International Journal of Analysis and Applications

Volume 18, Number 2 (2020), 262-276

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

DOI: 10.28924/2291-8639-18-2020-262

SEMICLASSICAL RESONANCES VIA MEROMORPHY OF THE RESOLVENT AND

THE S-MATRIX

SOUMIA BELMOUHOUB1,2,∗, BEKKAI MESSIRDI2 AND ABDERRAHMANE

SENOUSSAOUI1,2

1Department of Mathematics, Faculty of Exact and Applied Sciences, University of Oran1 Ahmed Ben

Bella, Algeria

2Laboratory of Fundamental and Applicable Mathematics of Oran (LMFAO), University of Oran1 Ahmed

Ben Bella, Algeria

Corresponding author: belmsou@yahoo.fr

Abstract. The purpose of this paper is to describe the basic problems of resonances via meromorphic

continuation of the resolvent and the scattering matrix. An example from mathematical physics is given

by investigating the poles of the resolvent of semiclassical Schrödinger operators and Born-Oppenheimer

Hamiltonians. Mathematical techniques, dilation-analyticity and Feshbach reduction are used here for the

characterization of resonances of these Hamiltonians.

1. Introduction

The spectrum in the complex plane of Schrödinger operators P(h) = −h2∆ + V, is often the union

of the line Imz = 0 and at most finitely-many points of the form iλj(h) on the positive imaginary axis

λj(h) > 0. These points correspond to the negative eigenvalues of P(h) so that z(h) = −λ2j(h) belongs

to the discrete spectrum of P(h). The resolvent
(
P(h) −λ2

)−1
is an operator-valued function defined for

Imλ > 0 and λ 6= λj(h). We would like to find the largest region in the complex λ-plane on which the

Received 2020-01-11; accepted 2020-01-28; published 2020-03-02.

2010 Mathematics Subject Classification. 35J10, 35P25, 47A56, 47A75.

Key words and phrases. Schrödinger operators; Born-Oppenheimer Hamiltonians; Meromorphic continuation; Resonances;

Scattering matrix; Feshbach reduction; Dilation operator.

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.

262

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-262


Int. J. Anal. Appl. 18 (2) (2020) 263

resolvent can be defined. For several types of potential V , the spectrum of −h2∆ + V is continuous and

equals [0, +∞[, and hence contains no (further) information about V. In this setting resonances replace the

discrete data of eigenvalues. Precisely, the poles of the meromorphic continuation of the resolvent are called

resonances or scattering poles. They constitute a natural remplacement of discrete spectral data for problems

on non-compact domains. The multiplicity of a pole λ0 is given in terms of multiplicity of the corresponding

resonance z0 = λ
2
0, multiplicity of λ0 = dim Imπλ20

(
L2comp(R

n)
)

where:

πλ20 =
1

2πi

∮
γ

(z −P(h))−1dz : L2comp(R
n) −→ L2loc(R

n)

γ : [0, 2π[ 3 t 7→ z0 + εeit.

The resonances are shown to be the same as the poles of the meromorphically continued scattering matrix.

Meromorphic extensions of resolvents have been studied in many frameworks and their finite rank poles

or resonances, serve in a sense as discrete data similar in character to eigenvalues of a compact manifold.

However, if the manifold has constant negative sectional curvature away from a compact, Guillopé and

Zworski [1] showed the meromorphic continuation of the resolvent to C with finite rank poles. For n odd

they are defined as the poles of the meromorphic continuation of
(
P(h) −λ2

)−1
: L2comp(R

n) −→ L2loc(R
n)

from {Imλ > 0} to C or to the Riemann surface (logarithmic covering of C) if n is even. The main advantage

of odd dimensions greater than one is the strong Huyghens principle for the wave equation. Effectively, one

consequence of the strong Huyghens principle is the analytic continuation of
(
−h2∆ −λ2

)−1
from {Imλ > 0}

to C.

Under suitable assumptions on V, the operator P(h) extends as a selfadjoint family of operators on

L2(Rn) with continuous spectrum [0, +∞[, for example if V is real-valued and lim
|x|→∞

V (x) = 0. For λ ∈

C�[0, +∞[, the resolvent RV (λ) = (P(h) −λ)
−1

is an holomorphic function from C�[0, +∞[ to B(L2(Rn))

the algebra of bounded operators on L2(Rn). As operator on L2(Rn), RV (λ) has no analytic extension across

its spectrum. But, if we replace L2(Rn) by a smaller dense subspace, like C∞0 (R
n), then RV (λ) might have

some continuation across [0, +∞[ to some Riemann surface above C�[0, +∞[. If the continuation turns out

to be meromorphic, we then obtain the resonance of P(h) which are exactly the poles of this continuation.

When V = 0, i.e. P(h) = −h2∆, is the free Hamiltonian, the resonances can be accessible using Fourier

analysis. If V 6= 0, many effective approaches combine the known extension of the free resolvent to properties

of V.

The mathematical study of resonances initiated for Schrödinger operators on Rn. Later, it was extended

to more geometric situations, such as the Laplacian on hyperbolic and asymptotically hyperbolic manifolds,

symmetric or locally symmetric spaces, and Damek-Ricci spaces, see e.g. [1] and [2]. In a typical situation,



Int. J. Anal. Appl. 18 (2) (2020) 264

one works on a complete Riemannian manifold X, for which the positive Laplacian −∆ is an essentially

self-adjoint operator on the Hilbert space L2(X) of square integrable functions on X.

The basic problems of resonances are described here for Schrödinger operators and Born-Oppenheimer

Hamiltonians with regular and singular potentials. We first show an holomorphic extension result via Fred-

holm operator theory in Hilbert spaces. In section 2, we review basic situations for meromorphic continuation

of the resolvent. We study in section 3, the meromorphy of the scattering matrix, it follows that the poles

of the meromorphic continuation of the S-matrix are exactly the poles of the continuation of the resolvent

and conversely. Some interesting characterizations of the resonances of semiclassical Schrödinger operators

and Hamiltonians in the Born-Oppenheimer approximation are obtained in section 4, by dilation-analyticity

and Feshbach reduction.

We start with the following definition.

Definition 1.1. Let Ω ⊆ C be open and connected and H a complex Hilbert space. Suppose that A(λ) is a

B(H)-valued analytic function on Ω except for isolated singularities. Then A(λ) is said meromorphic in Ω,

if for each λ0 ∈ Ω, there exist a neighbourhood Uλ0 of λ0, an integer p > 0 and some (Ai)1≤i≤p ⊂B(H) such

that for all λ ∈ Uλ0�{λ0} , we have the finite Laurent expansion:

A(λ) =

p∑
i=1

Ai
(λ−λ0)i

+ B(λ)

where B(λ) is an holomorphic function on Uλ0 with values in the algebra B(H) of bounded linear operators

on H.

It is easy to see that A(λ) is holomorphic in U�S where S is a discrete set of U whose elements are the

poles of A(λ). p is the order of the pole, A1 is the residue of A(λ) at λ0.

We essentially have the following result:

Theorem 1.1. Let Ω ⊆ C be a connected open set and (A(z))z∈Ω a holomorphic family of Fredholm operators

on H. If A(z0)
−1 exists at some z0 ∈ Ω, then z 7→ A(z)−1 is a meromorphic family in Ω of operators with

poles of finite rank.

Proof. For any z ∈ Ω, let n+ = dim ker A(z) and n− = dim HImA(z) and set H+ = C
n+ and H− = Cn−. Let{

e1, ...,en+
}

be a basis of H+. So define

R+ : H −→ H+

x 7→ R+x =
(
〈x,e1〉 , ...,

〈
x,en+

〉)
.



Int. J. Anal. Appl. 18 (2) (2020) 265

Next choose y1, ...,yn− whose images in
H

ImA(z)
form a basis of H

ImA(z)
and define

R− : H− −→ H

x 7→ R−
(
a1, ...,an−

)
=

n−∑
k=1

akyk.

We produce a Grushin problem for A(z) as decribed by Belmouhoub and Messirdi in [3]:

A(z) =


 A(z) R−

R+ 0


 .

A(z) is well-posed Grushin problem for z in some sufficiently small neighborhood V (z) of z with inverse

E(z) =


 E(z) E+(z)

E−(z) E−+(z)


 : H ⊕H+ −→ H ⊕H−

such that A(z) is invertible if and only if E−+(z) is invertible and
 A

−1(z) = E(z) −E+(z)E−1−+(z)E−(z)

E−1−+(z) = −R+A(z)−1R−
(1.1)

In particular, this is true for z = z0 and for z ∈ V (z). We know that the index of A(z) is constant

in V (z). Since the index vanishes at z0, then at any z we have n+ = n− = n and E−+(z) is an n × n

matrix with holomorphic coefficients. So for any z ∈ Ω the function fz(λ) = det E−+(λ) is holomorphic in

a neighborhood of z such that A(λ) is invertible if and only if fz(λ) 6= 0. As Ω is connected and since A(z0)

is invertible for at least one z0 ∈ Ω, none of the functions fz can be identically zero. Since det E−+(z) is

not identically zero, E−+(z) is a meromorphic family of matrices. From the Schur complement (1.1), then

z 7→ A(z)−1 is a meromorphic family of operators with poles of finite rank. �

2. Meromorphic continuation of the resolvent

2.1. Helmholtz operator. Consider the resolvent of Laplacian ∆ =
n∑
k=1

∂2

∂x2
k

defined by:

R0(λ) = (−h2∆ −λ2)−1 : L2(Rn) −→ L2(Rn) for Imλ > 0, h > 0.

Since the classical theorems on the usual Fourier transform extend to the semiclassical case, it suffices

to consider h = 1. Then the existence of R0(λ) follows from using the Fourier transform which provides an

explicit diagonalization of −∆ :

R0(λ)u(x) =
1

(2π)n

∫
Rn

eixξ

|ξ|2 −λ2
û(ξ)dξ, Imλ > 0. (2.1)

(2.1) is of course valid in all dimension but the resolvent operator R0(λ) has much nice properties when n

is odd. We establish the following important result concerning the meromorphic continuation of the resolvent

in odd dimensions.



Int. J. Anal. Appl. 18 (2) (2020) 266

Theorem 2.1. ( [4]) Suppose that n ≥ 3 is odd, then the resolvent operator R0(λ) defined on L2(Rn) for

Imλ > 0, extends analytically to an entire family of operators from L2comp(R
n) to L2loc(R

n).

For any χ ∈ C∞0 (Rn) :

‖χR0(λ)χ‖L2(Rn)→Hk(Rn) = O((1 + |λ|)
k−1

emax(−Imλ,0)R),

k = 0, 1, 2 and supp(χ) ⊂ B(0,R) (the open ball of radius R centered at 0).

Proof. By the functional calculus, we have (see e.g. [4]):

R0(λ) =

+∞∫
0

eiλtU(t)dt, Imλ > 0

where U(t) =
sin(t

√
−∆)

√
−∆ . Now

∂2U
∂t2

= ∆U(t) so U(t) is a operator solution of the wave equation. More

precisely,

u(x,t) = U(t)φ1(x) + U
′(t)φ0(x)

=
sin
(
t
√
−∆

)
√
−∆

φ1(x) + cos
(
t
√
−∆

)
φ0(x)

is the solution of the wave equation with the initial conditions u(x, 0) = φ0(x) and
∂u
∂t

(x, 0) = φ1(x). In

odd dimensions, the strong Huyghens principle (see [5]) implies that:

suppf ⊂ B(0,R) =⇒ (U(t)f) (x) = 0, |x| < t−R.

So, if χ ∈ C∞0 (Rn) with supp(χ) ⊂ B(0,R), then:

χR0(λ)χ =

2R∫
0

eiλtχU(t)χdt, Imλ > 0.

The right hand side is now defined and, as an operator L2(Rn) −→ L2(Rn), holomorphic for λ ∈ C. In

fact,

λχR0(λ)χ =

2R∫
0

Dt
(
eiλt
)
χU(t)χdt,

where Dt =
1
i
∂
∂t
. Since DtU(t) =

1
i

cos
(
t
√
−∆

)
, integration by parts shows that the right hand side is

bounded with a bound depending on R and α = max (−Imλ, 0) . So,

‖χR0(λ)χ‖L2→L2 ≤
eαR

|λ|
.

We can consider U(t) as a map from L2(Rn) to the Sobolev space H1(Rn). Indeed, since sup
λ∈R

∣∣sin tλ
λ

∣∣ = |t| ,
we have:

‖U(t)‖L2→H1 = ‖U(t)‖L2→L2 +
∥∥∥√−∆U(t)∥∥∥

L2→L2
= O (|t|) + O(1)

and integrating shows that:

‖χR0(λ)χ‖L2→H1 = O(e
αR).



Int. J. Anal. Appl. 18 (2) (2020) 267

We also get a bound for χR0(λ)χ as a map from L
2(Rn) to H2(Rn). Recall that the norm on H2(Rn) can

be taken as ‖u‖L2 + ‖∆u‖L2 . So, we have:

‖χR0(λ)χ‖L2→H2 ≤ ‖∆ (χR0(λ)χ)‖L2→L2 + ‖χR0(λ)χ‖L2→L2

≤ ‖χ∆ (R0(λ)χ)‖L2→L2 + ‖[∆,χ] (χ1R0(λ)χ1) χ‖L2→L2

+‖χR0(λ)χ‖L2→L2

where χ1 ∈ C∞0 (Rn) such that χ1 = 1 near supp(χ) and with suppχ1 in a ball of radius R1 > R. Since

(−∆ −λ2)R0(λ) = IL2(Rn) (as operators on functions of compact support), we have:

‖χ∆ (R0(λ)χ)‖L2→L2 = O(λ
2).

Since [∆,χ] is a first order operator, we obtain:

‖[∆,χ] (χ1R0(λ)χ1) χ‖L2→L2 ≤ C‖χ1R0(λ)χ1‖L2→H1 , C > 0.

So

‖χR0(λ)χ‖L2→H2 ≤ |λ|
2 ‖χR0(λ)χ‖L2→L2 + C‖χ1R0(λ)χ1‖L2→H1

+‖χR0(λ)χ‖L2→L2 .

χR0(λ)χ is a bounded operator from L
2(Rn) to L2(Rn) and its image consists of functions supported in

B(0,R). By Rellich’s lemma, the embedding of this space in L2(Rn) is compact. Hence:

‖χR0(λ)χ‖L2→H2 = O(|λ|e
αR1 ).

�

Remark 2.1. Suppose n is odd and R0(λ) : L
2(Rn) −→ L2(Rn) for Imλ > 0. Then the analytic continuation

of the Schwartz kernel R0(λ,x,y) is given by Stone’s formula [6]:

R0(λ,x,y) −R0(−λ,x,y) =
i

2

λn−2

(2π)
n−1

∫
Sn

eiλω(x−y)dω, λ ∈ C

where dω denotes the standard measure on the unit sphere Sn of Rn.

2.2. Schrödinger operators. Let’s study now the resolvent of the Schrödinger operator P = −∆ + V on

L2(Rn) with domain H2(Rn) where V ∈ L∞(Rn,C), n ≥ 3, odd. The resolvent operator RV (λ) =
(
P −λ2

)−1
exists at points λ ∈ C such that Imλ � 0.

Theorem 2.2. (Meromorphic continuation of the resolvent) Suppose that V ∈ L∞comp(Rn,C) (ie V is a.e.

bounded potential of compact support), n ≥ 3 is odd. Then



Int. J. Anal. Appl. 18 (2) (2020) 268

1) RV (λ) : L
2(Rn) −→ L2(Rn) for Imλ > 0, is a meromorphic family of operators with finitely many

poles.

2) RV (λ) extends to a meromorphic family of operators RV (λ) : L
2
comp(R

n) −→ L2loc(R
n) for λ ∈ C.

3) If χ ∈ C∞0 (Rn) then ‖χRV (λ)χ‖L2→L2 ≤
C
λ
, λ > 0.

Proof. We write:

RV (λ) −R0(λ) =
(
−∆ + V −λ2

)
−
(
−∆ −λ2

)
= −RV (λ)V R0(λ)

where χ ∈ C∞0 (Rn) such that χV = V. Multiply the above equation on the right by χ to get

RV (λ)χ−R0(λ)χ = −RV (λ)χV R0(λ)χ,

so

RV (λ)χ (I + V R0(λ)χ) = R0(λ)χ.

But, ‖R0(λ)‖L2→L2 ≤
1

|Imλ|2 , so for large values of Imλ, (I + V R0(λ)χ) is invertible and

RV (λ) −R0(λ) = −R0(λ) (I + V R0(λ)χ)
−1
V R0(λ).

Now V R0(λ)χ = V χR0(λ)χ is compact being the product of a bounded operator with a compact op-

erator. Then, V R0(λ)χ is analytic family of compact operators and we can apply the analytic Fredholm

theory (Theorem 1.2) to conclude that RV (λ) extends as a meromorphic operator valued family of operators

L2comp(R
n) −→ L2loc(R

n) for λ ∈ C.

On the other hand, we have shown that RV (λ)χ = R0(λ) (I + V R0(λ)χ)
−1

is a meromorphic family of

operators. For λ � 1, we have
∥∥∥(I + V R0(λ)χ)−1∥∥∥

L2→L2
≤ 2 and hence the estimate on R0(λ) implies part

(3) of the theorem. �

Remark 2.2. In the even-dimensional case similar results are valid, except that the resolvent operator only

extends to be entire on the logarithmic covering of the complex plane. Precisely, for n even, RV (λ) extends

to be entire as a function of log λ, i.e. entire on the logarithmic covering Λ of C.

3. Meromorphy of the scattering matrix

We have just studied above the meromorphic continuation of the resolvent to C for odd dimensions and

to Λ for even dimensions. From this it can be deduced that the scattering matrix has a similar continua-

tion. Indeed, the meromorphic continuation of the cut-off resolvent χRV (λ)χ permits us to mermorphically

continue the scattering matrix S(λ) as a bounded operator on L2loc(S
n) on C or on Λ depending on the

parity of n. In this section, we will define and describe the scattering matrix of P(h) = −h2∆ + V for

V ∈ L∞comp(Rn,R), n ≥ 3, where L∞comp(Rn,R) is the space of real essentially bounded functions of compact



Int. J. Anal. Appl. 18 (2) (2020) 269

support. P(h) is self-adjoint with domain H2(Rn) and generates a one-parameter strongly continuous uni-

tary group R 3 t 7→ UV (t) = e−itP(h). The unitary group UV (t) provides solutions ψ(t) = UV (t)ψ0, to the

initial value problem: 
 i

∂ψ
∂t

= P(h)ψ

ψ(0) = ψ0 ∈ H2(Rn)

U0(t) is the one-parameter strongly continuous unitary group associated to −h2∆.

Proposition 3.1. 1) For any f ∈ L2(Rn), n ≥ 3, the limits lim
t→±∞

UV (t)
∗U0(t)f exist and define bounded

transformations Ω±(P(h),−∆) or Ω± called wave operators:

Ω±f = lim
t→±∞

UV (t)
∗U0(t)f

‖Ω±‖L2→L2 = 1

2) For any f,g ∈ L2(Rn),

〈Ω±f, Ω±g〉L2 = 〈f,g〉L2(
Ω∗±
)∗

Ω∗± = Ω±Ω
∗
±.

3) The operator F± = Ω±Ω
∗
± satisfies:

F 2± = F±, F
∗
± = F±, F±Ω

∗
± = Ω

∗
±,

∥∥Ω∗±f∥∥ = ‖F±f‖ , ImΩ± = ImF±.
Ω±U0(t) = UV (t)Ω± and U0(t)Ω

∗
± = Ω

∗
±UV (t).

4) The pair (−∆,P(h)) is asymptotically complete in the sense that the wave operators Ω±(−∆,P(h))

exist.

Proof. The existence of wave operators comes from an explicit estimate for the free propagation given by

U0(t) (see e.g. [7]). The relations (2), (3) and (4) follow from the existence of Ω± and the simple properties

of the unitary evolution groups. �

The existence of the wave operators Ω± gives the limits lim
t→±∞

UV (t)
∗U0(t)f = f±. The scattering operator

S maps f− to f+. It is a bounded operator on L
2(Rn) since

Sf− = f+ = Ω
∗
+f = Ω

∗
+Ω−f−,

S = Ω∗+Ω− : L
2(Rn) −→ L2(Rn).

Furthermore, the S-operator commutes with the free time evolution U0(t) :

SU0(t) = Ω
∗
+Ω−U0(t) = Ω

∗
+UV (t)Ω− = (UV (−t)Ω+)

∗
Ω−

= (Ω+U0(−t))
∗

Ω− = U0(t)S.



Int. J. Anal. Appl. 18 (2) (2020) 270

Remark 3.1. It’s simple to show that if ImΩ− (P(h),−∆) = ImΩ+ (P(h),−∆) , then, the S-operator is a

unitary operator on L2(Rn). Hence, the S-operator is invertible and S−1 = S∗.

This allows for a reduction of the S-operator to a family of operators S(λ) defined on L2(Sn) called the

S-matrix. Effectively, for λ ∈ R we can define the scattering operator S (λ) : L2(Sn) −→ L2(Sn) by (see [2]):

S(λ) = IL2(Sn) + A(λ)

with a trace class operator A(λ), and it continues meromorphically to the entire complex plane. Its poles

coincide with the poles of the resolvent with multiplicities, mS(λ), related to the multiplicities of the poles

of the resolvent, mR(λ), by the formula mS(λ) = mR(λ) −mR(−λ). Which gives the following fundamental

result:

Theorem 3.1. The scattering matrix S(λ) admits a meromorphic continuation to C if n is odd, or to the

Riemann surface Λ, if n is even, with poles precisely at the resonances of P(h). The multiplicity of the poles

are the same as the multiplicity of the poles for the resolvent of P(h) and the residues at these poles have

the same finite rank.

Remark 3.2. For the semiclassical Schrödinger operator P(h), the resonances may be defined as the poles

of the meromorphic continuation of the resolvent RV (λ) or in terms of the meromorphic continuation of

the S-matrix S(λ). From [8], it follows that the poles of the meromorphic continuation of the S-matrix are

exactly the poles of the continuation of the resolvent and conversely. However, the scattering poles differ

from the resolvent poles for example for hyperbolic spaces.

4. Resonances of semiclassical Schrödinger Operators and Born-Oppenheimer

Hamiltonians

There are various models for which one can prove the existence of resonances for example that of Stark

hydrogen Schrödinger operator, and also those of semiclassical approximation and especially the resonances

in the Born-Oppenheimer approximation.

4.1. Resonances of semiclassical Schrödinger operators. The theory developed by Hunziker [9], iden-

tify the resonances with the eigenvalues of the deformed hamiltonian Pθ(h), in the lower complex half-plane,

of the Schrödinger operator P(h) = −h2∆+V (x) defined on L2(Rn) with domain D(P(h)) = H2(Rn)∩D(V ).

The resonances do not depend on θ and they are associated with the poles of the meromorphic extension from

the upper complex half-plane of the resolvent of Pθ(h). In order to prove the existence of such continuation

we operate an explicit construction assuming appropriate conditions on the potential.

Let the potential V (x) be smooth real function, extends analytically in |Imθ| < δ0, δ0 > 0, and such that

V (−∆ + 1)−1 is compact. We introduce the resonances of P(h) = −h2∆ + V (x) on L2(Rn) by using the



Int. J. Anal. Appl. 18 (2) (2020) 271

analytic dilation operator:

(Uθϕ) (x) = e
nθ/2ϕ(eθx), ϕ ∈ C∞0 (R

n).

Uθ has a unitary extension on L
2(Rn). It follows that V (eθx)(−∆ + 1)−1 is a compact operator-valued

analytic function of θ in the strip |Imθ| < δ0. Then,

Pθ(h) = U
∗
θ P(h)Uθ = −h

2e−2θ∆ + V (eθx)

is an analytic family of non selfadjoint operators where θ runs in the strip |Imθ| < δ0, since for z ∈ C+

and ϕ,ψ ∈ L2(Rn),

〈R(z)ϕ,ψ〉 = 〈UθR(z)ϕ,Uθψ〉 = 〈[UθR(z)U∗θ ] Uθϕ,Uθψ〉 , (4.1)

where UθR(z)U
∗
θ = Rθ(z) = (Pθ(h) −z)

−1.

Let σess = σ�σdisc be the essential spectrum where the discrete spectrum σdisc is the set of isolated

points of the spectrum such that the corresponding Riesz projectors are finite dimensional. Then, by Weyl

Theorem:

σess (Pθ(h)) = e
−2θσess

(
−h2∆ + e2θV (eθx)

)
= e−2θσess(−h2∆) = e−2θR+.

The definition of resonances is adapted here as follows:

Definition 4.1. A complex number ρ is a resonance of P(h) if Reρ > inf σess(P(h)) and if there exists θ

small enough, Imθ > 0, such that ρ ∈ σdisc(Pθ(h)). Σ(h) denotes the set of resonances of P(h).

It is well known, see for example the works of Messirdi [10], [11], that the resolvent operator R(z) =

(P(h)−z)−1, z ∈ C�R, admits an analytic extension in C+ = {z ∈ C : Imz > 0} and under the assumption

that V (eθx) is analytic, we can extend R(z) to a meromorphic function in a larger domain, the set of poles

of this extension is precisely Σ(h). Effectively,

Pθ(h) −z = −h2e−2θ∆ + V (eθx) −z

=
[
I + V (eθx)

(
−h2e−2θ∆ −z

)−1]
(−h2e−2θ∆ −z).[

I + V (eθx)
(
−h2e−2θ∆ −z

)−1]
is invertible for |Imz| → ∞ and

(
−h2e−2θ∆ −z

)−1 ∈ B(L2(Rn)) if
z ∈ C�e−2θR+. Furthermore, it is easily shown that

∂

∂θ

[〈
Rθ(z)Uθϕ,Uθψ

〉]
= 0 (4.2)

(4.2) implies that
〈
Rθ(z)Uθϕ,Uθψ

〉
is an holomorphic function, for all ϕ,ψ ∈ H2(Rn) and z ∈ Ωδ an

non-empty open subset of C+ such that Ωδ ∩ σ(Pθ(h)) = ∅, |θ| < δ0. However, (4.1) is true for θ ∈ R and

z ∈ C+, the two sides of this equality are holomorphic with respect to θ ∈{|θ| < δ0} . The two holomorphic

functions 〈R(z)ϕ,ψ〉 and 〈UθR(z)ϕ,Uθψ〉 coincide on a subset of R, so equality is still true in {|θ| < δ0} . Now

consider (4.1) with respect to the variable z, the function z 7−→ Rθ(z) is meromorphic on C�σess (Pθ(h)) =



Int. J. Anal. Appl. 18 (2) (2020) 272

C�e−2θR+, and the poles of Rθ(z) are exactly the elements of σdisc (Pθ(h)) . More precisely, we have for all

θ complex such that |θ| < δ0,

σdisc (Pθ(h)) =⋃
ϕ,ψ

{poles of z 7−→ 〈R(z)ϕ,ψ〉}∩
{
z ∈ C : −2Imθ < arg z <

π

2

}
(4.3)

and

Σ(h) =
⋃

Imθ>0, |θ|<δ0

σdisc (Pθ(h)) .

4.2. Resonances of Born-Oppenheimer Hamiltonians. The quantum Hamiltonian in the Born-

Oppenheimer approximation is written as:

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

on L2(Rnx ×Rpy) when h tends to 0+. ∆x (resp. ∆y) is the Laplace operator with respect to x (resp. y),

x ∈ Rn and y ∈ Rp, n = 3m,p = 3q, m,q ≥ 1. Q(x) is the electronic Hamiltonian defined on L2(Rpy). The

purpose of this section is to show that using a general dilation operator:

Uθϕ(x,y) = e
nθ/2ϕ(eθx,y),ϕ ∈ C∞0 (R

n ×Rp)

and the Feshbach reduction scheme, the study of resonances of P(h) is reduced to the discrete spectrum

of a matrix of operators Fθ(z) defined on L2(Rnx) ⊕L2(Rnx) (the so-called effective Hamiltonian) such that:

z is a resonance of P(h) ⇐⇒∃θ ∈ C, Imθ > 0, z ∈ σdisc
(
Fθ(z)

)
.

The dilated Hamiltonian is

Pθ(h) = U
∗
θ P(h)Uθ = −h

2e−2θ∆x + Q(e
θx).

Assume that:

(H1) V ∈ L∞(Rnx ×Rpy,R) and can be analytically extended on the complex strip:

Dδ0 =

{
x ∈ Cn : |Imx| < δ0

(
1 + |Rex|2

)1/2}
.

Thus, P(h) and Q(x) are selfadjoint on their respective natural domains H2(Rnx ×Rpy) and H2(Rpy). In

particular, the domain of Q(x) is independent of x. We suppose furthermore:

(H2) For every x ∈ Rn, the spectrum of Q(x) contains at least two eigenvalues λ1(x) and λ2(x) where

λ2(x) is simple and satisfies:

inf
λ∈σ(Q(x))�{λ2(x)}

|λ−λ2(x)| ≥ δ.

In particular, this last assumption implies that the spectral projector π(x) of Q(x) associated to to the

wave packet {λ1(x),λ2(x)} is C2-regular with respect to x (see [12]). Furthermore, by the mini-max principle,



Int. J. Anal. Appl. 18 (2) (2020) 273

we deduce that λ1(x) and λ2(x) are uniformly bounded with respect to x and can be analytically extended

on Dδ0.

We also assume that λ2(x) has a potential well above the maximum level of λ1(x) :

(H3)

λ2(x) ≥ 0, sup
x∈Rn

λ1(x) < 0, lim
|x|→Rn

λ2(x) > 0, λ
−1
2 (0) = {0} , λ

′′
2 (0) > 0.

By virtue of (H1), Q(eθx) and Pθ(h) can be extended to small enough complex values of θ as analytic

families. We then put a Viriel hypothesis to avoid the resonances coming from the effective potential λ1(x)

near level 0 :

(H4)

sup
x∈Rn

(2λ1(x) + x.∇λ1(x)) < 0.

Let uj(x,y), j = 1, 2, the two eigenfunctions of Q(x) associated to λ1(x) and λ2(x) respectively, real

and normalized in L2(Rpy). We then consider a Grushin problem that will lead to the Feshbach reduction

( [3]). For z,θ ∈ R, let Aθz(h) the matrix operator defined from H2(Rnx × Rpy) ⊕ L2 (Rnx) ⊕ L2 (Rnx) to

L2(Rnx ×Rpy) ⊕L2 (Rnx) ⊕L2 (Rnx) by:

Aθz(h) =




Pθ(h) −z uθ1 uθ2〈
.,uθ1

〉
y

0 0〈
.,uθ2

〉
y

0 0




where uθj (x,y) = uj(xe
θ,y), j = 1, 2, and 〈., .〉y denotes the inner product in L

2(Rpy). We shall study the

extension of Aθz(h) to z and θ complexes, for this we have the following results:

Q(eθx) −Q(x) =
(
V (xeθ,y) −V (x,y)

)
∈B(L2(Rn))

and by the assumption (H1) :

∂

∂θ

(
V (xeθ,y) −V (x,y)

)
= eθx.∇xV (xeθ,y) = O(1),

uniformly with respect to x,y and θ complex such that |θ| small enough. Furthermore, for j = 1, 2,

λj(xe
θ) −λj(x) = O(|θ|),

and uj extends into a holomorphic function on Dδ0 with values in H
2(Rpy), such that:

∥∥∂αxuθj∥∥y = O((1 + |x|2)−|α|/2),∥∥∂αxuθj −∂αxuj∥∥y = O(|θ|(1 + |x|2)−|α|/2)
uniformly with respect to x and θ complex, |θ| small enough.



Int. J. Anal. Appl. 18 (2) (2020) 274

We now define on L2(Rnx ×Rpy), for θ complex, |θ| small enough, the projector πθ by:

πθu =
〈
u,uθ1

〉
y
uθ1 +

〈
u,uθ2

〉
y
uθ2

and π̂θ = 1−πθ. It is therefore simple to show, using previous results, that there is a constant C > 0 such

that for θ complex, |θ| small enough,

Re〈π̂θ (Pθ(h) −z) π̂θu,π̂θu〉y ≥ C‖π̂θu‖
2

(4.4)

for all u ∈ L2(Rnx ×Rpy) and |z| small enough. In particular, the estimate (4.4) shows the existence of a

bounded inverse for (P ′θ(h) −z) the restriction of π̂θ (Pθ(h) −z) to
{
u ∈ L2(Rnx ×Rpy) : π̂θu = u

}
. It is then

elementary to verify that Aθz(h) is invertible, and its inverse A
θ
z(h)

−1 is given by ( [3]):


Xθ(z) u
θ
1 −Xθ(z)Pθ(h)

(
.uθ1
)

uθ2 −Xθ(z)Pθ(h)
(
.uθ2
)〈

(1 −Pθ(h)Xθ(z)) (.),uθ1
〉
y

z −
〈
Yθ(z)

(
.uθ1
)
,uθ1

〉
y

〈
Yθ(z)

(
.uθ2
)
,uθ1

〉
y〈

(1 −Pθ(h)Xθ(z)) (.),uθ2
〉
y

〈
Yθ(z)

(
.uθ1
)
,uθ2

〉
y

z −
〈
Yθ(z)

(
.uθ2
)
,uθ2

〉
y


 (4.5)

where Xθ(z) = (P
′
θ(h) −z)

−1
π̂θ and Yθ(z) = Pθ(h) −Pθ(h)Xθ(z)Pθ(h).

Fθ(z) =



〈
Yθ(z)

(
.uθ1
)
,uθ1

〉
y

〈
Yθ(z)

(
.uθ2
)
,uθ1

〉
y〈

Yθ(z)
(
.uθ1
)
,uθ2

〉
y

〈
Yθ(z)

(
.uθ2
)
,uθ2

〉
y


 (4.6)

is called the Feshbach operator, it reduces the initial spectral problem to a problem in L2 (Rnx)⊕L2 (Rnx) .

It will also serve to show that we have a theory of resonances for P(h). Indeed, we have:

Fθ(z) = UθFU
−1
θ + R

θ(z,h)

F =


 −h2∆x + λ1(x) 0

0 −h2∆x + λ2(x)




and

Rθ(z,h) =
 −h2

〈
∆xu

θ
1,u

θ
1

〉
y
−
〈
Zθ(z)

(
.uθ1
)
,uθ1

〉
y

〈
Yθ(z)

(
.uθ2
)
,uθ1

〉
y〈

Yθ(z)
(
.uθ1
)
,uθ2

〉
y

−h2
〈

∆xu
θ
2,u

θ
2

〉
y
−
〈
Zθ(z)

(
.uθ2
)
,uθ2

〉
y




where Zθ(z) = Pθ(h) −Yθ(z) = Pθ(h)Xθ(z)Pθ(h), and for all m ∈ Z,∥∥Rθ(z,h)∥∥B(Hm(Rn)⊕Hm(Rn),Hm−1(Rn)⊕Hm−1(Rn)) = O(h2).
On the other hand, using (H3) and (H4),

(
F̃θ −z

)
is boundedly invertible from L2 (Rnx) ⊕ L2 (Rnx) to

H2 (Rnx) ⊕ H2 (Rnx) , for z complex, |z| small enough, where F̃θ = UθFU
−1
θ + W, W ∈ C

∞
0 (R

n) such that

Re
(
W + λθ2(x)

)
> 0.



Int. J. Anal. Appl. 18 (2) (2020) 275

Thus, Kθ(z,h) =
(
Rθ(z,h) −W(x)

)(
F̃θ −z

)−1
is a compact operator on L2 (Rnx)⊕L2 (Rnx) , for θ and z

complex small enough. Kθ(z,h) depends analytically on z and lim
z∈R,z→−∞

∥∥Kθ(e−2θz,h)∥∥ = 0. By Theorem
1.2, we deduce that (I + Kθ(z,h))−1 is a z-meromorphic family for z in a complex neighborhood of 0. So it

is the same for (
Fθ(z) −z

)−1
=
(
F̃θ −z

)
(I + Kθ(z,h))−1

and (see [3]):

(Pθ(h) −z)−1 = Xθ(z) + Aθ+(z)
(
Fθ(z) −z

)−1
Aθ−(z) (4.7)

with

Aθ+(z) =
((

1 −Xθ(z)Pθ(h))(.uθ1
)
,
(
1 −Xθ(z)Pθ(h))(.uθ2

))
Aθ−(z) = A

θ
+(z)

∗.

We can also deduce by construction that the spectra of Pθ(h) is discrete near 0 and z ∈ Σ(h) if and only

if there exists θ ∈ C, Imθ > 0, |θ| small enough such that z ∈ σdisc
(
Fθ(z)

)
.

(4.7) shows that (Pθ(h)−z)−1 extends into a meromorphic function in z near 0, and we have exactly like

in (4.3) :

⋃
θ∈C,|θ|<δ

⋃
ϕ,ψ

{
Poles of

〈
(Pθ(h) −z)−1ϕ,ψ

〉
L2

⋂
(]−ε,ε[ + i ]−ε,ε[)

}
=

⋃
θ∈C,|θ|<δ

{σ (Pθ(h)) ∩ (]−ε,ε[ + i ]−ε,ε[)}

is the set of resonances Σ(h) of the Hamiltonian P(h) in ]−ε,ε[ + i ]−ε,ε[ , ε > 0.

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

of this paper.

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Int. J. Anal. Appl. 18 (2) (2020) 276

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	1. Introduction
	2. Meromorphic continuation of the resolvent
	2.1. Helmholtz operator
	2.2. Schrödinger operators

	3. Meromorphy of the scattering matrix
	4. Resonances of semiclassical Schrödinger Operators and Born-Oppenheimer Hamiltonians
	4.1. Resonances of semiclassical Schrödinger operators
	4.2. Resonances of Born-Oppenheimer Hamiltonians

	References