A TRACE FORMULA AND APPLICATION TO STARK HAMILTONIAN WITH NONCONSTANT MAGNETIC FIELDS

A TRACE FORMULA AND APPLICATION TO STARK HAMILTONIAN
WITH NON-CONSTANT MAGNETIC FIELDS
ANH TUAN DUONG

Abstract. In this paper, we generalize a trace formula due to D. Robert concerning the
spectral shift function (SSF for short). We then give an application to study the semiclassical
asymptotics of the SSF for the Stark Hamiltonian with non-constant magnetic field.

1. Introduction
Let H0 and H be two self-adjoint operators acting on the same Hilbert space H and assume
that the operator (H + i)−N − (H0 + i)−N is of trace class for some N ≥ 1. Then there exists
a unique function ξ, modulo a real number, such that
ξ ′ , f = tr (f (H) − f (H0 )) for all f ∈ C0∞ (R).
Here, ξ ′ is the derivative of ξ in the distribution sense. The function ξ is called the spectral
shift function (see [23]).
The SSF was first introduced by the physicist I.M.Lifshits in the fifties of the last century
and brought in mathematical use by M.G.Krein in [15]. Notice also that a deep connection
between the SSF and the scattering matrix is known since the work of M.Birman and M.Krein
(see [1] and also [3, 23]). Several formula representations for the spectral shift function are
discussed and intercompared in [2].
The study of spectral shift functions is an important problem in the spectral theory which
covers both discrete and continuous spectrum. For the Schr¨
odinger operators (H = H0 + V,
H0 = −∆ and V is an electric potential), there are a lot of works treating the SSF in many
aspects, see, e.g., [4, 7, 8, 9, 10, 6, 12, 14, 16, 17, 19, 20, 21, 24] and references therein. For
the Schr¨
odinger operator with constant magnetic field, we refer to the survey of G.Raikov
[17] in three-dimensional case, in which, the singularities and the expansion at high energy of
the SSF at Landau levels were obtained. In [17], the author exploited a trace formula due to
A. Pushnitski and the Birman- Schwinger principle.
The behavior of the spectral shift function corresponding to the non-semi-bounded operators (H = H0 + V and H0 = −∆ + βx1 ) was studied in [21]. In particular, they obtained the

following trace formula
(1.1)

tr (f (H) − f (H0 )) = −tr (∂x1 V f (H)) , for all f ∈ C0∞ (R).

Recently, M. Dimassi and V. Petkov have generalized (1.1) for Stark Hamiltonian with constant magnetic field in dimension two, see [9, 10].
2010 Mathematics Subject Classification. 81Q10, 35J10, 35P25, 35C20, 47A55,47N50.
Key words and phrases. Schr¨
odinger operators, non-constant magnetic fields, qualitative properties, asymptotic expansions, spectral shift function .
1


2

ANH TUAN DUONG

In this paper, we first generalize the trace formula (1.1) to the abstract setting. We then give
an application to the study of the semiclassical asymptotics of the SSF for Stark Hamiltonian
with non-constant magnetic fields.
In [9], the authors exploited the shift operator (the semi-group generated by A := ∂x )
instead of A. Nevertheless, the use of semi-group generated by A does not work in the
abstract case, even in the case of pseudodifferential operators. This is because one cannot
exploit the passage of limit as in the case of Stark Hamiltonian. Another difficulty appears
in the case where the operator A does not generate any semi-group. Here, to overcome these
difficulties, we need to construct a sequence of suitable bounded operators (Aj ) and use some
properties of trace class operators.
For the Stark Hamiltonian, we do not require that the dimension of phase space is equal
to two or the magnetic field is constant. Hence, our result is a generalization of some results
in [7, 9, 10, 21]. The proof is based on the trace formula and some Tauberian arguments.
The rest of this paper is organized as follows: In Section 2, we recall some definitions and

auxiliary results. The main results are given in Section 3. Section 3.1 is devoted to the trace
formula. The application to the Stark Hamiltonian is given in Section 4.
2. Preliminaries
We recall some notations and auxiliary results which will be used in the next sections. Let
h > 0 be the semiclassical parameter.
Definition 2.1. A function m : R2n → (0, ∞) is called an order function if there exist
constants C, N such that
m(w) ≤ C w
˜−w

N

m(w)
˜ for all w,
˜ w ∈ R2n .

1

Here w = (1 + |w|2 ) 2 .
Definition 2.2. Let m be an order function on R2n . For k ∈ R, 0 ≤ l <
symbols Slk (m, R2n ) is defined by

1
2,

the class of

Slk (m, R2n ) := a ∈ C ∞ (R2n )| ∀β ∈ N2n , ∃Cβ > 0 s.t. |∂ β a| ≤ Cβ h−k−l|β| m .
If l = 0, we omit the index l and only write S k (m, R2n ).
We will use the standard Weyl quantization of symbols defined, for a ∈ Slk (m, R2n ), by

aw (x, hDx )u(x) :=
for u ∈

S(Rn )-

1
(2πh)n

i

eh

x−y,ξ

a

R2n

x+y
, ξ u(y)dydξ,
2

the space of rapidly decreasing functions.

Lemma 2.3. ([11, Chapter 9]) Let aw (x, hDx ) be a h- pseudodifferential operator with a ∈
S 0 (m, R2n ). If m ∈ L1 (R2n ) then aw (x, hDx ) is of trace class and
1
a(x, ξ)dxdξ.
tr (aw (x, hDx )) =
(2πh)n

R2n
.
Lemma 2.4. ([23, Lemma 3, page 188]) Let A be a trace class operator on H and (Tn ) be a
sequence of bounded linear operators which converges strongly to 0. Then lim Tn A tr = 0.
n→∞


A TRACE FORMULA AND APPLICATION

3

Lemma 2.5. ([22, Corollary 3.8]) Let A, B ∈ L(H) have the property that both AB and BA
lie in the space of trace class operators, then trAB = trBA.
3. Trace formula
In this section, we state a generalization of the trace formula given in [21, 9].
3.1. Abstract result. Let H be a separable Hilbert space. Consider a pair of self-adjoint
operators H0 , H which are densely defined on the same domain D(H) ⊂ H. Suppose that
the difference V = H − H0 is a bounded, self-adjoint operator. We introduce the following
hypotheses:
H.1. There exists a closed operator A such that the resolvent set ρ(A) is unbounded in C.
Moreover, the commutator [A, H0 ] = AH0 − H0 A is densely defined and equal to the identity
operator.
H.2. The commutator [V, A] is self-adjoint and bounded on the domain D([V, A]) ⊃ D(H).
In addition, there is m ∈ N \ {0} such that V (H0 + i)−m and [V, A](H0 + i)−m are trace class
operators.
Our result concerning the trace formula is the following.
Theorem 3.1. Assume that Hypotheses (H.1) and (H.2) hold. Then we have
(3.1)

tr (f (H) − f (H0 )) = tr ([V, A]f (H)) , for all f ∈ C0∞ (R).

Under the above hypotheses, we can follow the general setup (see [23, Chapter 8] or [20])
to define the SSF related to the pair of operators H and H0 by
′
ξH,H
, f := tr (f (H) − f (H0 )) , for f ∈ C0∞ (R).
0

Applying Theorem 3.1, it is easy to obtain a relation between the derivative of the SSF and
the spectral resolution EH (λ).
Corollary 3.2. Under Hypotheses in Theorem 3.1, we have the following representation of
the derivative of the SSF in the sense of distributions
(3.2)

′
ξH,H
(λ) = η(λ) = tr [V, A]
0

∂EH
(λ) ,
∂λ

where η is defined by
η, ϕ = tr [V, A]ϕ(H) , for all ϕ ∈ C0∞ (R).
Remark 3.3.
i) If we replace Hypothesis [A, H0 ] = 1 in (H.1) by [A, H0 ] = P , where P is invertible
and commutes with H0 , then our arguments still work. Indeed, by setting A˜ = P −1 A
˜ H0 ] = 1.
it is reduced to the case [A,

ii) Trace formula (3.1) was proved in [10, 9] in a special case, i.e., for the constantmagnetic Stark Hamiltonian in dimension two with A = ∂x .


4

ANH TUAN DUONG

3.2. Proof of Theorem 3.1. Note that under Hypotheses of Theorem 3.1, the operators
(H0 + z)−m V, V (H + z)−m and (H + z)−m V are also of trace class, for all z ∈ C \ R.
We split the proof into three steps.
Step 1. Prove that f (H) − f (H0 ) is a trace class operator for f ∈ C0∞ (R).
Let z ∈ C \ R. The resolvent identity follows
(H − z)−1 − (H0 − z)−1 = −(H − z)−1 V (H0 − z)−1 .

(3.3)

Differentiating with respect to z both sides of the equality (3.3) (N − 1) times, we obtain
(H − z)−N − (H0 − z)−N is equal to a linear combination of finite terms (H − z)−j V (H0 −
z)−N −1+j which are of trace class for N > 2m. Consequently, (H − z)−N − (H0 − z)−N is a
trace class operator for N > 2m.
Fix N > 2m. Let f ∈ C0∞ (R) and g(t) = f (t)(t − i)N . We denote f˜ an almost analytic
extension of f . Applying the so-called Helffer-Sj¨
otrand formula (see [11, Chapter 8]), one
obtains
f (H) = g(H)(H − i)−N = −
(3.4)

1
π

f (H0 ) = g(H0 )(H0 − i)−N = −

∂¯z f˜(z)(z − i)N (z − H)−1 (H − i)−N L(dz),
1
π

∂¯z f˜(z)(z − i)N (z − H0 )−1 (H0 − i)−N L(dz).

Here L(dz) = dxdy is the Lebesgue measure on C. Thus,
1
π
1
=−
π
1
−
π

f (H) − f (H0 ) = −

∂¯z f˜(z)(z − i)N (z − H)−1 (H − i)−N − (z − H0 )−1 (H0 − i)−N L(dz)
∂¯z f˜(z)(z − i)N (z − H)−1 (H − i)−N − (H0 − i)−N L(dz)
∂¯z f˜(z)(z − i)N (z − H)−1 − (z − H0 )−1 (H0 − i)−N L(dz).

Moreover, (z − H)−1 − (z − H0 )−1 (H0 − i)−N is a trace class operator due to the following
identity
(z − H)−1 − (z − H0 )−1 (H0 − i)−N = −(z − H)−1 V (z − H0 )−1 (H0 − i)−N
= −(z − H)−1 V (H0 − i)−N (z − H0 )−1 .
Therefore, the operator f (H) − f (H0 ) is of trace class.
Step 2. Construction of auxiliary operators.

It follows from the hypothesis (H.1) that there is an unbounded sequence (λj )j≥0 ⊂ ρ(A)
verifying |λj | → +∞ as j → ∞. Put µj = λ1j , Aj = (1 − µj A)−1 and Tj (t) = exp(Aj t) for
t ≥ 0, ( Tj (t) is the semi-group generated by bounded operator Aj ).
We construct the following operators:
Gj (t) := [Tj (t), H]f (H) − [Tj (t), H0 ]f (H0 ), j ∈ N.


A TRACE FORMULA AND APPLICATION

5

Fix j ∈ N. We show that Gj (t) is of trace class and tr (Gj (t)) = 0. A straightforward
computation gives
Gj (t) = Tj (t)Hf (H) − HTj (t)f (H) − Tj (t)H0 f (H0 ) + H0 Tj (t)f (H0 )
(3.5)

= Tj (t)(Hf (H) − H0 f (H0 )) + H0 Tj (t)f (H0 ) − HTj (t)f (H)
= Tj (t)(Hf (H) − H0 f (H0 )) + (H0 − H)Tj (t)f (H0 ) + HTj (t)(f (H0 ) − f (H))
= Tj (t)(Hf (H) − H0 f (H0 )) − V Tj (t)f (H0 ) + HTj (t)(f (H0 ) − f (H)).

The first term Tj (t)(Hf (H) − H0 f (H0 )) is of trace class because the function sf (s) has
compact support and the operator Tj (t) is bounded. Using [A, H0 ] = 1, we get
(3.6)

[Aj , H0 ] = µj Aj [A, H0 ]Aj = µj A2j and [Anj , H0 ] = nµj An+1
.
j

Thus,
[Tj (t), H0 ] = [exp(tAj ), H0 ] =

n≥1

(3.7)

tn

= µj
n≥1

n!

tn n
A , H0
n! j

nAn+1
= µj tA2j
j
n≥0

tn n
A = µj tA2j Tj (t).
n! j

We now decompose
V Tj (t)f (H0 ) = V (H0 + i)−1 (H0 + i)Tj (t)f (H0 )
= V (H0 + i)−1 (Tj (t)(H0 + i) + [H0 , Tj (t)])f (H0 )
= V (H0 + i)−1 (Tj (t)(H0 + i) − µj tA2j Tj (t))f (H0 ).
Repeating this decomposition N times, we can show that V Tj (t)f (H0 ) is equal to a finite sum
of trace class operators of the form V (H0 + i)−N Pj (t)g(H0 ), where Pj (t) is some bounded

operator and g ∈ C0∞ (R). Therefore, the second term V Tj (t)f (H0 ) is of trace class.
For the third term HTj (t)(f (H0 ) − f (H)) we shall prove a stronger result saying that
HTj (t)(f (H0 )−f (H))(H +i) is a trace class operator. Notice that HTj (t)(f (H0 )−f (H))(H +
i) = H0 Tj (t)(f (H0 ) − f (H))(H + i) + V Tj (t)(f (H0 ) − f (H))(H + i) and V Tj (t)(f (H0 ) −
f (H))(H + i)is of trace class. Then it is sufficient to prove that H0 Tj (t)(f (H0 )− f (H))(H + i)
is a trace class operator. Indeed,
H0 Tj (t)(f (H0 ) − f (H))(H + i)
= Tj (t)H0 (f (H0 ) − f (H))(H + i) + [H0 , Tj (t)](f (H0 ) − f (H))(H + i)
=: (I) + (II).
By (3.7) and using the fact that (f (H) − f (H0 ))H = (Hf (H) − H0 f (H0 )) − f (H0 )V which
is the sum of two trace class operators, one obtains (II) is of trace class. Similarly, since
H0 (f (H0 ) − f (H))(H + i) = (H0 f (H0 )(H0 + i) − Hf (H)(H + i)) + H0 f (H0 )V + V f (H)(H + i)
which is the sum of four trace class operators, (I) is a trace class operator. Therefore,
H0 Tj (t)(f (H0 ) − f (H))(H + i) is of trace class. Accordingly, Gj (t) is also of trace class.
Next we shall show that tr(Gj (t)) = 0. It follows from (3.5) that
(3.8)
tr(Gj (t)) = tr(Tj (t)(Hf (H) − H0 f (H0 ))) − tr(V Tj (t)f (H0 )) + tr(HTj (t)(f (H0 ) − f (H))).


6

ANH TUAN DUONG

The cyclicity of the trace of operators follows
tr(HTj (t)(f (H0 ) − f (H))) = tr(HTj (t)(f (H0 ) − f (H))(H + i)(H + i)−1 )
= tr((H + i)−1 HTj (t)(f (H0 ) − f (H))(H + i))
= tr(Tj (t)(f (H0 ) − f (H))(H + i)(H + i)−1 H)

(3.9)

= tr(Tj (t)(f (H0 ) − f (H))H)
= tr(Tj (t)(H0 f (H0 ) − Hf (H))) + tr(Tj (t)f (H0 )V ).
Substituting (3.9) into (3.8), we have
tr(Gj (t)) = tr(V Tj (t)f (H0 )) − tr(Tj (t)f (H0 )V ) = 0 (see Lemma 2.5).
Step 3. Passage of limit
Consider the operator Gj (t)
Gj (t) = [Tj (t), H0 ](f (H) − f (H0 )) + [Tj (t), V ]f (H)
= µj tA2j Tj (t)(f (H) − f (H0 )) + [Tj (t), V ]f (H).
Recall that tr(Gj (t)) = 0, then
tr µj tA2j Tj (t)(f (H) − f (H0 )) = −tr([Tj (t), V ]f (H)).

(3.10)
It is equivalent to

tr µj A2j Tj (t)(f (H) − f (H0 )) = −tr
(3.11)

= −tr
= −tr

Tj (t) − 1
, V f (H)
t
Tj (t) − 1
Tj (t) − 1
V f (H) + tr V
f (H)
t
t
Tj (t) − 1

Tj (t) − 1
V f (H) + tr
f (H)V .
t
t

T (t)−1

As t → 0, the operator Tj (t) (resp. j t ) converges strongly to 1 (resp. Aj ). Therefore, we
get (see Lemma 2.4 and also [10, Proposition 1])
(3.12)

tr(µj A2j (f (H) − f (H0 ))) = −tr(Aj V f (H)) + tr(Aj f (H)V ).

On the other hand,
1
1
[Aj , V ]f (H0 ) = Aj [µj A, V ]Aj f (H0 ) = Aj [A, V ](H0 +i)−1 (Aj (H0 +i)f (H0 )−µj A2j f (H0 )),
µj
µj
where in the last equality, we use the decomposition
Aj f (H0 ) = (H0 + i)−1 (H0 + i)Aj f (H0 ) = (H0 + i)−1 (Aj (H0 + i) + [H0 , Aj ])f (H0 )
= (H0 + i)−1 (Aj (H0 + i)f (H0 ) − µj A2j f (H0 )).
Repeating this decomposition N times, we obtain
i)−N Aj (H0 + i)N f (H0 ) and finite
C0∞ (R) and supj Pj ≤ constant.
trace class operator.

1
µj [Aj , V

]f (H0 ) is the sum of Aj [A, V ](H0 +

terms
](H0 + i)−N Pj g(H0 ), where k ≥ 1, g ∈
It results from this and (H.2) that µ1j [Aj , V ]f (H0 ) is a
µkj Aj [A, V


A TRACE FORMULA AND APPLICATION

7

Note that Aj converges strongly to 1 as j → +∞. Then, Aj [A, V ](H0 + i)−N Aj (H0 +
i)N f (H0 )( resp. µkj Aj [A, V ](H0 + i)−N Pj g(H0 )) converges to [A, V ]f (H0 ) (resp. 0) in the
trace norm (see Lemma 2.4). This shows that
(3.13)

1
[Aj , V ]f (H0 )
µj

tr

→ tr ([A, V ]f (H0 )) as j → +∞.

We write

(3.14)

1
1
1
[Aj , V ]f (H)) = [Aj , V ]f (H0 )) + [Aj , V ](f (H) − f (H0 ))
µj
µj
µj
1
= [Aj , V ]f (H0 )) + Aj [A, V ]Aj (f (H) − f (H0 )).
µj

This decomposition and (3.13) gives
(3.15)

tr

1
[Aj , V ]f (H)
µj

→ tr([A, V ]f (H)) as j → ∞.

It follows from (3.13), (3.14) and (3.15) that V Aj f (H) = Aj V f (H) − [V, Aj ]f (H) is a trace
class operator and then tr(V Aj f (H)) = tr(Aj f (H)V ). From this, it is easy to see that (3.12)
is equivalent to
tr(A2j (f (H) − f (H0 ))) = tr(

(3.16)

1

[V, Aj ]f (H)).
µj

Letting j → ∞ in (3.16) and using (3.13), we finally achieve
tr(f (H) − f (H0 )) = tr([V, A]f (H)).
The proof is finished.

4. Semiclassical asymptotics of the spectral shift function
In this section, we only concentrate on the application of Theorem 3.1 for Stark Hamiltonian although we believe that the above trace formula can be applied to study Schr¨
odinger
operators with or without Stark potential and Dirac operators. Consider
P (h) = P0 (h) + V (x) = (hDx − A(x′ ))2 + x1 + V (x), x = (x1 , x′ ) ∈ R × Rn−1 ,
where h is a positive constant, V is a real smooth function, A(x′ ) = (A1 (x′ ), ..., An (x′ )) is the
magnetic potential and is independent of the first coordinate. We always suppose that
|∂ α Aj (x′ )| ≤ Cj,α for all α ∈ Nn \ {0} and V ∈ C0∞ (Rn ).
Then the operators P( h) and P (h) are essentially self-adjoint on C0∞ (Rn ) (see [11]).
We are concerned with the semiclassical asymptotics of the SSF associated to the pair
(P (h), P0 (h)).


8

ANH TUAN DUONG

4.1. Semiclassical asymptotics.
In this subsection, we give the weak and the point-wise asymptotic expansion of the SSF.
Theorem 4.1. Let f ∈ C0∞ (R) and V ∈ C0∞ (Rn ). Then we have
ξP′ (h),P0 (h) , f ∼ (2πh)−n

(4.1)

aj (f )hj , h ց 0,
j≥0

where we give explicitly the first three coefficients
a0 (f ) = −

a2 (f ) =

1
12

R2n

∂x1 V (x)f (ξ 2 + x1 + V (x))dxdξ, a1 (f ) = 0 and


∂x1 V (x) 



j,k

2
(x′ ) + ∆x V (x) f ′′ (ξ 2 + x1 + V (x))dxdξ,
Bjk

Here the magnetic field Bjk (x′ ) is given by Bjk (x′ ) =

∂Ak

′
∂xj (x )

−

∂Aj
′
∂xk (x ).

We say that λ is not a critical value of a smooth function V if ∇V = 0 on the set
{x ∈ Rn ; V (x) = λ}.
The semiclassical asymptotics of the SSF is given in the next theorem.
Theorem 4.2. Let V ∈ C0∞ (Rn ). Assume that a, b are not critical values of x1 + V (x). Then
as h ց 0 we have
ξP (h),P0 (h) (b) − ξP (h),P0(h) (a) = (2πh)−n ((c0 (b) − c0 (a)) + O(h)) ,
where
c0 (λ) = −
Here σn is the area of the unit

σn
n

n

∂x1 V (x)(λ − x1 − V (x))+2 dx.

Rn
sphere S n−1

and t+ = max(0, t).

4.2. Sketch of proofs of Theorems 4.1 and 4.2.
We now apply Theorem 3.1 to give the trace formula for Stark Hamiltonian. Choosing
the operator A = ∂x1 , then it is easy to see that [A, P0 (h)] = [∂x1 , (hDx − A(x′ ))2 + x1 ] = 1
and [A, V ] = ∂x1 V . Moreover, since V ∈ C0∞ (Rn ), there exists m > 0 such that V (P0 (h) +
i)−m , ∂x1 V (P0 (h) + i)−m are of trace class (see [9, Lemma 1]). Thus, Theorem 3.1 follows
that
Lemma 4.3. For all f ∈ C0∞ (R), we have
(4.2)

tr (f (P (h)) − f (P0 (h))) = −tr (∂x1 V (x)f (P (h))) .

We denote by
˘ )= 1
θ(τ
2π

eitτ θ(t)dt,
R

1˘ t
θ˘ǫ (t) = θ(
).
ǫ ǫ

The proof of the following results is similar to that in [7, 9]. So we omit the detail here.


A TRACE FORMULA AND APPLICATION

9

Proposition 4.4. Let f ∈ C0∞ (R) and ϕ ∈ C0∞ (Rn ). Then we have
tr (ϕ(x)f (P (h))) ∼ (2πh)−n

(4.3)

bj (f, ϕ)hj , h ց 0,
j≥0

where
b0 (f, ϕ) =

R2n

ϕ(x)f (ξ 2 + x1 + V (x))dxdξ, b1 (f, ϕ) = 0

and
1
b2 (f, ϕ) = −
12

R2n



ϕ(x) 



j,k

2
(x′ ) + ∆x V (x) f ′′ (ξ 2 + x1 + V (x))dxdξ.
Bjk

Proposition 4.5. Let ϕ ∈ C0∞ (R). Let C be a fixed, large constant and θ ∈ C0∞ − C1 , C1 ; R ,
θ = 1 near zero. Assume that λ is not a critical value of x1 + V (x). Then there is σ > 0 such
that for f ∈ C0∞ ((λ − σ, λ + σ); R) we have


N −1

tr ϕ(x)θ˘h (τ − P (h))f (P (h)) = (2πh)−n f (τ )

cj (τ )hj + O hN

j=0

τ

−M 

, ∀M, N ∈ N,

uniformly for τ ∈ R and
c0 (τ ) = (2πi)−1

ϕ(x) (τ − i0 − p(x, ξ))−1 − (τ + i0 − p(x, ξ))−1 dxdξ.
R2n

We notice that in [9], the authors only gave b0 (f ). Hence, let us explain how to obtain
b1 (f, ϕ) and b2 (f, ϕ). Firstly, we use as usual the Helffer-Sj¨
ostrand formula
(4.4)

tr (ϕ(x)f (P (h)) = tr −

1
π

∂¯z f˜(z)ϕ(x)(z − P (h))−1 L(dz) .

Since f ∈ C0∞ (R) and ϕ ∈ C0∞ (Rn ), thanks to the pseudodifferential calculus, modulo O(h∞ )
we only need to work with (x, ξ) in some compact set of R2n . This means that for some
χ ∈ C0∞ (R2n ),
tr −

1
π

= tr −

∂¯z f˜(z)ϕ(x)(z − P (h))−1 L(dz)
1
π

∂¯z f˜(z)ϕ(x)(z − P (h)χw (x, hDx ))−1 L(dz) + O(h∞ ).

Note that the operator P (h)χw (x, hDx ) is compact, then by using the computations as in [13]

and [18, Proposition II-56] we obtain b1 (f, ϕ) = 0 and
b2 (f, ϕ) = −

1
24

ϕ(x)f ′′ (p(x, ξ))
j,k

∂2p
∂2p
∂2p
∂2p
−
∂ξj ∂ξk ∂xj ∂xk
∂xj ∂ξk ∂ξj ∂xk

dxdξ,


10

ANH TUAN DUONG

where p(x, ξ) = (ξ −A(x′ ))2 +x1 +V (x) is the symbol of P (h). A straightforward computation
of partial derivatives gives
1
2

j,k

=
j,k

∂2p
∂2p
∂2p
∂2p
−
∂ξj ∂ξk ∂xj ∂xk
∂xj ∂ξk ∂ξj ∂xk
∂Aj
∂Ak
−
∂xj
∂xk

∂Aj
∂Ak
−
∂xj
∂xk

=2
j,k

∂Ak
∂xj

∂Ak

∂Aj
−
∂xj
∂xk

+ ∆x V (x)

2
Bjk
(x′ ) + ∆x V (x).

+ ∆x V (x) =
j,k

Consequently, by using a change of variable ξ˜ = (ξ − A(x)) we get the formula of b2 (f, ϕ).
Proof of Theorem 4.1
Theorem 4.1 is immediate consequence of Proposition 4.4 and Lemma 4.3.
Proof of Theorem 4.2
The main idea of the proof is as follows:
• Decompose the spectral shift function into two parts which are monotone,
• Apply Tauberian arguments.
Since the proof is similar to that in [9], we only give the main lines of the proof.
Firstly, we choose ψ ∈ C0∞ (Rn ) such that ψ = 1 on the support of V .
∂x1 V L∞ (Rn ) then

Let M >

ξh′ , f = tr(f (P (h)) − f (P0 (h))) = −tr(∂x1 V f (P (h)))
= −tr(∂x1 V ψf (P (h))ψ)
= tr((M − ∂x1 V )ψf (P (h))ψ) + tr(M ψf (P (h))ψ)

1

1

= tr((M − ∂x1 V ) 2 ψf (P (h))ψ(M − ∂x1 V ) 2 ) + tr(M ψf (P (h))ψ)
=: µ′h , f − νh′ , f .
Next, observing that if f ≥ 0 then µ′h , f , νh′ , f ≥ 0. This means that µ′h , νh′ have the same
sign as f . In consequence, the functions λ → νλ ), µh (λ) are monotone. Therefore, given
Propositions 4.4 and 4.5 we can apply Tauberian arguments (see [5, 9, 18]) and follow exactly
the arguments in [9] to obtain the results. We also omit the detail here.
Acknowledgments
The author would like to thank the Vietnam Institute for Advanced Study in Mathematics
(VIASM) for the hospitality during the completion of this work. He is grateful to Professor
M.Dimassi for many useful discussions and Dr Q.H.Phan for valuable comments. This research is funded by Vietnam National Foundation for Science and Technology Development
(NAFOSTED) under Grant No. 101.02-2014.06.
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Anh Tuan DUONG, Department of Mathematics, Hanoi National University of Education,
136 Xuan Thuy, Cau Giay, Ha noi, Viet Nam
E-mail address: