International Journal of Analysis and Applications

Volume 18, Number 2 (2020), 183-193

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-183

BIRKHOFF NORMAL FORMS FOR BORN-OPPENHEIMER OPERATORS

NAWEL LATIGUI1,4, BEKKAI MESSIRDI2,4,∗ AND KAOUTAR GHOMARI3,4

1Department of Mathematics, Faculy of Exact and Applicable Sciences, University of Oran1 Ahmed Ben

Bella, Algeria

2High School of Electrical Engineering and Energetics-Oran, Algeria

3Department of Mathematics and Informatics, ENPOran, Algeria

4Laboratory of Fundamental and Applicable Mathematics of Oran (LMFAO), Algeria

∗Corresponding author: bmessirdi@yahoo.fr

Abstract. We describe in this paper a significant spectral reduction method for Born-Oppenheimer oper-

ators with regular potentials, which leads to an adaptable Birkhoff normal form theorem for the associated

effective Hamiltonians. As illustration of the established results, we compute the Birkhoff normal form in

Fermi resonance.

1. Introduction

For a molecular system with N electrons and N′ nuclei, the Hamiltonian, under the Born-Oppenheimer

approximation, can be written as:

P(h) = −h2∆x + Q(x) , Q(x) = −∆y + V (x,y)

on L2(Rnx × Rpy) where n = 3N and p = 3N′ and where h > 0 is a small parameter playing the role of

the semi-classical parameter. ∆x (resp. ∆y) is the Laplace operator with respect to x (resp. y), x ∈ Rn and

y ∈ Rp, N,N′ ≥ 1, V is the interaction potential between particles. P(h) is called the Born-Oppenheimer

Received 2019-10-12; accepted 2019-11-11; published 2020-03-02.

2010 Mathematics Subject Classification. 58K50, 81S10, 47A20.

Key words and phrases. Born-Oppenheimer operator; effective Hamiltonian; Birkhoff normal form; harmonic oscillator;

Fermi resonance.

c©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

183

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-183


Int. J. Anal. Appl. 18 (2) (2020) 184

Hamiltonian. Q(x) is the electronic Hamiltonian defined on L2(Rpy). It is well known that if V (x,y) is smooth

verifying suitable assumptions, then

z ∈ σ(P(h)) ⇐⇒ z ∈ σ(F(z))

where F(z) is a semiclassical analytic pseudodifferential operator on L2 (Rnx) and σ stands for the spectrum,

(see e.g. [10], [11], [3]), the main idea, due to Born and Oppenheimer in [5], is to replace, for fixed x, the

operator Q(x) by its eigenvalues. This reduction is possible thanks to the pseudodifferential calculus with

operator valued symbols. Then we are led to consider, the reduced operator (called the effective Hamiltonian

in the Born-Oppenheimer approximation):

Peff (h) = −h2∆x + λ1(x)

where λ1(x) is the lowest eigenvalue of Q(x), by the minimax principle λ1(x) is simple and analytic if

V is sufficiently smooth. Motivated by various physical questions we consider the connected problems in

the asymptotic h → 0+, note that through standard semiclassical analysis Peff (h) can explain the complete

spectral picture of P(h) modulo errors in h.

We wish to describe the Birkhoff normal form near an equilibrium point of P(h). It is well known that

a more precise description of the vibrational energies of a molecule is given by the harmonic oscillator, our

approach here is to replace Q(x) in the Born-Oppenheimer Hamitonian P(h) by its lowest eigenvalue λ1(x)

and thus, we are reduced to an effective h-pseudodifferential operator Op(eλ), with symbol eλ depending

only on (x,ξ). The normal forms in the Born-Oppenheimer approximation, are introduced here as being

those of the Schrödinger effective operator Peff (h) on L
2 (Rnx) . Birkhoff normal form is one of the basic

tools in quantum and semiclassical mechanics (see e.g. [7], [8]), it has already been used by Birkhoff [4] to

study some problems of dynamical systems.

Precisely, the goal of this paper is to analyze the notion of the Birkhoff normal form near an equilibrium

point and discuss the dynamical consequences for the Schrödinger Hamiltonian P(h). Suppose 0 is a non-

degenerate local minimum of λ1(x), by applying semiclassical techniques especially the pseudodifferential

calculus with operator valued symbols and the classical quantum formal Birkhoff normal form theorem,

we show that we can find a canonical transformation putting P(h) as a reasonable perturbation of −∆x +
1
2
〈λ′′1 (0)x,x〉 modulo O(h2). Our approach is natural, it consists in computing the normal form of the effective

Hamiltonian −h2∆x + λ1(x) after reduction of the operator P(h) to Peff (h). To our knowledge this is the

first attempt to determine the Birkoff normal forms for the Born-Oppenheimer Hamiltonians.

In Section 2, we recall some results on pseudodifferential operators with operator valued symbols. Then,

we give a representation of the effective Hamiltonian and obtain WKB solutions of the Hamiltonian P(h).

In section 3, we investigate the theorem of Birkhoff normal form near an equilibrium point in infinite



Int. J. Anal. Appl. 18 (2) (2020) 185

dimension in the Born-Oppenheimer approximation for P(h) via the effective Hamiltonian Peff (h). In the

fourth section we compute the Birkhoff normal form of Peff (h) in the Fermi resonance.

2. Reduction to an effective operator

In this section we explain the construction of WKB solutions for the Hamiltonian P(h) and several

mathematical results concerning the pseudodifferential calculus with operator valued symbols of the Born-

Oppenheimer approximation. For further informations about the pseudodifferential calculus and BKW

method we refer the reader to the works of Balazard-Konlein [2], Messirdi [10], Baklouti [1] and other

authors.

2.1. Pseudodifferential calculus with operator valued symbols. For m ∈ R, Ω a bounded open subset

of Rnx and H a complex Hilbert space, consider the space of formal power series:

Sm(Ω,H) =



∞∑
j=0

h−m+j/2sj(x) : sj ∈ C∞(Ω,H)




where C∞(Ω,H) is the space of C∞-functions mapping Ω into H.

Given ψ ∈ C∞(Ω,R) and U a neighborhood of 0 in Rnx, we set:

Ω∗ = {(x,ξ) ∈ Ω ×Cn : ξ − i∇ψ(x) ∈U}

and

S0(Ω∗,L(H,K)) =



∞∑
j=0

hjaj(x,ξ) : aj ∈ C∞(Ω∗,L(H,K))




where K is Hilbert space and L(H,K) is the algebra of all continuous linear operators from H into K.

The operator valued functions in S0(Ω∗,L(H,K)) are called symbols.

For any symbol a = a(x,ξ; h) in S0(Ω∗,L(H,K)), by analogy with the action of differential oper-

ators on the space e−ψ(x)/hSm(Ω,H), one can define an operator Op(a) from e−ψ(x)/hSm(Ω,H) into

e−ψ(x)/hSm(Ω,K) by the formula:

Op(a)
(
e−ψ(x)/hs(x,h)

)
=

e−ψ(x)/h
∑
α∈Nn

h|α|

i|α|α!
∂αξ a(x,i∇ψ(x); h)∂

α
y

(
s(y,h)eχ(x,y)/h

)
y=x

(2.1)

χ(x,y) = ψ(y) −ψ(x) − (y −x).∇ψ(x) = O
(
|x−y|2

)
, s ∈ Sm(Ω,H).

Op(a) is called h-pseudodifferential operator with operator valued symbol a(x,ξ; h) =
∞∑
j=0

hjaj(x,ξ). The

function a0(x,ξ) (coefficient of h
0) is called principal symbol of the h-pseudodifferential operator Op(a).

Furthermore, such operators verify:

eψ(x)/hOp(a)
(
e−ψ(x)/hs(x,h)

)
∈ Sm(Ω,H)



Int. J. Anal. Appl. 18 (2) (2020) 186

and can be composed using the formula:

Op(b) ◦Op(a) = Op(b]a)

where a ∈ S0(Ω∗,L(H,K)), b ∈ S0(Ω∗,L(K,L)) (L is a third Hilbert space), the range of Op(a) is

contained in the domain of Op(b), and

b]a(x,ξ,h) =
∑
α∈Nn

h|α|

i|α|α!
∂αξ b(x,ξ; h)∂

α
xa(x,ξ; h) ∈ S

0(Ω∗,L(H,L)). (2.2)

This formula makes it possible to inverse asymptotically operators Op(a) whose principal symbol a0(x,ξ)

is invertible as a linear operator from H into K.

2.2. BKW solutions (scalar case). Let us take H = C and recall the following result:

Theorem 2.1. ( [9]) Let a(x,ξ; h) =
∞∑
j=0

hjaj(x,ξ) ∈ S0(Ω∗,C) be such that a0(x,ξ) = ξ2 + λ(x) where

λ ∈ C∞(Ω,R), λ ≥ 0, λ−1(0) = {0} , λ′(0) = 0 and λ′′(0) > 0. Let C0 > 0 and N0 be the number of

eigenvalues of

−∆x +
1

2
〈λ′′(0)x,x〉

in the compact interval [0,C0]. Denote by e1, ...,eN0 these eigenvalues. Then there are formal series:

Ek(h) = ekh +

∞∑
j=1

ek,jh
1+j/2 and ak(x,h) ∈ Smk(Ω,C),

ek,j, mk ∈ R, k ∈{1, ...,N0} , such that

(Op(a) −Ek(h))
(
e−ψ(x)/hak

)
= 0 in e−ψ(x)/hSmk(Ω,C)

where ψ(x) is the Agmon distance associated to the metric λ(x)dx2. The functions e−ψ(x)/hak(x,h) are

called the BKW solutions.

2.3. BKW solutions (general case). Let V ∈ C∞(Ω,L(H2(Rpy),L2(Rpy))) be ∆-compact:

V (x,y) (−∆y + 1)
−1 ∈ C∞(Ω,L(L2(Rpy)))

where Ω is a bounded open subset of Rnx. Thus, P(h) is selfadjoint on L
2(Rnx×Rpy) with domain H2(Rnx×Rpy)

as well as the operator Q(x) on L2(Rpy) with domain H
2(Rpy). Denote

λ1(x) = inf(σ(Q(x)))

the lowest energy level (ground state) of operator Q(x). Suppose that λ1(x) is an isolated eigenvalue of

finite multiplicity of Q(x), having unique and non-degenerate minimum at 0 :

λ1(x) ≥ 0, λ−11 (0) = {0} , λ
′
1(0) = 0, λ

′′
1 (0) > 0, (2.3)



Int. J. Anal. Appl. 18 (2) (2020) 187

and that λ1(x) is separated from the rest of the spectrum σ(Q(x)), i.e.,

inf
x∈Rn

(inf (σ(Q(x))\{λ1(x)})) > 0. (2.4)

We also denote by u1(x,y) the first eigenfunction of Q(x) associated with λ1(x) and normalized by

‖u1(x,.)‖L2(Rpy) = 1 for all x ∈ R
n. It can be shown that λ1 ∈ C∞(Ω,R) and u1 ∈ C∞(Ω,H2(Rpy)) (cf. [10]).

In particular, the assumption (2.4) implies that the orthogonal projection Π(x) on the subspace of L2(Rpy)

spanned by u1(x,.), x ∈ Ω, is C2-regular with respect to x (see [6]). To construct BKW solutions of P(h),

the idea here is to use the pseudodifferential calculus with operator valued symbols developed in subsection

2.1.

Consider, for λ ∈ C, the following symbol:

aλ(x; ξ) =


 ξ2 + Q(x) −λ u1

〈.,u1〉y 0


 ∈ S0(Ω∗,L(H2(Rpy) ⊕C,L2(Rpy) ⊕C)),

where 〈.,u1〉y is the inner product in L
2(Rpy). It follows from the assumptions and (2.1) that:

Op(aλ) =


 P(h) −λ u1
〈.,u1〉y 0




is h-pseudodifferential operator from e−ψ(x)/hSm(Ω,H2(Rpy)) into e
−ψ(x)/hSm(Ω,L2(Rpy)), with operator

valued symbol aλ, where ψ(x) is the Agmon distance associated to the metric λ1(x)dx
2.

We now describe a method for finding the inverse of Op(aλ). Using the fact that (∇ψ)2(x) = λ1(x) and

the gap assumption (2.4), one can easily show that for |λ| small enough and ξ close enough to i∇ψ(x),

Re
(

Π̂(x)Q(x)Π̂(x) −λ
)
> 0

and aλ is invertible with inverse:

b0(x,ξ; λ) =


 Π̂(x)

(
ξ2 + Π̂(x)Q(x)Π̂(x) −λ

)−1
Π̂(x) u1

〈.,u1〉y λ− ξ
2 −λ1(x)




where Π̂(x) = 1 − Π(x) (see e.g. [3]).

In particular,

b0(x,ξ; λ) ∈ S0(Ω∗,L(L2(Rpy) ⊕C,H
2(Rpy) ⊕C)).

Then using the composition formula (2.2), it is easy to construct a symbol:

bλ(x,ξ; h) = b0(x,ξ; λ) + hb1(x,ξ; λ) + h
2b2(x,ξ; λ) + ...

bλ(x,ξ; h) ∈ S0(Ω∗,L(L2(Rpy) ⊕C,H
2(Rpy) ⊕C)),



Int. J. Anal. Appl. 18 (2) (2020) 188

such that

aλ]bλ(x,ξ; h) = 1

Op(aλ) ◦Op(bλ) = I

I is the identity operator on e−ψ(x)/hSm(Ω,L2(Rpy) ⊕C). Let us pose:

Op(bλ) =


 E(λ) E+(λ)

E−(λ) E∓(λ)


 .

By Lemma 3.1 in [3], we also know that E∓(λ) = Op(eλ(x,ξ; λ)) is h-pseudodifferential operator with

symbol eλ(x,ξ; λ) ∈ S0(Ω∗,C) and its principal symbol is e0(x,ξ; λ) = λ−ξ2 −λ1(x). In particular, F(λ) =

λ−E∓(λ) is a scalar h-pseudodifferential operator with principal symbol ξ2 + λ1(x). Moreover, we have the

following fundamental spectral reduction:

λ ∈ σ(P(h)) ⇐⇒ λ ∈ σ(F(λ)).

Hence, the spectral study of the Hamiltonian P(h) on L2(Rnx × Rpy) is reduced to that of the h-

pseudodifferential operator F(λ) on L2(Rnx) so-called effective Hamiltonian of P(h). Now use Theorem 2.1

with F(λ), |λ| small enough, we find BKW solutions of P(h) as formal series Ek(h) = ekh +
∞∑
j=1

ek,jh
1+j/2

and ak ∈ Smk(Ω,C), such that:

(F(Ek(h)) −Ek(h))
(
e−ψ(x)/hak

)
= 0

in the exponentially weighted symbol space e−ψ(x)/hSmk(Ω,C).

In fact, one can show in many situations that F(λ) = Peff (h) + O(h2), which makes it easy to compare

(using, for example, the maximum principle) the eigenvalues of P(h) and those of Peff (h), and then identify

them when h decays to zero fast enough [6]. This reduction will justify in the next section our definition of

the normal Birkhoff forms of P(h) as those of the effective Hamiltonian Peff (h).

3. Reduction to Birkhoff normal form for the effective Hamiltonian

There exists a very convenient way of constructing a canonical transformation such that we conserve the

Hamiltonian structure of P(h) by using the Birkhoff normal form theorem via the effective Hamiltonian

Peff (h).

Definition 3.1. We call normal forms of the semi-classical operator P(h), the Birkhoff normal forms of the

associated effective Hamiltonian Peff (h).

The general philosophy will consist in transforming Peff (h) in such a way that the new Hamiltonian

becomes Ĥ2 + Λ where Ĥ2 is the harmonic oscillator and Λ is a reasonable perturbation term who commut



Int. J. Anal. Appl. 18 (2) (2020) 189

with Ĥ2. We consider here Ω = Rnx and assume that the hessian matrix λ
′′
1 (0) is diagonal, let

(
2ν21, ..., 2ν

2
n

)
be its eigenvalues, with νj > 0 and ν = (ν1, ...,νn). The rescaling xj →

√
νjxj, x = (x1, ...,xn) , transforms

P(h) as well as Peff (h) into:

Peff (h) = Ĥ2 + Γ(x)

where Ĥ2 is the harmonic oscillator
n∑
j=1

νj

(
−h2 ∂

2

∂x2
j

+ x2j

)
and Γ(x) is a smooth function such that

Γ(x) = O( |x|3) as |x| → 0. In general, Γ does not commute with Ĥ2, on the other hand we do not have

enough information on this perturbation, for that we will use the Birkhoff normal form of P(h) which is a

transformation of the previous type but more adapted and less restrictive.

Let:

Sd (m) =




a (x,ξ; h) : Rnx ×Rnξ × ]0, 1] −→ C, depends smoothly on x and ξ and

for all α ∈ N2n,
∣∣∣∂α(x,ξ)a (x,ξ; h)∣∣∣ ≤ Cαhd (1 + |x|2 + |ξ|2)m/2 ,

Cα > 0, uniformly with respect x,ξ and h




where m,d ∈ R. Sd (m) is called the semiclassical space of symbols of order d and degree m.

For a ∈ Sd (m) and u ∈ C∞0 (R2n), we set:

(Au) (x) = (Op~ (a) u) (x)

= (2πh)
−n
∫

R2n

eih
−1〈x−x′,ξ〉a

(
x + x′

2
,ξ; h

)
u (x′) dx′dξ. (3.1)

A is unbounded linear operator on L2 (Rn) with domain C∞0 (R
2n) the space of infinitely differentiable

functions on R2n with compact support, A : C∞0 (R
2n) −→ C∞(R2n) is called a semiclassical pseudodifferen-

tial operator with h-Weyl symbol a of order d and degree m. Ψd (m,Rn) denotes the set of all semiclassical

pseudodifferential operators with symbols in the class Sd (m) .

Different classes of symbols can also be defined, but for our purpose this class is enough. For example,

the h-Weyl symbol of the harmonic oscillator Ĥ2 is the polynomial H2 =
n∑
j=1

νj
(
ξ2j + x

2
j

)
.

Now, we introduce the space S to be the set of formal series:

S =


 ∑
α,β∈Nn,`∈N

tα,β,lx
αξβh` : tα,β,l ∈ C for all α,β ∈ Nn,` ∈ N




where the degree of xαξβh` is defined by |α| + |β| + 2`, α,β ∈ Nn, ` ∈ N, for technical reasons that of h is

double-counted. Let N ∈ N. Let DN be the finite dimensional vector space:

DN =


 ∑
α,β∈Nn,`∈N

tα,β,lx
αξβh` : tα,β,l ∈ C, α,β ∈ Nn,` ∈ N such that |α| + |β| + 2` = N




and

ON =


 ∑
α,β∈Nn,`∈N

tα,β,lx
αξβh` : tα,β,l = 0 if |α| + |β| + 2` < N


 .



Int. J. Anal. Appl. 18 (2) (2020) 190

Note that, for all N ∈ N, DN and ON are subspaces of S and

S = O0 ⊃O1 ⊃, ...,
⋂
N

ON = {0} .

Let 〈., .〉W be the Weyl bracket defined on S by:

〈f,g〉W = f̂ĝ − ĝf̂

where f̂ and ĝ are the h-Weyl quantizations of symbols f and g, respectively. Precisely,

〈fT ,gT〉W = σW (〈f,g〉W ) = σW
(
f̂ĝ − ĝf̂

)
where fT and gT are formal Taylor series at the origin of f and g in S, respectively and σW denotes the

h-Weyl symbol. Then, 〈., .〉W is antisymmetric satisfying the Jacobi identity:

〈〈fT ,g〉W ,hT〉W + 〈〈hT ,fT〉W ,gT〉W + 〈〈gT ,hT〉W ,fT〉W = 0

and the Leibniz identity:

〈fT ,gThT〉W = 〈fT ,gT〉W hT + gT 〈fT ,hT〉W .

Thus, S equipped with the Weyl bracket is a Lie algebra such that:

〈h,xj〉W = 〈h,ξj〉W = 0 and 〈ξj,xj〉W = −ih, for all j = 1, ...,n

x = (x1, ...,xn) and ξ = (ξ1, ...,ξn) ∈ Rn.

For and any S ∈S, we define a map:

adS : S −→S

P 7→ adS(P) = 〈S,P〉W

which is called the adjoint action. S has a representation on itself, the adjoint representation defined via

the map ad.

Let us consider the important special case of this concept, which is the adjoint action adS for S ∈D2, and

especially adH2. Let C [z,z,h] be the C-linear space of polynomials spanned by z
αzβh` of degree |α|+|β|+2`;

α,β ∈ Nn,` ∈ N, where z = (x1 + iξ1, ...,xn + iξn) ∈ Cn and z is the complex conjugate of z. Then,

B =
{
zβzγ : z ∈ Cn; β,γ ∈ Nn

}
is a natural basis of C [z,z,h] .

The next proposition gives some important properties and results on adH2.

Proposition 3.1. ( [7]) 1) ih−1adH2 (P) = {H2,P} , where {H2,P} =
n∑
j=1

∂H2
∂ξj

∂P
∂xj
− ∂H2

∂xj
∂P
∂ξj

is the classical

Poisson bracket.

2) adH2 is diagonal on B, in the sense that adH2
(
zβzγ

)
= h〈γ −β,ν〉zβzγ.



Int. J. Anal. Appl. 18 (2) (2020) 191

The assumption (2.3) implies that λ1(x) ∈O3, and since H2 + λ1(x) ∈D2, the quantum Birkhoff normal

form theorem for Peff (h) can now be formulated as follows:

Theorem 3.1. For R ∈O3, there exist S and T in the subspace O3 with real coefficients such that:

eih
−1adS (H2 + R) = H2 + T

where T = T3 + T4 + ... and Tj ∈Dj commutes with H2 : 〈H2,T〉W = 0.

This result is a direct consequence of the Birkoff normal form theorem shown for example in the article

by Ghomari and Messirdi [7].

We gain, compared with the BKW constructions developed in the second section, the commutative prop-

erty for Weyl product between the harmonic oscillator and the rest of reduction in Birkhoff normal form of

the Hamiltonian. The Birkhoff normal form is a more usable semi-classical reduction involving other inter-

esting spectral properties, especially it conserve the Hamiltonian structure and contains enough informations

to study the quantum resonances.

4. Birkhoff normal form for P(h) in Fermi resonance

It has been established in the previous sections that P(h) can be reduced to the effective Hamiltonian

Peff (h) modulo O(h2). Thus, it is natural to define the Birkhoff normal forms of P(h) as those of Peff (h)

modulo O(h2).

Let us recall the definitions of the different relations of resonances for the frequencies (ν1, ...,νn) of Ĥ2

associated with the eigenvalues of the matrix λ′′1 (0). We say that the frequencies (ν1, ...,νn) are resonant if

they are dependent over Z, i.e., there exist integers d1, ...,dn ∈ Z, not all zero, such that d1ν1 +...+dnνn = 0.

The number d =
n∑
j=1

|dj| is called the degree of resonance of Peff (h). In the particular resonant case where

νj = νcdj for all j = 1, ...,n with νc > 0 and d1, ...,dn ∈ N, (νj)j are said to be completely resonant.

As an application we study the structure of Birkhoff normal form in Fermi resonance (νj, 2νj). We

compute the Birkhoff normal form of Peff (h) in the case of 1 : 2 resonance. Fermi resonances provide an

essential mechanism for intramolecular vibrational energy flow and often dominate the vibrational dynamics

in highly excited molecules. First discovered for the CO2 molecule, Fermi resonances are seen for many

molecules.

Fermi resonance. The harmonic oscillator in Fermi resonance is given by:

Ĥ2 =

(
−h2

∂2

∂x21
+ x21

)
+ 2

(
−h2

∂2

∂x22
+ x22

)
(4.1)



Int. J. Anal. Appl. 18 (2) (2020) 192

with symbol H2 = |z1|
2
+2 |z2|

2
where zj = xj+iξj, j = 1, 2. We construct K3 ∈D3 such that 〈H2,K3〉W = 0,

so

K3 =
∑

2`+|α|+|β|=3

h`zαzβ

with 〈ν,β −α〉 = 0. Thus, K3 is generated by the monomials z21z2 and z2z
2
1 and since K3 is real, we can

write:

K3 = ρRe(z
2
1z2), ρ ∈ R.

Consequently, the Birkhoff normal form of the effective hamiltonian Peff (h) in the Fermi resonance,

H2 + W is equal to H2 + K3 + O4, with

ρ =
1

2
√

2

∂3W

∂x21∂x2
(0)

see [8]. The Weyl quantization K̂3 of K3 is given by:

K̂3 = ρOph
(
Re(z21z2)

)
= ρOph

(
x21x2 + 2x1ξ1ξ2 − ξ

2
1x2
)

= ρ

[
x21x2 −h

2

(
2x1

∂2

∂x1∂x2
−x2

∂2

∂x21
+

∂

∂x2

)]
and finally,

Ĥ2 + K̂3 =

(
−h2

∂2

∂x21
+ x21

)
+ 2

(
−h2

∂2

∂x22
+ x22

)
(4.2)

+ρ

[
x21x2 −h

2

(
2x1

∂2

∂x1∂x2
−x2

∂2

∂x21
+

∂

∂x2

)]
.

Acknowledgements: This work is supported by Laboratory of Fundamental and Applicable Mathematics

of Oran (LMFAO) and is dedicated to Professor Bekkai Messirdi on the occasion of his 61th birthday.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

References

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[4] G.D. Birkhoff, Dynamical Systems, AMS Colloq. Publ. 9, AMS New York. (1927).

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J. Contemp. Math. Sciences. 4(35) (2009), 1701-1707.

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187-204.


	1. Introduction
	2. Reduction to an effective operator
	2.1. Pseudodifferential calculus with operator valued symbols
	2.2. BKW solutions (scalar case)
	2.3. BKW solutions (general case)

	3. Reduction to Birkhoff normal form for the effective Hamiltonian
	4. Birkhoff normal form for P(h) in Fermi resonance
	References