. [Tra
e formula℄ Let be as in Theorem 2.2. Suppose that is
Ω φ ∈ C (Ω ∩ R) φ = 1 Ω ∩ R
holomorphi
on a neighborhood of aned that 0∞ satis(cid:28)es near .
Then, under the assumptions of Theorem 2.2, we have the following tra
e formula e
H −2bq−λ H −2bq−λ w−2bq−λ
0
(φf)( )−(φf)( ) = f( )+E (r)
f,φ
tr(cid:18) r r (cid:19) r
X e
w (H) 2bq+λ+rΩ
∈Res ∩
with
|E (r)|≤ M sup{|f(z)| : z ∈ Ω\Ω, z ≤ 0}×N (r),
f,φ φ q
Im
e 2
Nq(r) = n+(r,νpqWpq)|lnr|+ n1(r/ν)+ n2(r/ν) = O(|lnr|r−δ⊥), Mφ
where and depends
φ
only on . e e
MAGNETIC SCHRÖDINGER RESONANCES AND SSF 7
3. Definition of resonan
es via distortion analyti
ity
In this se
tion, we start with the de(cid:28)nition oRf the deformation for the ele
tromagneti
S
hrödinger operator by analyti
distortion on x3. We
al
ulate the essential spe
trum
H H
θ θ
of the distorted S
hrödinger operator . We prove that the dis
rete eigenvalues of are
independentofthedistortion, thisjusti(cid:28)esthede(cid:28)nitionofaresonan
easadis
reteeigenvalue
H H
θ
of the distorted operator . We will also prove that the resonan
es of repeated with their
det (I + A)
2
multipli
ity
oin
ide with the zeros of a regularized determinant de(cid:28)ned for a
A
Hilbert-S
hmidt operator by
(I +A) := det((I +A)e A),
2 −
(3.1) det
(see Krein [13℄). Let us now introdu
e the one-parameter family of unitary distortions in the
x
3
magneti
(cid:28)eld dire
tion :
1
U f(x)= J2 f(φ (x)), θ ∈ R, f ∈ S(R),
(3.2) θ φθ(x) θ
φ (x) = x+θg(x) g : R7−→ R J = det(I+θg (x))
where θ , is a smooth fun
tion and φθ(x) ′ is the
φ (x) g
θ
Ja
obian of . We suppose that satis(cid:28)es the assumption
|g (x)| < 1,
x R ′
(Ag) (i) sgu(px)∈= 0, [−R0,R0], ( ( )),
 (ii) g(x) = x, in the
ompa
t set K(⊃ [−Rsee,R2.]8).
0 0
(iii) outside a
ompa
t set

We re
all that H := (I ⊗U )H(I ⊗U 1) = H +V .
θ θ θ− 0,θ θ
⊥ ⊥
From (2.9) and using Kato's theorem [11, Theorem 4.5.35℄ we have the following (see also
Hunziker [10℄ for S
hrödinger operator and [12, Se
tion 3℄ for the Dira
operator).
V
Proposition 3.1. We suppose that the potential satis(cid:28)es all the assumptions of Se
tion 2.
Then we have
θ ∈ D 7−→ H = H +V
ǫ θ 0,θ θ
(i) σ (H ) =σ (H ) = 2bNis+aσn(Hanaly(θti)
).family of type A.
ess θ ess 0,θ 0,
(ii) k
H (θ)
0,
Lemma 3.1. The essential spe
trum of k is
µ
σ (H (θ))= ∈ C; µ ∈ [0,+∞[ .
(3.3) ess 0,k n(1+θ)2 o
The rest of the spe
trum is
σ (H )∪{z ,z ,...},
disc 0, 1 2
k
σ (H ) H z ,z ,...
disc 0, 0, 1 2
where k denotes the dis
rete spe
trum of k and are the
omplex eigen-
H (θ)
0,
values of k .
σ (H ) ∪ {z ,z ,...} H (θ)
disc 0, 1 2 0,
Remark 3.1. The part k of the spe
trum of k
orrespond to
I ⊗H (θ)
0,
eigenvalues with in(cid:28)nite multipli
ity of ⊥ k .
q ∈ N Ω 2bq+λ
q
In the following we (cid:28)x and a
ompa
t set
entered at su
h that
Ω ∩σ (H ) = {2bq+λ}.
q ess θ
(3.4)
Repeatingargumentsintheproofof[4,Proposition1℄,[4,Lemma1℄andusingtheresolvent
equation (H −z) 1 = (H −v −z) 1 I −v (H −z) 1 ,
0 − 0 0 − 0 0 −
(cid:0) (cid:1)
we obtain the following lemma.
8 ABDALLAHKHOCHMAN
V(H − z) 1 ∂ (V(H − z) 1) {z ∈
0 − z 0 −
LCe;mm(za)3>.20.}The operators and S are holomoSrphi
on
2 1
Im with values intheHilbert-S
hmidt
lass andinthe tra
e
lass respe
tively.
V (H −z) 1 ∂ (V (H −z) 1) z ∈
θ 0,θ − z θ 0,θ −
Lemma 3.3. The operators and are holomorphi
for
Ω \{2bq+λ} S S
q 2 1
with values in the Hilbert-S
hmidt
lass and in the tra
e
lass respe
tively.
Ω z 7−→
q
Proof. A

ording to (i) of Proposition 3.1 and to the de(cid:28)nition of , the fun
tion
V (H −z) 1 z ∈ Ω \{2bq +λ}
θ 0,θ − q
is analyti
for . Moreover, from the resolvent equation, for
z ∈Ω \{2bq +λ}
q
, we have
V (H −z) 1 = V (H −i) 1 1+(H −H +z−i)(H −z) 1 .
θ 0,θ − θ 0 − 0 0,θ 0,θ −
(3.5)
(cid:0) (cid:1)
p H := (H −2bq)
q q 0,
If we denote by the orthogonal proje
tion onto ker ⊥ we have
(H −z) 1 = p ⊗(H (θ)+2bq−z) 1, z ∈ Ω \{2bq +λ}.
0,θ − q 0, − q
(3.6) Xq N k
∈
Sin
e
H −H = I ⊗(H −H (θ))
0 0,θ 0, 0,
⊥ k k
(H −H )(H −z) 1
0 0,θ 0,θ −
the operator is bounded. Consequently, a

ording to Lemma 3.2 an(cid:3)d
equation (3.5), we obtain the lemma.
Let us now introdu
e the fun
tion
z ∈Ω \{2bq+λ} −→ d (z) = (I +T (z)),
q θ 2 V,θ
(3.7) det
T (z) = V (H −z) 1 d (z)
V,θ θ 0,θ − θ
with . The determinant is well de(cid:28)ned a

ording to Lemma 3.3
V v H Ω
0 q
Proposition 3.2. Let and as in Se
tion 2. The resonan
es of in are the zeros of
d (z) = (I +T (z)) Ω \{2bq +λ}
θ 2 V,θ q
the regularized determinant det in , and are independent
θ ∈D Ω ∩σ (H ) = {2bq+λ}
ǫ q ess θ
on su
h that .
z f(z) z z
0 0
If is a resonan
e, there exists a holomorphi
fun
tion , for
lose to , su
h that
f(z ) 6= 0
0
and
(I +T (z)) = (z−z )l(z0)f(z),
2 V,θ 0
det
0 < l(z )= (z ) (z )
0 0 0
with mult where mult is the multipli
ity of the resonan
e de(cid:28)ned by (2.13).
H Ω \{2bq +λ}
0,θ q
Proof. Sin
e the operator has no spe
trum in , we have
H −z = I +V (H −z) 1 (H −z).
θ θ 0,θ − 0,θ
(3.8)
(cid:0) (cid:1)
z ∈ Ω \{2bq +λ} H
q
Then, if is a resonan
e of whi
h is by de(cid:28)nition a dis
rete eigenvalue of
H d (z) = (I +V (H −z) 1) = (I +T (z))
θ θ 2 θ 0,θ − 2 V,θ
, the determinant det det vanishes.
A
Let us re
all that, if is a bounded operator and B is a tra
e
lass operator on some
det(I +AB) = det(I +BA) A
separable Hilbert spa
e, we have . Moreover, for bounded and
B
Hilbert-S
hmidt, we have
(I +AB)= (I +BA).
2 2
(3.9) det det
d (z) = det (I+T (z)) det (I+T (z)) = (I+V(H −
θ 2 V,θ 2 V,0 2 0
zT)he1n), theθfu∈n
Rti,on z > 0
oin
idewith det θ ∈ D
− ǫ
for Im and by uniqueness of the extension, it is independent on .
H d (z) θ ∈ D
θ ǫ
Sin
e the resonan
es of are the zero of the resonan
es are independents on .
MAGNETIC SCHRÖDINGER RESONANCES AND SSF 9
z d (z) l(z ) d (z) = (z −
0 θ 0 θ
In a neighborhood of a zero of with multipli
ity , we write
z )l(z0)G(z), G(z) z G(z ) 6= 0
0 0 0
where is a holomorphi
fun
tion in a neighborhood of with .
l (z)
0
Then, by the de(cid:28)nition of ,
1
l (z) = ∂ ln (1+T (z))dz,
0 z 2 V,θ
2iπ Z det
Γ
Γ z
0
where is a small positively oriented
ir
le
entered at . Further, we have
∂ lndet(1+T(z)) = (1+T(z)) 1∂ T(z) , z ∈ Ω
z − z q
tr
(cid:0) (cid:1)
T(z) S
1
for any operator-valued holomorphi
fun
tion in the tra
e
lass . Therefore,
∂ ln (1+T (z)) = (1+T (z)) 1∂ T (z) − (∂ T (z)).
z 2 V,θ V,θ − z V,θ z V,θ
det tr tr
(cid:0) (cid:1)
∂ T (z) Γ
z V,θ
A

ording to Lemma 3.3, is holomorphi
in the tra
e
lass, then its integral on
vanishes and (3.8) yields
1
l (z) = (H −z) 1V (H −z) 1 dz
0 θ − θ 0,θ −
2iπ Z tr
Γ (cid:0) (cid:1)
1
= − (H −z) 1−(H −z) 1 dz
θ − 0,θ −
2iπ Z tr
Γ (cid:0) (cid:1)
1
= (z−H ) 1−(z−H ) 1 dz
θ − 0,θ −
rank2iπ Z
Γ(cid:0) (cid:1)
1
= (z−H ) 1dz.
θ −
rank2iπ Z
Γ
In the two latter equ(Halitie−s,zw)e1haveΓused that the tra
e of the proje
tor
oΩin
ide with its ran(cid:3)k
0,θ − q
and the integral of on vanishes sin
e it is holomorphi
in .
2bq+λ
4. Upper bound for the number of resonan
es near
Ω
q
In this se
tion, we establish an upper bound on the number of resonan
es in a ring of
2bq +λ z ∈ Ω z = 2bq +λ+η η
q

entered at (see (3.4)). For , we write where is a
omplex
number in a domain
entered at 0 and 0 < r < |η|. Let W = supx3∈Cǫ,0|hx3iδkV|. There
M(x)
exists a bounded fun
tion su
h that
V(x) = W(X )hx i 2δ3M(x), δ = δ /2.
3 − 3
⊥ for k
Ω
q
A

ording to the previous se
tion, the resonan
es in
an be identi(cid:28)ed with the points
z ∈Ω d (z) = det (I +T (z))
q θ 2 V,θ
where the determinant vanishes. Using (3.9), we have
d (z) = (I +T (z))
θ 2 V,θ
det
with
(4.1) TV,θ(z) = W21Mθhx3i−θδ3(H0,θ −z)−1W21hx3i−θδ3
where Mθ = M(X⊥,φθ(x3)) and hx3iθ := Uθhx3iUθ−1 = (1+(φθ(x3))2)12.
z > 0
Using spe
tral theorem, for Im , we
an write
(H −z) 1 = p ⊗(H (θ)−z+2bj) 1.
0,θ − j 0, −
(4.2) Xj N k
∈
10 ABDALLAHKHOCHMAN
2bq+λ T (z)
V,θ
In order to study the resonan
es near , we split into two parts:
T (z) = T +T+,
V,θ J−,θ J,θ
T (z) = T T+ = T J > q kT+k<
w1herekTJ−,θ+k< 1Pj≤J j,θ and J,θ hPj>J j,θ for su(cid:30)
iently large su
h that J,θ
8 and J,0 8 (for that weusethe -pseudo-di(cid:27)erential
al
ulusand thespe
traltheorem).
Here,
Tj,θ = MθBj ⊗hx3i−θδ3Rj,θhx3i−θδ3,
Bj = W21pjW21 Rj,θ = (H0, (θ)−z+2bj)−1 hx3i−δ3Rj,0hx3i−δ3
with and k . The operator is of

lass tra
e (see [2℄).
B
j
Further, let us de
ompose the self-adjoint operator into a tra
e-
lass operator whose
ε/2 ε > 0 r
norm is bounded by for some and an operator of (cid:28)nite-rank independent on ,
namely
B = B 1 (B )+B 1 (B ).
j j [0,ε/2] j j ]ε/2,+ [ j
(4.3) ∞
j 6= q
Then, for , we have
T = M B 1 (B )⊗hx i δ3R hx i δ3 +M B 1 (B )⊗hx i δ3R hx i δ3
j,θ θ j [0,ε/2] j 3 −θ j,θ 3 −θ θ j ]ε/2,+ [ j 3 −θ j,θ 3 −θ
∞
= T< +T>.
j,θ j,θ
T p (θ) H (θ)−
q,θ 0,
λ Letusnowpan(aθl)y·z=ethh·,eψteirψm . Dψen=oteUby1ψk thψespe
tralproje
tionontoKer( k
). We have k θ¯ θ with θ θ− and is an eigenfun
tion satisfying
H0, ψ =λψ, kψkL2(R) = 1, ψ = ψ¯ R.
k on
η = z−2bq−λ
Then we have, for
T =M B ⊗hx i δ3R p (θ)hx i δ3 +M B ⊗hx i δ3R (I −p (θ))hx i δ3
q,θ θ q 3 −θ q,θ 3 −θ θ q 3 −θ q,θ 3 −θ
k k
1
=− τ +T˜ ,
q q,θ
η
with τq = MθBq ⊗hx3i−θδ3pk(θ)hx3i−θδ3. We also have
T˜q,θ = MθBq1[0,ε/2](Bq)⊗hx3i−θδ3Rq,θ(I −p (θ))hx3i−θδ3
k
+ MθBq1]ε/2,+ [(Bq)⊗hx3i−θδ3Rq,θ(I −p (θ))hx3i−θδ3
∞ k
= T˜< +T˜>.
q,θ q,θ
A>(z) = T˜> + T>, A<(z) = T˜< + T< + T+.
We denote by q,θ j6=q, j≤J j,θ and q,θ j6=q, j≤J j,θ J,θ
P P
Then, we have
1
T (z) = − τ +A>(z)+A<(z).
V,θ q
(4.4) η
τ rν 1
q −
We de
ompose the operator into a tra
e-
lass operator whose norm is bounded by for
ν > 0
and an operator of (cid:28)nite-rank:
τq = MθBq1[0,rν−1](Bq)⊗hx3i−θδ3p (θ)hx3i−θδ3 +MθBq1]rν−1,+ [(Bq)⊗hx3i−θδ3p (θ)hx3i−θδ3
k ∞ k
= τ +τ .
q,1 q,2
ε ν > 0
If we take su(cid:30)
iently small and su(cid:30)
iently large, we have the two following lemmas