CUBO A Mathematical Journal
Vol.14, No¯ 01, (29–47). March 2012

Spectral shift function for slowly varying perturbation of
periodic Schrödinger operators.

Mouez Dimassi

Univ. Paris 13,

LAGA, (UMR CNRS 7539),

F-93430 Villetaneuse, France,

email: dimassi@math.univ-paris13.fr

and

Maher Zerzeri

Univ. Paris 13,

LAGA, (UMR CNRS 7539),

F-93430 Villetaneuse, France,

email: zerzeri@math.univ-paris13.fr

ABSTRACT

In this paper we study the asymptotic expansion of the spectral shift function for

the slowly varying perturbations of periodic Schrödinger operators. We give a weak

and pointwise asymptotic expansions in powers of h of the derivative of the spectral

shift function corresponding to the pair
(
P(h) = P0 + ϕ(hx),P0 = −∆ + V(x)

)
, where

ϕ(x) ∈ C∞(Rn,R) is a decreasing function, O(|x|−δ) for some δ > n and h is a small
positive parameter. Here the potential V is real, smooth and periodic with respect to

a lattice Γ in Rn. To prove the pointwise asymptotic expansion of the spectral shift
function, we establish a limiting absorption Theorem for P(h).



30 M. Dimassi and M. Zerzeri CUBO
14, 1 (2012)

RESUMEN

En este art́ıculo estudiamos la expansión asintótica de la función shift espectral para

perturbaciones de variación lenta de operadores periódicos de Schrödinger. Propor-

cionamos una expansión débil y puntual en potencias de h de la derivada de la función

shift espectral que corresponde al par
(
P(h) = P0 + ϕ(hx),P0 = −∆ + V(x)

)
, donde

ϕ(x) ∈ C∞(Rn,R) es una función decreciente, O(|x|−δ) para algún δ > n y h un
parámetro positivo pequeño. Aqúı el potencial V es real, suave y periódico con re-

specto a un ret́ıculo Γ en Rn. Para demostrar la expansión asintótica puntual de la
función shift espectral establecemos un teorema de absorción ĺımite para P(h).

Keywords and Phrases: Periodic Schrödinger operator, spectral shift function, asymptotic

expansions, limiting absorption theorem.

2010 AMS Mathematics Subject Classification: 81Q10 (35P20 47A55 47N50 81Q15)

1 Introduction

The aim of this paper is to give an asymptotic expansion of the spectral shift function for the

slowly varying perturbations of periodic Schrödinger operator:

P(h) = P0 + ϕ(hx), h > 0, (1.1)

P0 = −∆x + V(x),

Here V is a real-valued, C∞ function and periodic with respect to a lattice Γ of Rn.
The Hamiltonian P(h) describes the quantum motion of an electron in a crystal placed in

an external field. There are many works devoted to the spectral properties of this model, see

[1, 3, 5, 8, 9, 10, 12, 13, 15, 16, 18, 21, 36].

We assume that ϕ ∈ C∞(Rn; R) and satisfies the following estimate: for all α ∈ Nn, there
exists Cα > 0 such that

|∂αxϕ(x)| ≤ Cα(1 + |x|)
−δ−|α|, ∀x ∈ Rn, with δ > n . (1.2)

The operators P0,P(h) are self-adjoint on H
2(Rn). Under the assumption (1.2) we show in

Theorem 2.2 below that the operator
[
f(P(h))−f(P0)

]
belongs to the trace class for all f ∈ C∞0 (R).

Following the general setup we define the spectral shift function, SSF, ξh(µ) := ξ(µ;P(h),P0)

related to the pair (P(h),P0) by

tr
[
f(P(h)) − f(P0)

]
= −〈ξ′h(·),f(·)〉 =

∫
R
ξh(µ)f

′(µ)dµ, ∀f ∈ C∞0 (R). (1.3)
By this formula ξh is defined modulo a constant but for the analysis of the derivative ξ

′
h(µ) this

is not important. See [22] and [2].



CUBO
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SSF for perturbed periodic Schrödinger operators 31

The SSF may be considered as a generalization of the eigenvalues counting function. The

notion of SSF was first singled out by the outstanding theoretical physicist I-M.Lifshits in his

investigations in the solid state theory, in 1952, see [23]. It was brought into mathematical use in

M-G. Krĕın’s famous paper [22], where the precise statement of the problem was given and explicit

representation of the SSF in term of the perturbation determinant was obtained. The work of M-G.

Krĕın’s on the SSF has been described in detail in [2]. For more details about the interpretation of

SSF we refer to the survey by D. Robert [32] and to the monograph by D-R. Yafaev [42, Chapter

8].

In the case where V = 0, the asymptotic behavior of the SSF of the Schrödinger operator

has been intensively studied in different aspects (see [6, 17, 24, 25, 30, 31, 34] and the references

given there). In the semi-classical regime (i.e. H(h) = −h2∆x + ϕ(x),(h ↘ 0)) the Weyl type
asymptotics of ξh(·) with sharp remainder estimate has been obtained (see [30, 31, 34, 35]). On the
other hand, if an energy µ > 0 is non-trapping for the classical hamiltonian p(x,ζ) = |ζ|2 + ϕ(x)

(i.e. for all (x,ζ) ∈ p−1{µ}, |exptHp(x,ζ)| → ∞ when t → ∞) a complete asymptotic expansion in
powers of h of ξ′h(µ) has been obtained (see [30, 31, 34, 35]). Similar results are well-known for

the SSF at high energy (see [4, 6, 25, 26, 29]).

There are only few works treating the SSF in perturbed periodic Schrödinger operator. See [3],

[10] and also [16]. In [10] the connection between the resonances of P(h) and the SSF associated to

the pair
(
P(h),P0

)
were studied. Under the assumption that ϕ is analytic in some conic complex

neighborhood of the real axis and that P(h) has no resonances in a small complex neighborhood of

some interval I the first author obtained a full asymptotic expansion in powers of h of the derivative

of SSF:

ξ′(µ;h) ∼

∞∑
j=0

bj(µ)h
j−n, h ↘ 0, (1.4)

uniformly with respect to µ ∈ I.

Nevertheless, there is a lot of examples of perturbed periodic Schrödinger operator that the

perturbation ϕ does not satisfies the analyticity assumption.

In this paper, we improve the result of [10] concerning the behavior of the derivative of SSF

by removing the analyticity assumption on the potential ϕ. Our proof is based on a limiting

absorption principle and some arguments due to D. Robert and H. Tamura, see [33] and [35]. For

V 6= 0, the limiting absorption theorem is new (see Theorem 3.11). They are in harmony with
the physical intuition which argues that, when h sufficiently small, the main effect of the periodic

potential V consists in changing the dispersion relation from the free kinetic |k|2 to the modified

kinetic energy λp(k) given by the pth band.

By the method of effective hamiltonian spectral problems of P(h) can be reduced to similar

problem of system of h−pseudodifferential operators (See [9] and also [12]). Using a well-known

results on h−pseudodifferential calculus we get the asymptotic (1.4) in the sense of distributions.

If the values of the principal term of the effective hamiltonian are contained in non-trapping energy

region we prove a limiting absorption principle for P(h) (Theorem 3.11) and we get a pointwise



32 M. Dimassi and M. Zerzeri CUBO
14, 1 (2012)

asymptotic expansion for the derivative of the spectral shift function.

The paper is organized as follows: In the next section, we recall some well-known results con-

cerning the spectra of a periodic Schrödinger operator (subsection 2.1) and we state the assump-

tions and the results precisely (subsection 2.2). We give an outline of the proofs in subsection 2.3.

Section 3 is devoted to the proofs. Roughly, we introduce a class of symbols and the correspond-

ing h-Weyl operators (subsection 3.1). In the subsection 3.2 we recall the effective Hamiltonian

method and we give a representation of the derivative of the spectral shift function, denoted by

ζ′h(·). The proof of the weak asymptotic expansion of ξ
′
h is given in subsection 3.3. We establish

a limiting absorption principle for P(h) in the subsection 3.4 At last, the pointwise asymptotic

expansion of ξ′h is proved in subsection 3.5.

2 Statements

2.1 Preliminaries

Let Γ =
n
⊕
i=1

Zei be a lattice generated by some basis (e1,e2, · · · ,en) of Rn. The dual lattice Γ∗ is
given by Γ∗ := {γ∗ ∈ Rn; 〈γ|γ∗〉 ∈ 2πZ, ∀γ ∈ Γ}. A fundamental domain of Γ (resp. Γ∗) is denoted
by E (resp. E∗). If we identify opposite edges of E (resp. E∗) then it becomes a flat torus denoted

by T = Rn/Γ (resp. T∗ = Rn/Γ∗).

Let V be a real-valued potential, C∞ and Γ−periodic. For k ∈ Rn, we define the operator
P(k) on L2(T) by P(k) := (Dy +k)2 +V(y). The operator P(k) is a semi-bounded self-adjoint with
k-independent domain H2(T). Since the resolvent of P(k) is compact, P(k) has a complete set of
(normalized) eigenfunctions Φn(·,k) ∈ H2(T), n ∈ N, called Bloch functions. The corresponding
eigenvalues accumulate at infinity and we enumerate them according to their multiplicities, λ1(k) ≤
λ2(k) ≤ · · · . The operator P(k) satisfies the identity e−iy·γ

∗
P(k)eiy·γ

∗
= P(k + γ∗), ∀γ∗ ∈ Γ∗,

then for every p ≥ 1, the function k 7→ λp(k) is Γ∗−periodic.
Ordinary perturbation theory shows that λp(k) are continuous functions of k for any fixed

p, and λp(k) is even an analytic function of k near any point k0 ∈ T∗ where λp(k0) is a simple
eigenvalue of P(k0). The function λp(k) is called the band function and the closed intervals Λp :=

λp(T∗) are called bands. See [27], [39] and also [37, 38].

Consider the self-adjoint operator on L2(Rn) with domain H2(Rn):

P0 = −∆x + V(x), where ∆x =
n∑

j=1

∂2

∂x2
j

. (2.1)

The spectrum of P0 is absolutely continuous (see [41]) and consists of the bands Λp, p =

1,2, · · · . Indeed, σ(P0) = σac(P0) = ∪
p≥1

Λp. See also [40].

Definition 2.1. Let µ ∈ R and F(µ) =
{
k ∈ T∗; µ ∈ σ

(
P(k)

)}
the corresponding Fermi-surface.



CUBO
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SSF for perturbed periodic Schrödinger operators 33

a) We will say that µ ∈ σ(P0) is a simple energy level if and only if µ is a simple eigenvalue of
P(k), for every k ∈ F(µ).

b) Assume that µ is a simple energy level of P0 and let λ(k) be the unique eigenvalue defined on

a neighborhood of F(µ) such that λ(k) = µ, for all k ∈ F(µ). We say that µ is a non-critical
energy of P0 if dkλ(k) 6= 0 for all k ∈ F(µ).

Note that in one dimension case F(µ) is just a finite set of points.

Now, let us recall some well-known facts about the density of states associated with P0, see

[40]. The density of states measure ρ is defined as follows:

ρ(µ) :=
1

(2π)n

∑
p≥1

∫
{k∈E∗; λp(k)≤µ}

dk. (2.2)

Since the spectrum of P0 is absolutely continuous, the measure ρ is absolutely continuous

with respect to the Lebesgue measure dµ. Therefore the density of states, dρ
dE

(E), of P0 is locally

integrable.

2.2 Results

We now consider the perturbed periodic Schrödinger operator:

P(h) := P0 + ϕ(hx), h ↘ 0, (2.3)
where ϕ ∈ C∞(Rn; R) and satisfies:
(A1) There exists δ > 0 such that ∀α ∈ Nn, ∃Cα > 0 s.t.∣∣∂αxϕ(x)∣∣ ≤ Cα(1 + |x|)−δ−|α| uniformly on x ∈ Rn.
The operator P(h) is self-adjoint, semi-bounded on L2(Rn) with domain H2(Rn).
The assumption (A1) and the perturbation theory (Weyl theorem) give:

σess
(
P(h)

)
= σess(P0) = σ(P0) =

⋃
p≥1

Λp. (2.4)

Recall that σess(A), the essential spectrum of A, is defined by σess(A) = σ(A) \ σdisc(A), where

σdisc(A) is the set of isolated eigenvalues of A with finite multiplicity. Here A is an unbounded

operator on a Hilbert space.

Our first theorem in this section concerns the weak asymptotic of ζ′h(µ). Let I =]a,b[⊂ R.

Theorem 2.2 (Weak asymptotic). Assume (A1) with δ > n . For f ∈ C∞0 (I), the operator[
f(P(h)) − f(P0)

]
is of trace class and

tr
[
f(P(h)) − f(P0)

]
∼ h−n

+∞∑
j=0

aj(f)h
j, when h ↘ 0, (2.5)



34 M. Dimassi and M. Zerzeri CUBO
14, 1 (2012)

with

a0(f) = (2π)
−n

∑
p≥1

∫
Rnx

∫
E∗

[
f
(
λp(k) + ϕ(x)

)
− f
(
λp(k)

)]
dkdx. (2.6)

The coefficients f → aj(f) are distributions of finite order ≤ j + 1. Moreover, if µ is a non-critical
energy of P0 for all µ ∈ I, then aj(f) = −〈γj(·),f〉, for all f ∈ C∞0 (I). Here γj(µ) are smooth
functions of µ ∈ I. In particular,

γ0(µ) =
d

dµ

[∫
Rnx

{
ρ
(
µ
)
− ρ
(
µ − ϕ(x)

)}
dx

]
. (2.7)

The proof of Theorem 2.2 is contained in subsection 3.3.

Let [a,b] ⊂ R. Assume that:

(A2) for all µ ∈ [a,b], µ is a non-critical energy of P0.

For all µ ∈ [a,b], let λ(k) be the unique eigenvalue defined on a neighborhood of F(µ) such that
λ(k) = µ. We assume that for all (k,r) ∈ T∗ × Rn such that µ = λ(k) + ϕ(r) ∈ σ(P0) ∩ [a,b], µ is
a simple energy level, and that:

(A3) |∇λ(k)|2 − r∇ϕ(r)∆λ(k) > 0, for all (k,r) s.t. λ(k) + ϕ(r) ∈ [a,b].

Remark. Note that the assumption (A2) is fulfilled in the bottom of the spectrum of P0. More-

over, assuming (A2) the hypothesis (A3) is satisfied if ‖ϕ‖∞ +‖x∇ϕ‖∞ << 1, (see [27], [37, 38]).
Our main result concerning the derivative of the spectral shift function is the following.

Theorem 2.3 (Pointwise asymptotic). Assume (A2), (A3) and (A1) with δ > n . Then the

following asymptotic expansion holds:

ζ′h(µ) ∼ h
−n

∑
j≥0

γj(µ)h
j as h ↘ 0, (2.8)

uniformly for µ ∈ [a,b].
The coefficients

(
γj(µ)

)
j≥0 are given in Theorem 2.2. Furthermore, this expansion has derivate in

µ to any order.

Remark 2.4. Theorems 2.2 and 2.3 still true also in the case when the potential ϕ(x) depend on

h, i.e. ϕ(x,h) = ϕ(x) + hϕ1(x) + h
2ϕ2(x) · · · in Sδ(1). See the next section for the definition of

Sδ(1). (subsection 3.1).

2.3 Outline of the proofs

Let Q1(h) = Q
w
1 (x,hDx), Q0(h) = Q

w
0 (x,hDx) be two h-pseudodifferential operators such that

Qj(h) = Q
∗
j (h), (j = 0,1), and

(Q1(h) + i)
−1 − (Q0(h) + i)

−1



CUBO
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SSF for perturbed periodic Schrödinger operators 35

is an operator of trace class. In this case, Theorem 2.2 is well-known see [28] and the references

therein. On the other hand, in the case of non-trapping geometrie, the asymptotic follows from

the results of Robert-Tamura (see [33, 34, 35] and also [31]). The main ingredient in the Robert-

Tamura method is the limiting absorption theorem and the construction of a long-time parametrix

for the time-dependent equation(
hDt − Q

w
j (x,hDx)

)
Uj(t) = 0, Uj(0) = I.

In our case P(h) = −∆ + V(x) + ϕ(hx) is not an h-pseudodifferential operator. In fact, when

(h ↘ 0), there are two spatial scales in equation (1.1). The first one of the order of the linear
dimension γ of the periodicity cell and the second one of order γ

h
on which the perturbation of the

potential varies appreciably. To remedy this we reduce the study of P(h), to the one of a system

of h-pseudodifferential operators. More precisely, following [7, 9, 12, 15, 18], we can reduce the

spectral study of P(h) near any fixed energy z to the study of a finite system of h-pseudodifferential

operators, E−+(z,h), acting on L
2(T∗n; CN). In general, for the reduced problem, the dependence

on the spectral parameter is non-linear. However, in the case of simple band (see assumption

(A2)) we show that

E−+(z,h) = z −
(
λ(k) + ϕ(r) + hK1(k,r) + h

2K2(k,r;z,h)
)
,

where K1 ∈ Sδ+1(T∗ × Rn) and K2(·;z,h) ∈ Sδ+2(T∗ × Rn), holomorphic with respect to z in a
small complex neighborhood Ω of a bounded interval I. (See 3.24). Now, considering s ∈ Ω∩ R as
a parameter and assuming (A3), we can apply the Robert-Tamura approach to the hamiltonian

Bs(k,−hDk;h) := λ(k) + ϕ(−hDk) + hK
w
1 (k,−hDk) + h

2Gw(k,−hDk;s,h) where G satisfies the

same properties as K2, and we obtain the theorem 2.3. Here we use the following crucial argument:

the assumption (A3) implies that the interval I is a non-trapping region of the classical hamiltonian

associated to Bs for all s in the compact set Ω∩R. In fact, r·Ki(k,r) ∈ Sδ(T∗×Rn) ⊂ S0(T∗×Rn),
(i = 1,2) then the corresponding operators are bounded uniformly for s ∈ Ω ∩ R and moreover,
the principal symbol of Bs does not depend on s.

3 Proofs

3.1 Definitions and notations

Let H be a Hilbert space. The scalar product in H will be denoted by 〈·, ·〉. The set of linear
bounded operators from H1 to H2 is denoted by L(H1,H2).
For (m,N) ∈ R × N we denote by Sm(T∗ × Rn; MN(C)) the space of P ∈ C∞(R2nk,r; MN(C)),
Γ∗-periodic with respect to k, such that for all α and β in Nn there exists Cα,β > 0 such that

‖∂αr ∂
β
k
P(k,r)‖MN(C) ≤ Cα,β〈r〉

−m−|α|, 〈r〉 =
(
1 + |r|2

)1
2 , (3.1)

where MN(C) is the set of N×N-matrices. In the special case when N = 1 (i.e., P is real valued),
we will write Sm(T∗ × Rn) instead of Sm(T∗ × Rn; M1(C)).



36 M. Dimassi and M. Zerzeri CUBO
14, 1 (2012)

If P depends on a semi-classical parameter h ∈]0,h0] and possibly on other parameters as well,
we require (3.1) to hold uniformly with respect to these parameters. For h dependent symbols, we

say that P(k,r;h) has an asymptotic expansion in powers of h, and we write

P(k,r;h) ∼

∞∑
j=0

Pj(k,r)h
j,

if for every N ∈ N, h−(N+1)
(
P −

N∑
j=0

Pjh
j
)
∈ Sm

(
T∗ × Rn; MN(C)

)
.

For P ∈ Sm(T∗ × Rn; MN(C)), the h-Weyl operator P = Pw(k,hDk;h) = Opwh (P) is defined
by:

Pw(k,hDk;h)u(k) = (2πh)
−n

∫ ∫
e

i
h
(k−y)rP(

k + y

2
,r;h)u(y)dydr. Here Dk =

1

i

∂

∂k
.

3.2 Effective Hamiltonian

In this subsection, we recall the effective Hamiltonian method. More precisely, we will construct a

suitable auxiliary (so-called Grushin) problem associated with the operator
(
P(h) − z

)
for z in a

small complex neighborhood of I, where I = [a,b] ⊂ R is some bounded interval. The reader can
find more details and the proofs of the results of this subsection in [20] (see also [11, 12, 15]). For

the reader convenience, let us point out the main change in our situation and fix the notations.

Denote by TΓ the distribution in S ′(R2n) defined by TΓ (x,y) =
1

vol(E)hn

∑
β∗∈Γ∗

ei(x−hy)
β∗
h .

We recall that E is a fundamental domain of Γ.

For m ∈ N, put Lm := {u(x)TΓ (x,y); ∂αxu ∈ L
2(Rn), ∀α, |α| ≤ m}.

It was shown in [11, Chapter 13, Proposition 13.5], that the operator P(h) acting on L2(Rn)
with domain H2(Rn) is unitary equivalent to

P1(h) :=
(
Dy + hDx

)2
+ V(y) + ϕ(x), (3.2)

acting on L0 with domain L2, and the following proposition holds.

Proposition 3.1. Assume (A1). There exist N ∈ N, a complex neighborhood Ω of I, and a
bounded operator R+ in L

(
L0;L2(T∗; CN)

)
such that for all z ∈ Ω and 0 < h < h0 small enough,

the operator

P1(z,h) :=

(
P1(h) − z R∗+
R+ 0

)
: L2 × L2(T∗; CN) → L0 × L2(T∗; CN), (3.3)

is bijective with bounded two-sided inverse

E1(z,h) :=

(
E1(z,h) E1,+(z,h)

E1,−(z,h) E1,−+(z,h)

)
. (3.4)



CUBO
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SSF for perturbed periodic Schrödinger operators 37

Here E1,−+ := E
w
1,−+(k,−hDk;z,h) is an h−pseudodifferential operator with symbol

E1,−+(k,r;z,h) ∼
∑
l≥0

El1,−+(k,r;z)h
l, ∀0 < h < h0, (3.5)

in S0
(
T∗ × Rn; L(CN,CN)

)
.

Remark 3.2. (1) We denote by P0(z,h) and E0(z,h) :=

(
E0(z,h) E0,+(z,h)

E0,−(z,h) E0,−+(z,h)

)
the opera-

tors given by Proposition 3.1 when ϕ = 0.

(2) Note that, R+ depends only on the non-perturbed periodic Schrödinger operator P0. See [15,

Proposition 2.1] and [11, Chapter 13]. Therefore, we may take the same R+ for P1(z,h) and
P0(z,h).

The following well-known formulas are a consequence of Proposition 3.1 (see also [20]), for

j = 0,1. (
Pj(h) − z

)−1
= Ej(z,h) − Ej,+(z,h)Ej,−+(z,h)

−1Ej,−(z,h), (3.6)

Ej,−+(z,h)
−1 = −R+

(
Pj(h) − z

)−1
R∗+ , (3.7)

and

∂zEj,−+(z,h) = Ej,−(z,h)Ej,+(z,h). (3.8)

Here P0(h) :=
(
Dy + hDx

)2
+ V(y).

We observe that Pj(z,h)∗ = Pj(z,h), which implies that Ej(z,h)∗ = Ej(z,h). From this, we
deduce the following identity:

Ej,−+(z,h)
∗ = Ej,−+(z,h), j = 0,1. (3.9)

In the following, we write [aj]
1
j=0 = a1 − a0.

Lemma 3.3. We have [
Ej,+(z,h)

]1
j=0

= E1(z,h)ϕ(r)E0,+(z,h), (3.10)

[
Ej,−(z,h)

]1
j=0

= E0,−(z,h)ϕ(r)E1(z,h), (3.11)

and [
Ej,−+(z,h)

]1
j=0

= E1,−(z,h)ϕ(r)E0,+(z,h). (3.12)

In particular, if (A1) is satisfied then

[
Ej,−+

(
k,r;z,h

)]1
j=0

∈ Sδ
(
T∗k × R

n
r ; MN(C

N)
)
. (3.13)



38 M. Dimassi and M. Zerzeri CUBO
14, 1 (2012)

Proof. Identities (3.10)-(3.12) follow from the first resolvent equation[
Ej(z,h)

]1
j=0

= E1(z,h)
[
P0(z,h) − P1(z,h)

]
E0(z,h)

= −E0(z,h)
[
P1(z,h) − P0(z,h)

]
E1(z,h)

and the fact that
[
Pj(z,h)

]1
j=0

=

(
ϕ(r) 0

0 0

)
.

Formula (3.13) is a simple consequence of (3.12) and standard h-pseudodifferential calculus.

Lemma 3.4. Assume (A1) with δ > n , the operator

ϕ(r)E0,+(z,h) : L
2(T∗; CN) → L2(Rn), (3.14)

and

E0,−(z,h)ϕ(r) : L
2(Rn) → L2(T∗; CN), (3.15)

are of trace class.

Proof. Since
(
E0,−(z,h)ϕ(r)

)∗
= ϕ(r)E0,+(z,h) it suffice to prove 3.14. Without any loss of

generality, we may assume that N = 1.

Consider the operator A =
(
Id − h2∆T∗

)− δ
2 on L2(T∗; C). Set B = ϕ(r)E0,+(z,h), C = B∗B

and D = A−1CA−1.

Since ϕ ∈ Sδ(R2n) and E0,+(k,r;z,h) ∈ S0, a standard result of h-pseudodifferential calculus
shows that D ∈ S0(T∗k × R

n
r ). Therefore, D extends to a bounded operator from L

2(T∗; C) into
L2(T∗; C), see [11, Chapter 13]. Combining this with the fact that C is positif, we get:

0 ≤ C = ADA ≤ ‖D‖A2,

which implies

0 ≤ C
1
2 ≤

√
‖D‖A.

Since δ > n then A : L2(T∗; C) → L2(T∗; C) is of trace class and the lemma follows from the above
inequality.

Remark 3.5. Notice that if P ∈ Sδ(T∗ × Rn; MN(C)) with δ > n, then the operator Pw(k,hDk)
is a trace class, see [11].

Proposition 3.6. Assume (A1) with δ > n . For z ∈ Ω such that =(z) 6= 0, the operator

[
Ej,+(z,h)Ej,−+(z,h)

−1Ej,−(z,h)
]1
j=0



CUBO
14, 1 (2012)

SSF for perturbed periodic Schrödinger operators 39

is of trace class from L2(Rn) to L2(Rn) and

tr
([
Ej,+(z,h)Ej,−+(z,h)

−1Ej,−(z,h)
]1
j=0

)
= tr

([
Ej,−+(z,h)

−1∂zEj,−+(z,h)
]1
j=0

)
. (3.16)

Here the operator in the right member of (3.16) is defined on L2(T∗; CN).

Proof. Let z ∈ Ω such that =(z) 6= 0, we have the following identity:

[
Ej,+(z,h)Ej,−+(z,h)

−1Ej,−(z,h)
]1
j=0

= (3.17)[(
[Ej,+(z,h)]

1
j=0

)
E1,−+(z,h)

−1E1,−(z,h)
]
+[

E0,+(z,h)E0,−+(z,h)
−1
(
[Ej,−(z,h)]

1
j=0

)]
−[

E0,+(z,h)E1,−+(z,h)
−1
(
[Ej,−+(z,h)]

1
j=0

)
E0,−+(z,h)

−1E1,−(z,h)
]
.

According to Lemmas 3.3 and 3.4, all the term of the right member in the last equality are of trace

class. Using the cyclicity of the trace and identity (3.8), we obtain the proposition.

Using again the cyclicity of the trace in (3.17) and the identity (3.8) we obtain

tr
([
Ej,−+(z,h)

−1∂zEj,−+(z,h)
]1
j=0

)
= tr

([
∂zEj,−+(z,h)Ej,−+(z,h)

−1
]1
j=0

)
. (3.18)

The main result in this subsection is

Proposition 3.7. Assume (A1) with δ > n . Let ψ ∈ C∞0 (R) and let ψ̃ be an almost analytic
extension of ψ. Then the operator [ψ(P(h)) − ψ(P0)] is of trace class as an operator from L

2(Rn)
to L2(Rn) and

tr[ψ(P(h)) − ψ(P0)] = tr[ψ(P1(h)) − ψ(P0(h))] = (3.19)

−
1

π

∫
C
∂ψ̃(z)tr

([
Ej,−+(z)

−1∂zEj,−+
]1
j=0

)
L(dz).

Here ∂ =
∂

∂z
and L(dz) = dxdy denotes the Lebesgue measure on C.

Recall that ψ̃ ∈ C∞0 (C) is an almost analytic extension of ψ, i.e. ψ̃|R = ψ and ∂ψ̃ = O(|=(z)|N)
for all N ∈ N. We refer to [11] for the existence of ψ̃.

Proof. By Helffer-Sjöstrand formula (see [19]), we have

ψ(P1(h)) − ψ(P0(h)) = −
1

π

∫
C
∂ψ̃(z)

[
(z − P1(h))−1 − (z − P0(h))−1

]
L(dz).



40 M. Dimassi and M. Zerzeri CUBO
14, 1 (2012)

Combining this with (3.6), we obtain

ψ(P1(h)) − ψ(P0(h)) =
1

π

∫
C
∂ψ̃(z)

[
Ej(z,h)]

1
j=0 L(dz) (3.20)

−
1

π

∫
C
∂ψ̃(z)

[
Ej,+(z,h)Ej,−+(z,h)

−1Ej,−(z,h)
]1
j=0

L(dz).

Since Ej(z,h), j = 0,1 is holomorphic in a neighborhood of supp(ψ̃), the first term in the right

member of (3.20) vanishes. Consequently,

ψ(P1(h)) − ψ(P0(h)) = −
1

π

∫
C
∂ψ̃(z)

[
Ej,+(z,h)Ej,−+(z,h)

−1Ej,−(z,h)
]1
j=0

L(dz).

Using Proposition 3.6, we conclude that [ψ(P1(h))−ψ(P0(h))] is of trace class and applying (3.16),
we obtain the second equality of (3.19). The first equality follows from the fact that P1(h) (resp.
P0(h)) is unitary equivalent to P(h) (resp. P0).

Now, we recall a representation of the derivative of the spectral shift function in term of the

effective Hamiltonian Ej,−+(z,h) (see [10, Lemma 1]).

Let I ⊂ R be some bounded interval and Ω be the complex neighborhood of I given by the
proposition 3.1. Put Ω± = Ω ∩ {z ∈ C; ±=(z) > 0}.
We introduce the functions e±(z) = tr

([
Ej,−+(z,h)

−1∂zEj,−+(z,h)
]1
j=0

)
, for z ∈ Ω±.

Since Ej,−+(z,h) is holomorphic on z, we deduce from (3.9) that

∂zEj,−+(z,h)
∗ = ∂zEj,−+(z,h).

Using the fact that tr(A) = tr(A∗) with (3.9), (3.18) and the above equality we obtain

e+(z) = e−(z), for all z ∈ Ω+. (3.21)

Lemma 3.8. [10, Lemma 1] Assume (A1) with δ > n . In D ′(I), we have

ζ′h(µ) = lim
�↘0

1

2πi

[
e+(µ + i�) − e+(µ + i�)

]
. (3.22)

For the reader convenience we give the proof of the lemma.

Proof. Let ψ ∈ C∞0 (I). In the previous proposition, we proved that
− 〈ζ′h(·),ψ〉 = tr[ψ(P(h)) − ψ(P0)] = −

1

π

∫
C
∂ψ̃(z)tr

([
Ej,−+(z)

−1∂zEj,−+
]1
j=0

)
L(dz).

Since e±(z) = O
(
h−n|=(z)|−2

)
and ∂ψ̃(z) = O

(
|=(z)|2

)
, the integral in the identity above

converge. Thus the r.h.s. of the previous equality can be written as

− 〈ζ′h(·),ψ〉 = lim
�↘0 −

1

π

[∫
=(z)>0

∂ψ̃(z)e+(z + i�)L(dz) +

∫
=(z)<0

∂ψ̃(z)e−(z − i�)L(dz)
]
.



CUBO
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SSF for perturbed periodic Schrödinger operators 41

Since e±(z ± i�) is holomorphic in Ω±, Green’s formula then gives

− 〈ζ′h(·),ψ〉 = lim
�↘0

i

2π

∫
R
ψ(µ)

[
e+(µ + i�) − e−(µ − i�)

]
dµ. (3.23)

Thus Lemma 3.8 follows from (3.21) and (3.23).

In the following we will use the result concerning the expression of the effective hamiltonian

Ej,−+(z,h), j = 0,1 given in [7] (see also [10]). In fact, under the assumptions (A1)-(A2), the two

leading terms of the symbol E1,−+(k,r;z,h) are computed in [7, section 4, formulas (4.5)-(4.7)], it

was shown that:

E1,−+(k,r;z,h) = z −
(
λ(k) + ϕ(r) + hK1(k,r) + h

2K2(k,r;z,h)
)
, (3.24)

where K1 ∈ Sδ+1(T∗ × Rn) and K2(·;z,h) ∈ Sδ+2(T∗ × Rn), holomorphic with respect to z in Ω.
Note that

E0,−+(k,r;z) = z − λ(k), k ∈ T∗, z ∈ Ω. (3.25)

From now on, we consider the h-pseudodifferential operator Hw(k,−hDk;h) with the following

symbol:

H(k,r;h) = λ(k) + ϕ(r) + hK1(k,r). (3.26)

Remark that this operator is z-independent.

Corollary 3.9. Under Assumptions (A1) with δ > n and (A2), there exists G(k,r;z,h) ∼∞∑
j=0

gj(k,r;z)h
j in Sδ+2(T∗ × Rn) such that for µ ∈ I and h small enough, we have:

ζ′h(µ) = lim
�↘0

1

2πi

[
tr
(
(z − Bµ)

−1 −
(
z − λ(k)

)−1)]z=µ+i�
z=µ−i�

, (3.27)

where Bµ := H
w(k,−hDk;h) + h

2Gw(k,−hDk,µ;h). Here H(k,r;h) is given by 3.26.

Proof. Identity (3.24) gives ∂zE1,−+(k,r;z,h) = 1+h
2∂zK2(k,r;z,h) and since ∂zK2 ∈ Sδ+2(T∗ ×

Rn) ⊂ S0(T∗ × Rn) it follows from the Calderon-Vaillancourt’s theorem and the Beal’s character-
ization (see [11]) that the corresponding operator ∂zE1,−+(z,h) is invertible for h small enough

and his inverse is given by [
∂zE1,−+(z)

]−1
= I + h2R(z),

where R(z) is an h-pseudodifferential operator with symbol satisfying the same properties as K2.
Combining this with (3.24) and using the composition formula of h-pseudodifferential operators

we see that there exists G ∼
∑∞

j=0 gj(k,r;z)h
j in Sδ+2(T∗ × Rn) such that(

∂zE1,−+(z,h)
)−1

E1,−+(z,h) = z − H
w(k,−hDk;h) + h

2Gw(k,−hDk;z,h),

which together with (3.25), Lemma 3.8 and the holomorphy of G on z give the corollary.



42 M. Dimassi and M. Zerzeri CUBO
14, 1 (2012)

3.3 Proof of the weak asymptotic expansion of ξ′h(·)

Let I =]a,b[∈ R and f ∈ C∞0 (I). The proof of Theorem 2.2 is a simple consequence of Proposition
3.7 (with ψ = f) and symbolic calculus. Here, we only give an outline of the proof. For the

details, we refer to [7]. Fix � in ]0, 1
2
[. The integral (3.19) over {z ∈ C; |=z| ≤ h�} is O(h∞), since

∂f̃(z) = O(|=z|∞) and ∥∥∥[Ej,−+(z)−1∂zEj,−+(z)]1
j=0

∥∥∥
tr
= O

(
|=z|−1

)
.

On the other hand,
[
Ej,−+(z,h)

−1∂zEj,−+(z,h)
]1
j=0

has an asymptotic expansion in powers

of h uniformly for z in {z ∈ suppf̃; |=(z)| ≥ h�} (see [7]). Therefore, as in [11, Theorem 13.28], we
have

−
1

π

∫
C
∂f̃(z)tr

([
Ej,−+(z)

−1∂zEj,−+(z)
]1
j=0

)
L(dz) ∼ h−n

∑
j≥0

ajh
j, (h ↘ 0)

with

a0 = (2π)
−n

∑
p≥1

∫
Rnx

(∫
E∗

[
f
(
λp(k) + ϕ(x)

)
− f
(
λp(k)

)]
dk
)
dx.

Note that, the sum in the last equality is finite, since lim
p→+∞ λp(k) = +∞ and ϕ is bounded.

The coefficient aj is a finite sum of term of the form
∫ ∫
cl(x,k)f

(l)
(
b(x,k)

)
dxdk, where cl

depends on ϕ and their derivatives and b(x,k) ∈
{
λp(k),λp(k) + ϕ(x)

}
, which complete the first

part of the theorem 2.2. See [11, Chapter 8, Identity (8.16)].

If µ is a non-critical energy of P0 for all µ ∈ I. Then d
(
λp(k)

)
6= 0 and d

(
λp(k) + ϕ(x)

)
6= 0

for all k ∈ F(µ). We recall that F(µ) is the Fermi surface. Therefore, aj(f) = −〈γj(·),f〉, for all f ∈
C∞0 (I) and γj(µ) are smooth functions of µ ∈ I, in particular, γ0(µ) = ddµ

[∫
Rnx
ρ
(
µ
)
− ρ
(
µ − ϕ(x)

)
dx
]
,

which complete the proof of the theorem 2.2. �

3.4 Limiting absorption Theorem

In this subsection we establish a limiting absorption principle for the operator P(h), see Theorem

3.11 below. We start by the following lemma, let [a,b] ∈ R and Ω given by Proposition 3.1.

Lemma 3.10. Assume that (A1), (A2) and (A3) are satisfied on [a,b]. Then, for l ∈ N∗, there
exists h0(l) > 0 small enough such that for all 0 < h < h0(l):∥∥ < hDk >−α Ej,−+(k,−hDk,µ ± i0;h)−l < hDk >−α ∥∥ = O(h−l), (j = 0,1), (3.28)
for all α > l− 1

2
and uniformly for µ ∈ [a,b], where ‖·‖ denotes the operator norm when considered

as an operator from L2(T∗k) into itself.

Proof. Recall that Ω is a small complex neighborhood of [a,b] such that the proposition 3.1 holds.

For s ∈ Ω∩ R, we consider the h-pseudodifferential operator P̃s(k,−hDk;h) := Hw(k,−hDk;h) +



CUBO
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SSF for perturbed periodic Schrödinger operators 43

h2Kw2 ((k,−hDk;s,h) with the following symbol:

P̃s(k,r;h) = H(k,r;h) + h
2K2(k,r;s,h),

where H(k,r;h) given by (3.26) and K2(·;z,h) ∈ Sδ+2(T∗ × Rn), holomorphic with respect to z in
Ω. Put A = Opwh

(
− ∇λ(k) · r

)
. Since K2 ∈ Sδ+2(T∗ × Rn) ⊂ S0(T∗ × Rn), it follows from the

h-pseudodifferential calculus and the Calderon-Vaillancourt’s theorem that[
P̃s,A

]
= Opwh

(∣∣∇kλ(k)∣∣2 − ( − r∇rϕ(−r)) · ∆λ(k)) + O(h),
in L(L2(T∗)), uniformly for s ∈ Ω ∩ R. Here [·, ·] denotes the commutator.

The assumption (A3) and the Gärding inequality (see [11, chapter 8]) imply that for f ∈
C∞0 (Ω ∩ R) there exists C > 0 such that

f(P̃s)
[
P̃s,A

]
f(P̃s) ≥ Cf(P̃s)2 for h small enough and uniformly for s ∈ Ω ∩ R.

Now applying [14, Theorem 1], we get, for all l ∈ N∗,∥∥ < hDk >−α (µ ± i0 − P̃s)−l < hDk >−α ∥∥ = O(h−l), for all α > l − 1
2
, (3.29)

uniformly for µ ∈ [a,b] and s ∈ Ω ∩ R. Take s = µ in (3.29) we obtain the estimation (3.28) for
j = 1.

Theorem 3.11 (Limiting absorption Theorem). With the same assumptions as Lemma 3.10. One

has, for l ∈ N∗, there exists h0(l) > 0 small enough such that for all 0 < h < h0(l):∥∥ < hx >−α (P(h) − µ ± i0)−l < hx >−α ∥∥
L
(
L2(Rnx )

) = O(h−l), (3.30)
for all α > l − 1

2
and uniformly for µ ∈ [a,b].

Proof. Recall that the operator P(h) acting on L2(Rn) with domain H2(Rn) is unitary equivalent
to

P1(h) :=
(
Dy + hDx

)2
+ V(y) + ϕ(x),

acting on L0 with domain L2, where Lm := {u(x)TΓ (x,y); ∂αxu ∈ L
2(Rn), ∀α, |α| ≤ m} for m ∈ N

and TΓ is a distribution in S ′(R2n) defined by

TΓ (x,y) =
1

vol(E)hn

∑
β∗∈Γ∗

ei(x−hy)
β∗
h .

Here E is a fundamental domain of Γ. Then we will prove (3.30) for P1(h).

It follows from identity (3.6) that:

〈hy〉−α
(
P1(h) − z

)−1〈hy〉−α = 〈hy〉−αE1(z,h)〈hy〉−α − 〈hy〉−αE1,+(z,h)〈hDk〉α·
· 〈hDk〉−αE1,−+(z,h)−1〈hDk〉−α · 〈hDk〉αE1,−(z,h)〈hy〉−α.



44 M. Dimassi and M. Zerzeri CUBO
14, 1 (2012)

Since E1(z,h) is holomorphic then the first term of the r.h.s is bounded. On the other hand, as in

Proposition 3.1, we prove that

(
〈hy〉−α

(
P1(h) − z

)
〈hy〉α 〈hy〉−αR∗+〈hDk〉α

〈hDk〉−αR+〈hy〉α 0

)
=

(
〈hy〉−α 0
0 〈hDk〉−α

)(
P1(h) − z R∗+
R+ 0

)(
〈hy〉α 0
0 〈hDk〉α

)

is well-defined as a bounded operator from L2 × L2(T∗; CN) to L0 × L2(T∗; CN) and is bijective
with bounded two-sided inverse given by:

(
〈hy〉−αE1(z,h)〈hy〉α 〈hy〉−αE1,+(z,h)〈hDk〉α

〈hDk〉−αE1,−(z,h)〈hy〉α 〈hDk〉−αE1,−+(z,h)〈hDk〉α

)
=

(
〈hy〉−α 0
0 〈hDk〉−α

)(
E1(z,h) E1,+(z,h)

E1,−(z,h) E1,−+(z,h)

)(
〈hy〉α 0
0 〈hDk〉α

)
.

Then 〈hy〉−αE1,+(z,h)〈hDk〉α and 〈hDk〉−αE1,−(z,h)〈hy〉α are well-defined and bounded. There-
fore combining this with Lemma 3.10 we obtain Theorem 3.11 for l = 1. With the same arguments

we obtain the result for l ≥ 2.

Remark 3.12. A simple consequence of Theorem 3.11 is that P(h) has no embedded eigenvalues

in [a,b].

3.5 Proof of the pointwise asymptotic expansion of ξ′h(·)

Recall that Ω is a small complex neighborhood of [a,b] such that the proposition 3.1 holds. For

s ∈ Ω ∩ R, we consider the h-pseudodifferential operator Bs(k,−hDk;h) := Hw(k,−hDk;h) +
h2Gw(k,−hDk;s,h) with the following symbol:

bs(k,r;h) = H(k,r;h) + h
2G(k,r;s,h),

where H(k,r;h) given by (3.26) and G(·;z,h) ∈ Sδ+2(T∗ × Rn), holomorphic with respect to z in
Ω (see Corollary 3.9).

Clearly, the principal symbol of Bs (i.e. b
0
s = λ(k) + ϕ(r)) is independent on s. Moreover,

the assumption (A3) implies that [a,b] is a non-trapping region for the classical hamiltonian bs.

Now, by constructing a long-time parametrix for the time-dependent equation

(
hDt − Bs

)
U(t) = 0, U(0) = I,



CUBO
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SSF for perturbed periodic Schrödinger operators 45

we can apply the Robert-Tamura method [33, 34, 35] (see also [31, Remark 6.1]) to prove that[
tr
(
(z − Bs)

−1 − (z − λ(k))−1
)]z=µ+i0

z=µ−i0
,

has a complete asymptotic expansion in powers of h uniformly for µ ∈ [a,b] and s ∈ Ω ∩ R.
Remembering (3.27) and take s = µ ∈ [a,b] ⊂ Ω we obtain (2.8).

Received: February 2011. Revised: March 2011.

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