Electronic Journal of Qualitative Theory of Differential Equations 2006, No. 18, 1-11; http://www.math.u-szeged.hu/ejqtde/

Location of Resonances Generated by Degenerate Potential Barrier

Hamadi BAKLOUTI and Maher MNIF

Keywords: Resonances; Asymptotic; Schrödinger operators Classification (MSC 2000) : 35P15, 35P20

Abstract

We study resonances of the semi-classical Schrödinger operator H = −h^{2}Δ+V on L^{2}(IR^{N}). We consider the case where the potential V have an absolute degenerate maximum. Then we prove that H has resonances with energies E = V_{0} + e^{−i \frac{π}{σ+1}} h^{\frac{2σ}{σ+1}} k_{j} + O(h^{\frac{2σ+1}{σ+1}} ), where k_{j} is in the spectrum of some quartic oscillator.

1 Introduction and main results.

We are interested in this paper to study the asymptotic behavior for Schrödinger operator:

H = −h^{2}Δ + V  (1)

on L^{2}(IR^{N}), where V is a bounded real function having an absolute maximum V_{0} realized at a unique point that we suppose to be x = 0. Under some assumptions on V, one can define on L^{2}(IR^{N}) the operator H_{θ}(x, hD) with domain the Sobolev space W^{2}(IR^{N}) obtained by analytic dilation :

H_{θ}(x, hD) = U_{θ}H(x, hD)U_{−θ} = e^{−2iθ}h^{2}D^{2} + V (xe^{iθ}).

Here U_{θ} denotes the group of unitary operators on L^{2}(IR^{N}), given by U_{θ}u(x) = e^{− \frac{iNθ}{2}} u(e^{iθ}x), for θ in IR such that 0 ≤ θ < θ_{0} < \frac{π}{2}. D denotes the differential operator (−i\frac{∂}{∂x_{1}} , −i\frac{∂}{∂x_{2}} , ..., −i\frac{∂}{∂x_{N}}). Isolated eigenvalues of H_{θ} with finite multiplicities which are not discrete eigenvalues of H are called resonances of H. In [2] and [3] Ph.Briet, J.M.Combes and P.Duclos consider the same problem with V having a non degenerate maximum at x = 0. They prove existence of resonance values of H near 0. They prove too that these values are located outside a ball of radius O(h). In the second paper they give the first factor in the asymptotic expansion of resonance values of H. We are interested here to generalize results of [2] and [3] to cases of degenerate potential barrier of order σ i.e V (x) ∼ −(λ_{1}x_{1}^{2σ} + ... + λ_{N}x_{N}^{2σ}) + O(x^{2σ+1}) as x → 0 in IR^{N}. Here σ > 1 is an integer and λ_{1}, … λ_{N} some strictly positive real constants. We prove that resonance values of H are located outside a ball centered at 0 and of radius O(h^{\frac{2σ}{σ+1}} ). This power of h exists already in the results of Martinez-Rouleux [5]. The authors study there asymptotic of eigenvalues of the operator H in the case of degenerate minimum of order σ for V. They prove that H has no eigenvalue inside a ball centered at 0 and of radius O(h^{\frac{2σ}{σ+1}} ). In fact this result is closely related to the Taylor expansion of V near 0. Our second aim is to get more precise result under more precise assumption on the Taylor expansion of V. We notice that in a recent paper [1] we study the resonances of H in the one dimensional case where the potential V have a degenerate maximum of quartic type. We give in that paper the full asymptotic expansion of resonance values near the barrier maximum. The method used there is the BKW techniques. This method is specific to the one dimensional case. Here we use local analysis on the resolvent of the operator. To state the results assume the following hypothesis on V: (A_{1}) V is a bounded real function having an absolute degenerate maximum of order σ at x = 0. (A_{2}) V_{θ}(x) = V (e^{θ}x), θ ∈ IR has an analytic continuation to complex θ in S_{α} = {θ ∈ I C, | Imθ |< α} as a family of bounded operators. (A_{3}) ∃θ_{0} = iβ_{0}, 0 < β_{0} < α, ∀δ > 0, ∃C_{δ} > 0, such that

∀x, | x |> δ, Im(e^{2θ_{0}}V_{θ_{0}}) < −C_{δ}β_{0}.

(A_{4}) Near x = 0 , V_{θ} has the Taylor expansion :

V_{θ}(x) = −e^{2σθ}(λ_{1}x_{1}^{2σ} + ... + λ_{N}x_{N}^{2σ}) + O(x^{2σ+1})

for all θ ∈ S_{α}, with λ_{1}, ...λ_{N} non vanishing positive real constants. Consider the operator given by :

K = −Δ + (λ_{1}x_{1}^{2σ} + ... + λ_{N}x_{N}^{2σ}).

It is well known that the spectrum of K is discrete. We use (k_{j})_{j≥1} for eigenvalues of K. We recall that the operator H_{θ} has a domain D(H_{θ}) = W^{2}(IR^{N}) independent of θ. This operator has an analytic continuation to an analytic family of type A in S_{α} (see [4] and [6], section XIII). We get :

Théorème 1 Consider the operator H_{θ_{0}} = −h^{2}e^{−2θ_{0}}Δ+V_{θ_{0}} and assume the hypothesis (A_{1,2,3,4}) on V. For all k_{j} ∈ σ(K) there exits a ball B(h) centered at e^{−i \frac{π}{σ+1}} h^{\frac{2σ}{σ+1}} k_{j} with radius O(h^{\frac{2σ+1}{σ+1}} ) such that for h small enough , H_{θ_{0}} has purely discrete spectrum inside B(h) with total algebraic multiplicity equal to the multiplicity of k_{j}.

In the direction to get the full asymptotic expansion of resonance values of H, we try more subtle approach and we get the following result: Assume (A_{5}) Near x = 0 , V_{θ} has the Taylor expansion :

V_{θ}(x) = −e^{2σθ}(λ_{1}x_{1}^{2σ} + ... + λ_{N}x_{N}^{2σ}) + O(x^{2σ+p})

with p > 0 an integer.

Théorème 2 Assume (A_{1,2,3,5}). For all k_{j} ∈ σ(K) there exits a ball B(h) centered at e^{−i \frac{π}{σ+1}} h^{\frac{2σ}{σ+1}} k_{j} with radius O(h^{\frac{2σ+p}{σ+1}} ) such that for h small enough, H_{θ_{0}} has purely discrete spectrum inside B(h) with the same algebraic multiplicity as k_{j}.

Finally to interpret the eigenvalues of H_{θ_{0}} mentioned in the Theorem 1 as resonance values of H we state the following theorem :

Théorème 3 Let ϕ be in the domain D of analytic dilation dense in L^{2}(IR^{N}). For all C > 0 and h small enough ((H − z)^{−1}ϕ, ϕ) has meromorphic continuation from I C^{+} = {z ∈ I C, Imz > 0} into a complex disk centered at 0 with radius Ch^{\frac{2σ}{σ+1}} . The poles of this continuation belong to the set of the eigenvalues of H_{θ_{0}} given in theorem 1.

This theorem shows in particular that there is no other part of the spectrum of H_{θ_{0}} in the disk other than the eigenvalues mentioned in the Theorem 1. The authors of [2] and [3] established the same result in the case σ = 1.

2 Localisation Formula

To prove theorems we decompose the operator H into a direct sum of an interior operator and an exterior one. To do this we use j_{i} and j_{e} ∈ C^{∞}(IR^{N}), with j_{i} = 0 outside the ball B(0, h^{\frac{1}{σ+1}} δ_{i}) et j_{e} = 0 inside the ball B(0, h^{\frac{1}{σ+1}} δ_{e}), satisfying to j_{i}^{2} + j_{e}^{2} = 1. Here 0 < δ_{e} < δ_{i} are two positive constants to be chosen later. The support of ∇j_{i} and ∇j_{e} is located in Ω_{0} = {x ∈ IR^{N}, 0 < h^{\frac{1}{σ+1}} δ_{e} <| x |< h^{\frac{1}{σ+1}} δ_{i}\}. Let H = L^{2}(IR^{N}) and H_{d} = L^{2}(IR^{N}) ⊕ L^{2}( support of j_{e}), we define the application J by

{J : H_{d} → H \atop J(u ⊕ v) = j_{i}u + j_{e}v.}

We have JJ^{∗} = id_{H}. For Imθ ≠ 0, we define H_{θ}^{d} = H_{θ}^{i} ⊕ H_{θ}^{e} where

H_{θ}^{i} = −h^{2}e^{−2θ}Δ − e^{2σθ}(λ_{1}x_{1}^{2σ} + ... + λ_{N}x_{N}^{2σ})  (2)

on L^{2}(IR^{N}) and

H_{θ}^{e} = −h^{2}e^{−2θ}Δ + V_{θ}

on L^{2}( support of j_{e}) with Dirichlet conditions on the boundary of the support of j_{e}. Hence we have J(D(H_{θ}^{d})) ⊂ D(H_{θ}) and we can write the resolvent equation:

(H_{θ} − z)^{−1} = J(H_{θ}^{d} − z)^{−1}J^{∗} − (H_{θ} − z)^{−1}Π(H_{θ}^{d} − z)^{−1}J^{∗}  (3)

where Π is the operator given by Π = H_{θ}J − JH_{θ}^{d} acting on H_{d} as follows:

Π(u ⊕ v) = wu − h^{2}e^{−2θ}([Δ, j_{i}]u + [Δ, j_{e}]v)

where

w = (V_{θ} + e^{2σθ}(λ_{1}x_{1}^{2σ} + … + λ_{N}x_{N}^{2σ}))j_{i}.

For the study of the interior operator H_{θ}^{i} one prove the following lemma:

Lemme 1 Let H_{θ}^{i} defined by (2) with domain D(H_{θ}^{i}) = W^{2}(IR^{N}) ∩ D(x^{2σ}) we have :

i) \{H_{θ}^{i}, 0 < Imθ < \frac{π}{σ+1}\} is an analytic family of type A of sector-operators with sector

Σ = {z ∈ I C, −π + 2Imθ ≤ argz ≤ −2Imθ}.

ii) The spectrum of H_{θ}^{i} is θ independent, purely discrete and given by :

σ(H_{θ}^{i}) = σ(H_{i \frac{π}{2σ+2}}^{i}) = e^{−i \frac{π}{σ+1}} h^{\frac{2σ}{σ+1}} σ(K).

Proof : The proof of i) can be reduced to prove the same properties for the one dimensional operator :

h_{θ} = −h^{2}e^{−2θ} \frac{∂^{2}}{∂x^{2}} − e^{2σθ}x^{2σ}.

the domain of h_{θ} is D(h_{θ}) = W^{2}(IR) ∩ D(x^{2σ}). To prove that the family (h_{θ}) is analytic of type A, one proves that h_{θ} is closed for all θ, 0 < Im(θ) < \frac{π}{σ+1}. For this we have to establish for all θ = iβ such that 0 <| β − \frac{π}{2σ+2} |< \frac{π}{2σ+2}, the following inequality :

∥ h_{θ}u ∥^{2} + ∥ u ∥^{2}≥ c(h^{4} ∥ u′′ ∥^{2} + ∥ x^{2σ}u ∥^{2})  (4)

on D(h_{θ}), where c is a positive constant. We have:

∥h_{θ}u∥^{2} = h^{4}∥u′′∥^{2} + ∥x^{2σ}u∥^{2} + 2h^{2}Re(e^{−i(2+2σ)β}(u′′, x^{2σ}u))

We write

h^{2}(u′′, x^{2σ}u) = −h^{2} ∫ x^{2σ}u′\overline{u}′dx − h^{2} ∫ 2σx^{2σ−1}u′\overline{u}dx.  (5)

One can estimate the second integral in this equation as follows :

h^{2}| ∫ 2σx^{2σ−1}u′\overline{u}dx| ≤ ∥h^{2}u′∥^{2} + ∥2σx^{2σ−1}u∥^{2}.

By integration by parts we get:

∥h^{2}u′∥^{2} ≤ h^{8}∥u′′∥^{2} + ∥u∥^{2}.  (6)

Let B(R) denotes the ball centered at 0 with radius R > 0 and B(R)^{c} the complementary set of this ball. We write :

∥2σx^{2σ−1}u∥^{2} = ∫_{B(R)∪B(R)^{c}}(2σx^{2σ−1})^{2}u\overline{u}dx

≤ 4σ^{2}R^{4σ−2}∥u∥^{2} + \frac{4σ^{2}}{R^{2}} ∥x^{2σ}u∥^{2}.

This yields in combination with (6):

h^{2}| ∫ 2σx^{2σ−1}u′\overline{u}dx| ≤ h^{8}∥u′′∥^{2} + ∥u∥^{2} + 4σ^{2}R^{4σ−2}∥u∥^{2} + \frac{4σ^{2}}{R^{2}} ∥x^{2σ}u∥^{2}. (7)

For the first integral in the equation 5 :

2h^{2}| ∫ x^{2σ}u′\overline{u}′dx| ≤ h^{4}∥u′′∥^{2} + ∥x^{2σ}u∥^{2} + 2h^{2}| ∫ 2σx^{2σ−1}u′\overline{u}dx|. (8)

We now write :

∥h_{θ}u∥^{2} = h^{4}∥u′′∥^{2}+∥x^{2σ}u∥^{2}+2Re[e^{−i(2+2σ)β}(−h^{2} ∫ x^{2σ}u′\overline{u}′dx−h^{2} ∫ 2σx^{2σ−1}u′\overline{u}dx)]

≥ h^{4}∥u′′∥^{2}+∥x^{2σ}u∥^{2}−2| cos((2+2σ)β)|h^{2} ∫ x^{2σ}u′\overline{u}′dx−2h^{2}| ∫ 2σx^{2σ−1}u′\overline{u}dx|

Therefore by (7) and (8)we get :

∥h_{θ}u∥^{2} ≥ [1 − | cos((2 + 2σ)β)| − 2h^{4}(| cos((2 + 2σ)β)| + 1)]∥h^{2}u′′∥^{2}+

(1 − | cos((2 + 2σ)β)| − 8(| cos((2 + 2σ)β)| + 1)\frac{σ^{2}}{R^{2}} )∥x^{2σ}u∥^{2}−

2(| cos((2 + 2σ)β)| + 1)(1 + 4σ^{2}R^{4σ−2})∥u∥^{2}.

Finally choosing R big enough we get (4).

To prove ii), notice that H_{θ}^{i} has a compact resolvent for 0 < Imθ < \frac{π}{σ+1}. By analyticity of (H_{θ}^{i})_{θ} the spectrum of H_{θ}^{i} is independent of θ, hence σ(H_{θ}^{i}) = σ(H_{i \frac{π}{2σ+2}}^{i}). Using the scaling of order h^{\frac{1}{σ+1}} we get:

H_{i \frac{π}{2σ+2}}^{i} = e^{−i \frac{π}{σ+1}} h^{\frac{2σ}{σ+1}} K.

Therefore:

σ(H_{θ}^{i}) = e^{−i \frac{π}{σ+1}} h^{\frac{2σ}{σ+1}} σ(K).

In the exterior domain our operator H here has the same shape as this one studied in [3]. Here we localize more closely to the boss of the barrier. So we get more precise results than lemma II-5 in [3].

Lemme 2 Under assumptions (A_{1,2,3,4}) and for θ_{0} = iβ_{0} we have :

i) The resolvent set of H_{θ_{0}}^{e} contains a complex neighborhood Υ of 0 in the form:

Υ = \{z ∈ I C, Im(e^{2θ_{0}}(−z)) < Cβ_{0}h^{\frac{2σ}{σ+1}} \}.

Here C > 0 is h-independent constant. ii) For all z ∈ Υ we have,

∥ (H_{θ_{0}}^{e} − z)^{−1} ∥≤ \frac{1}{dist(z, ∂Υ)}.

Proof: We first notice that by assumption A_{4} there exists ϵ > 0 independent of h such that for θ_{0} = iβ_{0} with β_{0} > 0 small enough we get for all | x |< ϵ:

\begin{aligned}Im(e^{2θ_{0}}V_{θ_{0}}(x)) & = −sin((2σ + 1)β_{0})(λ_{1}x_{1}^{2σ} + … + λ_{N}x_{N}^{2σ}) + O(x^{2σ+1}) \\ & ≤ −\frac{1}{2}sin((2σ + 1)β_{0})(λ_{1}x_{1}^{2σ} + ... + λ_{N}x_{N}^{2σ}). \end{aligned}

Therefore we get ∀x such that h^{\frac{1}{σ+1}} δ_{e} <| x |< ϵ :

Im(e^{2θ_{0}}V_{θ_{0}}) < −Cβ_{0}h^{\frac{2σ}{σ+1}} ,

for a convenient C > 0 independent of h. Hence by this and assumption A_{3} we get:

∀x, | x |> h^{\frac{1}{σ+1}} δ_{e}, Im(e^{2θ_{0}}V_{θ_{0}}) < −Cβ_{0}h^{\frac{2σ}{σ+1}} .  (9)

Let u ∈ D(H_{θ_{0}}^{e}) be such that ∥ u ∥= 1, and let z ∈ Υ. Since (Δu, u) has a real value, we get by (9):

\begin{aligned}Im(e^{2iβ_{0}}(H_{θ_{0}}^{e} − z)u, u) & = Im(e^{2iβ_{0}}(V_{θ_{0}} − z)u, u) \\ & ≤ Im(−e^{2iβ_{0}}z − Cβ_{0}h^{\frac{2σ}{σ+1}} ) \\ & ≤ −dist(z, ∂Υ) \end{aligned} ,

where (., .) denotes the inner product in L^{2}(IR^{N}). Hence we get:

\begin{aligned}∥ (H_{θ_{0}}^{e} − z)u ∥ & ≥| Im(e^{2iβ_{0}}(H_{θ_{0}}^{e} − z)u, u) | \\ & ≥ dist(z, ∂Υ). \end{aligned}  (10)

This proves that Ker(H_{θ_{0}}^{e} − z) = {0} and (H_{θ_{0}}^{e} − z) has a closed image. On the other hand (H_{θ_{0}}^{e} − z)^{∗} = He ; θ̄_{0} − z̄, and we get by the same way:

∥ (H_{θ_{0}}^{e} − z)^{∗}u ∥≥ dist(z̄, ∂Ῡ).

Therefore Ker(H_{θ_{0}}^{e} −z)^{∗} = {0}, and hence the image of (H_{θ_{0}}^{e} −z) is the whole L^{2}(supportj_{e}). This leads to i). For ii) it is immediately given by (10).

This lemma means that the operator valued function z ↦ (H_{θ_{0}}^{e} − z)^{−1} is holomorphic inside Υ. By the same argument one gets the following:

Lemme 3 Let g ∈ C_{0}^{∞}(IR^{N}) with support(g) ⊂ \{x ∈ IR^{N}, 0 < h^{\frac{1}{σ+1}} δ <| x | }. The operator valued functions z ↦ g(H_{θ_{0}}^{d} − z)^{−1}J^{∗} and z ↦ (H_{θ_{0}} − z)^{−1}g have no poles inside Υ for a convenient C.

Let now k_{j} ∈ σ(K) and γ denotes the closed curve given by:

γ = \{z ∈ I C, | z − E^{d} |= ρh^{\frac{2σ}{σ+1}} \}.  (11)

Here E^{d} = e^{−i \frac{π}{σ+1}} h^{\frac{2σ}{σ+1}} k_{j} and ρ > 0 is a constant independent of h to be chosen later small enough. We have:

Lemme 4 Let γ be the closed curve defined by (11), then for ρ small enough we have by assumptions (A_{1,2,3,4}) :

i) γ is in the resolvent set of H_{θ_{0}}. ii) ∥ (H_{θ_{0}} − z)^{−1} ∥= O(h^{\frac{−2σ}{σ+1}} ) uniformly for z ∈ γ as h ⟶ 0.

By our localization formula close to the boss of the barrier, the proof of this lemma seems to be much more complicated than if we do this using standard localization formula as in [3]. This is largely due to the contribution of the commutator part in Π in the equation (3). The estimation of this contribution becomes here not precise. By the same way as in [3] we get the same results. To avoid needless repetition we shall not rewrite proof.

3 Proof of Theorems

The following lemma shows that the spectrum of H_{θ_{0}} is discrete inside γ. The total algebraic multiplicity is equal to the multiplicity of E^{d} as an eigenvalue of H_{θ_{0}}^{i}.

Lemme 5 Let γ be the closed curve given by (11), with ρ small enough but independent of h. Then under assumptions (A_{1,2,3,4}) for h small enough, P_{θ_{0}} and P_{θ_{0}}^{i} have the same rank. where P_{θ_{0}} and P_{θ_{0}}^{i} denote the projectors defined respectively by:

P_{θ_{0}} = −\frac{1}{2iπ} ∮_{γ}(H_{θ_{0}} − z)^{−1}dz

and

P_{θ_{0}}^{i} = −\frac{1}{2iπ} ∮_{γ}(H_{θ_{0}}^{i} − z)^{−1}dz.

Proof : The lemma 4 shows that the operator P_{θ_{0}} is well defined for h small enough. By (3) we get

P_{θ_{0}} − j_{i}P_{θ_{0}}^{i}j_{i} = −\frac{1}{2iπ} ∮_{γ}(H_{θ_{0}} − z)^{−1}Π(H_{θ_{0}}^{d} − z)^{−1}J^{∗}dz.

We have for s = i and s = e, [Δ, j_{s}] = (Δj_{s} +2∇j_{s}∇) is supported in Ω_{0}. Let g ∈ C_{0}^{∞}(IR^{N}) with support(g) ⊂ \{x ∈ IR^{N}, 0 < h^{\frac{1}{σ+1}} δ <| x |\}, with δ < δ_{e} and satisfying to g(x) = 1 inside Ω_{0}. We have [Δ, j_{s}] = g[Δ, j_{s}]g. Then by lemma 3 the commutator part in the expression of Π has no contribution to this integral. We get

P_{θ_{0}} − j_{i}P_{θ_{0}}^{i}j_{i} = −\frac{1}{2iπ} ∮_{γ}(H_{θ_{0}} − z)^{−1}w(H_{θ_{0}}^{i} − z)^{−1}j_{i}dz.

Let us now prove that :

∥ (H_{θ_{0}} − z)^{−1}w(H_{θ_{0}}^{i} − z)^{−1}j_{i} ∥= O(h^{\frac{1−2σ}{σ+1}} )  (12)

uniformly for z ∈ γ. We have:

∥ (H_{θ_{0}} − z)^{−1}w(H_{θ_{0}}^{i} − z)^{−1}j_{i} ∥≤∥ (H_{θ_{0}} − z)^{−1} ∥ . ∥ w(H_{θ_{0}}^{i} − z)^{−1}j_{i} ∥ .

By lemma 1 ii) one can choose ρ such that dist(γ, σ(H_{θ_{0}}^{i})) > ch^{\frac{2σ}{σ+1}} . Hence one gets

∥ (H_{θ_{0}}^{i} − z)^{−1} ∥= O(h^{\frac{−2σ}{σ+1}} )  (13)

uniformly for z ∈ γ. Now by (A_{4}) we have w = O(h^{\frac{1+2σ}{σ+1}} ) on the support of j_{i}. Then we get:

∥ w(H_{θ_{0}}^{i} − z)^{−1} ∥= O(h^{\frac{1}{σ+1}} ).  (14)

This equation in combination with lemma 4 ii) leads to (12). Since | γ |= O(h^{\frac{2σ}{σ+1}} ), we get :

P_{θ_{0}} − j_{i}P_{θ_{0}}^{i}j_{i} = O(h^{\frac{1}{σ+1}} ).

This leads to :

P_{θ_{0}} − P_{θ_{0}}^{i} = O(h^{\frac{1}{σ+1}} ),

and then smaller than 1 for h small enough, and the lemma is proved.

To finish the proof of theorem 1, let E be an eigenvalue of H_{θ_{0}} inside γ, and let ϕ be the corresponding normalized eigenvector. We have :

(E − E^{d})ϕ = (H_{θ_{0}} − E^{d})ϕ = −\frac{1}{2iπ} ∫_{γ}(z − E^{d})(H_{θ_{0}} − z)^{−1}ϕdz.

By equation (3) and the fact that the commutator part has no contribution to the integral we get:

(E − E^{d})ϕ = −\frac{1}{2iπ} ∫_{γ}(z − E^{d})(H_{θ_{0}} − z)^{−1}w(H_{θ_{0}}^{i} − z)^{−1}j_{i}ϕdz.  (15)

Then

| E − E^{d} |≤ ρ^{2}h^{\frac{4σ}{σ+1}} sup_{z∈γ} ∥(H_{θ_{0}} − z)^{−1}w(H_{θ_{0}}^{i} − z)^{−1}j_{i}∥.

Therefore equation (12) leads to

\begin{aligned}| E − E^{d} | & ≤ ρ^{2}h^{\frac{4σ}{σ+1}} O(h^{\frac{1−2σ}{σ+1}} ) \\ & = O(h^{\frac{1+2σ}{σ+1}} ). \end{aligned}

This proves the theorem 1. To prove theorem 2, Assuming moreover (A_{5}), the equation (14) becomes

∥ w(H_{θ_{0}}^{i} − z)^{−1} ∥= O(h^{\frac{p}{σ+1}} ).

Then equation (12) becomes

∥ (H_{θ_{0}} − z)^{−1}w(H_{θ_{0}}^{i} − z)^{−1}j_{i} ∥= O(h^{\frac{p−2σ}{σ+1}} ).

Hence by equation (15) the following holds true:

| E − E^{d} |= O(h^{\frac{p+2σ}{σ+1}} ).

Therefore all eigenvalues of H_{θ_{0}} inside γ are inside the ball centered at E^{d} and of radius O(h^{\frac{p+2σ}{σ+1}} ). This in combination with the theorem 1 proves the theorem 2.

The proof of theorem 3 is similar to the proof of theorem (2.4) in [3], we have only to replace h by h^{\frac{2σ}{σ+1}} .

References

[1] Baklouti (H.) , Mnif (M.), – Asymptotique des résonances pour une barrière de potentiel dégénérée, Asymptotic Analysis V.47, N.1-2, pp 19-48 ,2006

[2] Briet (Ph.),Combes (J.M.), Duclos (P.) – On the location of resonances for Schrödinger operator in the semiclassical limit I: Resonances free domains, J. Math. anal. appl. 126, 90-99, 1987.

[3] Briet (Ph.),Combes (J.M.), Duclos (P.) – On the location of resonances for Schrödinger operator in the semiclassical limit II: Barrier top resonances, Commun. in Partial Differential Equations 12, 201-222, 1987.

[4] Kato (T.) – Perturbation theory for linear operators, Berlin, heidelberg, New York : Springer (1966).

[5] Martinez (A.), Rouleux (M.) – Effet tunnel entre puits dégénérés, Comm. in Partial Differential Equations 13, 1988, p. 1157-1187.

[6] Reed (M.), Simon (B.) – Methods of Modern Mathematical Physics IV , Academic Press, (1978).

(Received May 21, 2006)

Hamadi BAKLOUTI Département de Maths Faculté des Sciences de Sfax 3038 Sfax Tunisie e.mail: h baklouti@yahoo.com

Maher MNIF Département de Maths I.P.E.I.Sfax B.P.805 Sfax 3000 Tunisie e.mail: maher.mnif@ipeis.rnu.tn