Claude Billionnet
email: claude.billionnet@cpht.polytechnique.fr

Influence of the extent of the eigenstates of a system on the resonances formed through its coupling to a field

Keywords: numbers: 02.30.Tb, 03.70.+k, 11.10.St, 12.39.Mk

We examine resonances for two systems consisting of a particle coupled to a massless boson's field. The field is the free field in the whole space. In the first system, the particle is confined inside a ball. We show that besides the usual energy levels of the particle, which have become complex through the coupling to the field, other resonances are to be taken into account if the ball's radius is comparable to the particle's Compton wavelength. In the second system, the particle is in a finite-depth square-well potential. We study the way the resonances' width depends on the extent of the uncoupled particle's wave functions. In both cases, we limit ourselves to considering two levels of the particle only.

I. INTRODUCTION

In our preceding works [1,2], we introduced two types of resonances, which we called "standard" and "nonstandard", or "of R-type" and "of C-type". The term "nonstandard" is used with an analogous meaning in the expression "nonstandard multi-quark/gluon states". "C" is related to the fact that it is the continuum which yields nonstandard resonances whereas "R" refers to the real eigenvalues to which standard resonances are associated. These two types occur every time a quantum system S with discrete energylevels is coupled to a massless field. The most well-known of these resonances are of course those which are "attached" to the eigenvalues of the Hamiltonian of system S alone. By saying that they are attached to these eigenvalues, we mean precisely that they tend to these eigenvalues when the coupling of the system to the field tends to zero. We called them standard resonances, or of R-type.

Other resonances exist. We called them nonstandard resonances, or of C-type. The reason to distinguish them from the others is that they disappear at the limit where the coupling vanishes. Among those, some are also well-known, in certain experimental situations. For example, if an atom is placed inside a cavity with very good mirrors, the cavity being tuned to the transition between the first excited level of the atom and its fundamental level, one sees that the excited level is split into a doublet, the so-called Rabi doublet. Since the cavity is never perfect, the energies of the doublet are in fact complex, i.e. we get two resonances there; one can see that one is standard whereas the other one is nonstandard. This is an argument for considering both types of resonances. But there still exist other resonances besides this doublet (see for instance [2]). All of them are likely to connect to one another, this time for a mathematical reason: in the simplified models we studied, these resonances are the zeros of an explicit multivalued function. In an example [1], making the coupling constant and the width of the continuum seen by the atom vary along a closed real path in IR 2 and following the corresponding displacement of a resonance, we showed that its final position may differ from the initial one; in other words we get to another resonance. In particular, a standard resonance may become a nonstandard one or conversely. Therefore one may think that all resonances are to be put on the same footing, at least from a mathematical point of view.

In spite of these two theoretical reasons to consider the two types of resonances, there are many cases in which some of these resonances can be neglected, on account of their imaginary parts being very large compared to those of the others. This is the case for nonstandard resonances occurring when an atom is coupled to the transverse electromagnetic field in the vacuum, as can be seen for the hydrogen atom in a simplified treatment which only takes two levels into account [2]. On the contrary, besides the above mentioned Rabi doublet, in many cases the imaginary parts of standard and nonstandard resonances are comparable. It is the case in a rudimentary two-level model for the coupling of a (quark-antiquark)-type system to a massless boson [3].

In a first approach, one may be tempted to say that the importance of the nonstandard resonances with respect to the others is related to the size of the ratio χ of the coupling strength to the continuum's width, when the latter is defined which is the case for Lorentzian coupling functions. But it is also clear that χ being large does not imply that one can forget the standard resonance, since it may become an eigenvalue. Therefore some more thorough investigation has to be performed. The importance of the ratio χ has long been known. In the above quoted case of the Rabi doublet in a cavity, where the electromagnetic interaction is not strong, the importance of the nonstandard resonance is indeed due to the width of the continuum seen by the atom being small, as a result of the cavity's quality factor Q being large. Generally, in two-level cases, we could roughly expect some standard and nonstandard resonances being equally interesting when χ is of the order of 1. However, the position of the resonances is not a function of this ratio, neither even, strictly speaking, a function of these two variables, because of the multivaluedness. A precise analysis has therefore to be performed in each case. For example, we saw examples in which it cannot be decided which one of the standard or nonstandard resonance comes near to the reals when χ becomes large, unless a careful computation is performed: it may not be the same resonance whether it is the coupling which increases or the width which decreases [1].

The role of the continuum's width is seen in the following way, in one dimension, for a system S with two levels e and f , when the interaction with the field is given by the simplest Hamiltonian

λ | ψ e ψ f | ⊗ c(g)+ | ψ f ψ e | ⊗ c * (g) , (1)
where c(g) = c k g(k)dk, c k being the annihilator of a boson with wave number k. The existence of nonstandard resonances comes from the fact that the coupling function g becomes infinite, at points at a finite or infinite distance. If g has a pole in the lower half-plane (at a finite distance) and is bounded at infinity, then the limit of the nonstandard resonances when the coupling goes to zero is hc times this pole. This is why these resonances cannot be seen at zero coupling (since there is no coupling function any more). The pole and the width of g are clearly related in some cases. For example, in the Rabi doublet's case for an atom coupled to the field inside a large-Q cavity, the coupling function may be approximated by a Lorentzian whose width is twice the imaginary part of its pole in the complex plane. The nonstandard resonance is then interesting because the pole is close to the reals as a result of the coupling function's width being small. Note that the spreading of the coupling function, even if it makes the determination of a width problematic, does not prevent the nonstandard resonances from existing.

Let us now note that the poles of g have something to do with the extent in space of the eigenvalues of H S , the Hamiltonian of S, in certain types of coupling. This is mentioned in [4]. Let us recall the argument, in one dimension. Assume the field is of the form φ(x) = k A k e -ikx c * k + e ikx c k . To φ(x) we associate the operator Φ = k A k e -ikx ⊗ c * k + e ikx ⊗ c k acting in the product of L 2 (IR), the space of states of S, and the field state-space, e ±ikx acting through multiplication. Consider a coupling H I = λ (O ⊗ 1) Φ, where O denotes an operator in L 2 (IR). The coupling -e m p • A of electrodynamics is of this form, but in three dimensions, i.e.

H I = λ( n=1,2,3 (O n ⊗ 1)Φ n .
Let us restrict ourselves to the space spanned by | e and the | f, k 's, and replace the Hamiltonian of the S + f ield system by an operator H red in this space, H red having the same matrix elements as H. Setting

λ g(k) := ψ f , k | H red | ψ e = λ A k ψ f | Oe -ikx | ψ e ,
where the ψ's are here the wave functions in x-space, we can write the interaction Hamiltonian in the above form (1). If O is a polynomial P (x, d dx ), then ψ f | Oe -ikx | ψ e has the same analytical properties in k as the Fourier transform of ψ f ψ e . If |ψ f ψ e (x)| is upper bounded at infinity by Q(x)e -|x|/x 0 , with x 0 > 0 and Q a polynomial, then the Fourier transform is analytic inside the strip {k, |ℑ(k

)| < x -1 0 }. If ψ f ( d dx ) n ψ e (x) is exactly of the form Q n (x) e -|x|/x 0 , where Q n is a polynomial, then ψ f | Oe -ikx
| ψ e has a pole at ±x -1 0 i. Thus the pole in the lower half-plane is here directly related to the extent in space (in the exponential sense) of the pair of states (e, f ).

The extent of the wave functions, in that sense, is therefore an important parameter as regards the position of the resonances. When the coupling is small, the definition of nonstandard resonances tells us that the largest the wave-functions' extent is the most interesting the nonstandard resonances are, since the pole is the closest to the reals. The hydrogen atom is a concrete case on which this relation can be studied. However, in that case, we saw in [2] that considering pairs of very excited states is not sufficient to make these resonances interesting (unless perhaps if time scales much shorter than the life-time of the excited states are considered): the coupling is to small and the poles of the coupling functions are too far from the reals. Besides, the largeness of the imaginary parts is also due to the smallness of the coupling [2]. We are going to resume the study of the influence of that space extent in a strong coupling case, where the nonstandard resonances are going to be important. Unfortunately, we still will limit ourselves to considering two levels of S only, since the determination of the resonances for more than two levels is a problem too complicated to be dealt with at the moment [5]. When the coupling is strong (note that we do not use here the expression "strong coupling regime" which has a special meaning), knowing the poles of g , or its singularities, is not sufficient to get an idea of the position of the resonances. For example, we will see two cases, with coupling functions g 1 and g 2 , where poles of g 1 and g 2 respectively at k 1 and k 2 satisfying k 1 < k 2 yield resonances of widths Γ 1 and Γ 2 with Γ 1 > Γ 2 , whereas the coupling strengths are identical (see pairs (1,2) and (1,3) of Section IV). This might be due to the fact that g has other singularities than its poles: singularities at infinity. The resonances' widths for states of S of various extents should therefore be examined in each particular case.

In Section II, we will recall some results in cases where the coupling function is bounded at infinity, with poles at finite distances. In Section III, we will treat a simple case where the singularities of g are pushed to infinity. This will prepare ourselves for the study of the example of Section IV, for which g has singularities at infinity besides its poles. In this Section III, the extent of the wave functions will be fixed, if the physical parameters of the system are not changed; it will be zero, in the exponential sense. The ratio χ will therefore be zero. The system which is coupled to the transverse field is a particle inside a ball of radius a. Through indicating the role of parameter a/λ C ( λ C is the Compton wave-length of the particle) we will show that nonstandard resonances may be important although χ vanishes. This will prove that the only consideration of χ is insufficient to judge the interest of the resonances. Section IV also concerns the strong interaction domain. The particle which is coupled to the field is now put inside a finite-depth square-well. It is a simple system which allows the extent of the system to depend on the excitation level of the particle in the well. Our main aim is to discuss the position of the resonances as a function of this excitation level. It will also be interesting to have the well's geometry vary. It is a mixed case where the coupling function has both poles at finite distances and singularities at infinity.

II. EXAMPLES OF RESONANCES WITH A COUPLING FUNCTION BOUNDED AT INFINITY HAVING POLES AT FINITE DISTANCES

A. A case where a standard and a nonstandard resonance come to coincide.

We first recall two cases for which we can precisely calculate coupling constants and coupling function widths for which a standard and a nonstandard resonance coincide. In these two examples the above-mentioned ratio χ is of the order of 1. The coupling function g in Hamiltonian (1) is Lorentzian in both cases, with a width µ, expressed in E-units, E being the energy difference between the two levels. (We recall that resonances are zeros of an analytic continuation of f (z) = z -E -λ 2 ∞ -∞ g(k) 2 (z -hc|k|) -1 dk, if the energy of the fundamental is assumed to be 0; the equation for the eigenvalues of the Hamiltonian is f (z) = 0.)

In the first case, the (normalized) coupling function is

2/π ( hc Eµ ) 3/2 |k| 1 + ( hck Eµ ) 2
(see [1]). It has a pole at k = -iµE/(hc). Let κ = E -1 λ denotes the dimensionless coupling constant. Curves of Figures 8 and9 of [1] indicate that, for χ = κ/µ ≪ 1, one expects the width of the nonstandard resonances which are attached to the pole to be much larger than the standard resonance's one. We show in [1] that some nonstandard and standard resonances come to coincide for κ = 0.5 and µ = 1.08. The ratio χ (coupling strength/continuum's width) or (coupling strength/pole's imaginary part) is then about 1/2.

In the second case, the coupling function is now centered at a value which does not depend on µ. It is

2/π ( hc Eµ ) 1/2 1 1 + µ -2 ( ch E k -1) 2 .
Although this function is neither even nor vanishing at the origin, we used it in [2] since it allows simple calculations in a model approaching the coupling of a two-level atom S with the field in a cavity. The continuum is tuned to the transition in the atom. If κ still denotes the coupling strength, in E-units, some resonances almost coincide for µ = 0.01 and κ = 0.003, therefore χ ≃ 0.3.

B. Review of the hydrogen atom case

Matrix elements of interaction -e m p • A of the hydrogen atom with the transverse field are no longer Lorentzian functions (see for instance [6]). However they all have poles at finite distances. (Note that these poles may be multiple poles and, since the coupling function occurs through its square, these poles yield twice more resonances than their multiplicities.) The largest the mean extent of the states the closest to the reals the poles are. Indeed, each wave function decreases exponentially with the distance r, as e -r/r 0 for a certain r 0 , and, for two levels, one can define a mean extent r0 (harmonic mean of the r 0 's of the two levels). When one restrict oneself to a pair of two levels e 1 and e 2 , considering Hamiltonian's matrix elements between states e 2 and (e 1 , 1 photon) only, the coupling function has a (multiple) pole at a complex energy E = -2i hc r-1 0 . The ratio χ between the coupling strength and the imaginary part of this pole is less than about 10 -4 . Therefore we expect nonstandard resonances to be very far from the reals, in the coupling strength unit for instance; this is confirmed by the calculus [2]. In this example, it is the smallness of the fine structure constant which entails the smallness of χ. Indeed, on the one hand, the coupling strength, in E-units, is proportional to √ α (see formula (20') of [2]). On the other hand, the wave-functions' extent (in the above-considered exponential sense) is small with respect to the transition wavelength (ratio less than α/2), that is to say the imaginary part of the pole is large with respect to the energy difference between the two levels. Due to the very large widths of the nonstandard resonances, the study of the relation of these widths to the extent of the states is a somewhat academic question. It would of course be interesting to examine the resonances' position when the coupling is strong rather than weak, possibly changing also the mass of the particle. We chose to discuss here the simpler well case (Section IV), even if it has the drawback that the coupling function has singularities at infinity, besides the one at a finite distance. It has also the advantage that the width and the depth of the well are parameters which may be varied.

III. THE RESONANCES FOR A PARTICLE CONFINED INSIDE A BALL AND COUPLED TO A MASSLESS VECTOR FIELD

Before considering the well case, we are going in this section to deal with a case where the coupling function is not bounded at infinity but has no pole.

A particle is confined by a infinite potential barrier, and so as to get simple enough computations with the coupling -e m p • A, we will assume that this barrier is a sphere of radius a. The aim is twofold: prepare the study of Section IV, in which g has singularities at infinity, and show how the position of nonstandard resonances depends on the extension of the states via the ratio of a to λ C , (2π) -1 times the Compton wavelength of the particle. The term "extension" is not here taken in the exponential sense but in its ordinary one.

A. Setting up and physical parameters of the problem

The states of the system decoupled from the field are those of a particle of mass m in a spherical potential which vanishes for r < a and is infinite for r ≥ a. The coupling of the particle to the field is assumed to be given by the interaction Hamiltonian -α 1/2 (hc) 1/2 m -1 p • A. Let us make it clear that the field is not determined by the ball; it is the free field in the whole space. The wave functions Ψ l,m,p of angular momenta (l, m) are given by the following formulas. Let us set

ξ l,1 ≤ ξ l,2 ≤ • • • the zeros of j l (z) = π 2 z -1/2 J l+1/2 (z) and N l,p := ( ξ l,p 0 j l (x) 2 x 2 dx) 1/2 . If r < a, Ψ l,m,p ( r) = R l,p (r) Y lm (θ, φ), where R l,p (r) = N -1 l,p ξ 3/2 l,p a -3/2 j l (ξ l,p r/a); if r > a, Ψ l,m,p ( r) = 0. The energies of states (l, m, p), independent of m, are E l,p = 1 2 (λ C /a) (hc/a) ξ 2 l,p .
As we said before, we forget all the states but two, since we do not know how to calculate otherwise. The upper state, denoted by e, has an angular momentum l 2 and we assume m 2 = 0. The lower state denoted by f is assumed to have zero angular momentum. Besides, in the coupling of the particle to the field, we only keep matrix elements of the interaction Hamiltonian between (f, 1 boson)-states and state e. In Appendix A we show that the eigenvalues of the particle-field Hamiltonian are the solutions of the equation in z

z -E e - l 2 (l 2 + 1) π (λ C /a) 2 (hc/a) 2 α ∞ 0 G l 2 ,e,f (y) 2 z -E f -hc a y dy y = 0, (2)
where the (not normalized) coupling function is

G l 2 ,e,f (y) = a 4 1 0 j l 2 (yρ) ρ R ′ f (ρa)R e (ρa) dρ. (3)
y is variable ka, where k is the boson's wave number. G l 2 ,e,f has no dimension; it does not depend on a. The interaction strength is measured by the normalized coupling constant

g nor e,f = l 2 (l 2 + 1) 2π (λ C /a)(hc/a) ||G l 2 ,e,f || √ α, with ||G|| = 2 ∞ 0 G(y) 2 dy y 1/2 .
We call G nor = ||G|| -1 G the normalized coupling function. In this example all wave functions have the same extent a. Were the extent defined by an exponential-decrease rate, as in the hydrogen atom case or in the finite-depth well case that we will consider in Section IV, we should have to consider this extent as zero. The variable µ defined in the Lorentzian case or the hydrogen atom case, proportional to the inverse of that extension, would then be infinite. The coupling function's pole is pushed to infinity, becoming an essential singularity, and the ratio χ e,f = κ nor e,f /µ e,f vanishes. But here we are going to consider the wave functions' extent as being equal to a and, so as to differentiate the two points of view, the ratio of this extent to (2π) -1 the wavelength of the considered transition will be denoted by ν -1 e,f . We have

ν -1 e,f = 1 2 λ C a (ξ 2 l 2 ,p 2 -ξ 2 0,p 1 ).
When it is measured with respect to the transition energy, the coupling strength, κ nor e,f := E -1 g nor e,f , is

κ nor e,f = 2 l 2 (l 2 + 1) π √ α 1 ξ 2 l 2 ,p 2 -ξ 2 0,p 1 ||G l 2 ,e,f ||.
Defining χ dist e,f := κ nor e,f /ν e,f , we get

χ dist e,f = l 2 (l 2 + 1) π √ α ||G l 2 ,e,f || λ C a . (4)
We are going to see that increasing χ dist e,f from a value which is small with respect to 1 to a value of order 1 change uninteresting nonstandard resonances into interesting ones. Thus, besides χ, the value of a/λ C also plays a role as regards the importance of nonstandard resonances.

B. Choice of two states e and f

Through introducing ζ = z-E f E e -E f , the eigenvalue equation becomes ζ -1 -l 2 (l 2 + 1)(κ nor e,f ) 2 ∞ 0 G nor l 2 ,e,f (y) 2 ζ -ν e,f y dy y = 0. ( 5
)
From now on we assume l 2 = 1 and take the states of lowest energies as e and f , i.e we set p 1 = 1 and p 2 = 1. We forget the index m = 0. The wave functions read:

ψ 0,1 = 1 √ 4π N -1 0 ξ 3/2 0,1 a -3/2 sin(ξ 0,1 r/a) ξ 0,1 r/a , with ξ 0,1 = π and N 0 = π/2 ≃ 1.25, ψ 1,1 = 3 4π N -1 1 ξ 3/2 1,1 a -3/2 sin(ξ 1,1 r/a) ξ 1,1 r/a -cos(ξ 1,1 r/a) 1 ξ 1,1 r/a cos θ,
with ξ 1,1 ≃ 4.49 and N 1 ≃ 1.46. From (3), the coupling function is:

G(y) = (N 0 N 1 ) -1 ξ 3/2 0,1 ξ 1/2 1,1 y × 1 0 ( sin(yρ) yρ -cos(yρ) cos(πρ - sin(πρ) πρ sin(ξ 1,1 ρ) ξ 1,1 ρ -cos(ξ 1,1 ρ) ρ -2 dρ. (6)
Unless we state it otherwise, states e, f now remain fixed as was said before and we do not mention them in indices. The norm of G, in the above mentioned sense, is ||G|| ≃ 2.92. Since G does not have a more explicit expression than (6), we give the graph of ||G|| -1 |G| in Figure 1. It has to be noted that the width of the highest bump in the curve has nothing to do with the imaginary part of a pole of this function.

Although we cannot calculate G explicitly, through comparing this function to the coupling function (9) presented in Section IV for the case of a finite depth square well, we expect the former to be unbounded at infinity. Indeed (9) is explicit and its C 2 -part, which corresponds to the part of the wave function localized inside the well, do have a singularity The resonances are given by the equation

ζ -1 -2 (κ nor ) 2 ∞ 0 G nor (y) 2 ζ -νy dy y + 4iπ (κ nor ) 2 ζ G nor (ζ/ν) 2 = 0. ( 7
)
With the levels we chose, we get

g nor = 1.65 (λ C /a)(hc/a) √ α, ν ≃ 0.194 a/λ C , κ nor ≃ 0.32 √ α, χ dist ≃ 1.65 (λ C /a) √ α.
C. Numerical application We are going to give two numerical applications. One with the electron, a = 1 Angström and α = 1/137, the other one with a particle whose mass is the reduced mass of two light quarks, i.e. m = 1/2 m q = 110 MeV, a = 1 Fermi and α = 0.6. In the first case, λ C /a = 3.86 10 -3 and in the second one λ C /a = 1.79. The results are the following. (We recall that, with the ζ variable, the levels of the decoupled particle are ζ = 0 and ζ = 1.)

1. The electron and an electromagnetic coupling If a is expressed in Angström, we get E l,p = 0.985 (λ C /a) ξ 2 l,p KeV. Besides, χ dist = 0.14 λ C /a. With a = 1

• A, we get E 0,1 = 37.5 eV, E 1,1 = 76.7 eV, g nor ≃ 1.07 eV, κ nor = 2.74 10 -2 , ν = 50.35, χ dist = 5.44 10 -4 .

One resonance is given by

ζ = 53 -778 i, which gives z ≃ (92 -30 i) KeV.
This resonance is thus of no interest. Here a/λ C ≃ 260. However, since at the Fermi and GeV scales of Section III.C.2 we will encounter an analogous resonance, no longer negligible, let us give some precisions. Its origin is not unknown of us. It can be shown numerically that if we make ν tend to 0 in [START_REF] Cohen-Tannoudji | Processus d'interaction entre photons et atomes[END_REF], for example through making a tend to 0, then this resonance becomes an eigenvalue (solution of ( 5)) when ν gets smaller than a certain critical value ν c (≃ 4.18 10 -4 , depending on (e, f )). For ν = ν c , the eigenvalue is zero: the resonance passes below the fundamental E 0,1 . We already met and explained such a situation in Section 3.2 of [4]: for ν = 0, the coupling reduces to the coupling of two states of the particle-field system and it is easy to calculate the two eigenvalues ζ + ≃ 1.006 and ζ -≃ -0.006 of the total Hamiltonian. It is toward the latter that the resonance we consider goes if ν goes to 0. Besides, we check that this resonance is nonstandard: it does not tend to 1 but goes away from the reals when κ nor tends to 0, probably going to infinity in the lower complex half-plane.

To be complete and connect the case we study here with the one we will consider in Section III.C.2, or to a more general setting, let us mention that there are other nonstandard resonances which now tend to 0 when ν tends to 0. There is very likely an infinite number of them and they stem from the essential singularity at infinity in the complex plane of the oscillating coupling function G (behavior e -i(i∞) ). Here are some of them, expressed in variable ζ:

ζ = 263 -791 i, ζ = 461 -814 i, ζ = 649 -840 i, ζ = 829 -866 i,
The resonance close to 1 is more difficult to get exactly since it is very close to the reals, the integral in [START_REF] Cohen-Tannoudji | Processus d'interaction entre photons et atomes[END_REF] being difficult to handle numerically. An estimation at first order in κ nor gives ζ = 1 -9 10 -6 -3.6 10 -7 i.

An approached numerical calculation confirms the imaginary part. This yields a width of 1.4 10 -5 eV. Let us note that the computation shows that when κ nor grows beyond a critical value close to 9.5, it is this resonance which becomes a real negative eigenvalue, after having been zero at the critical point. The situation will be different in the case studied in the following section. If ν goes to 0, this resonance tends to the above-defined ζ + . We already met these three types of behavior when ν goes to 0, namely ζ(ν) → ζ -, ζ(ν) → ζ + and ζ(ν) → 0, in our preceding works. We will meet them again for the strong interaction at the Fermi scale that we are now going to consider. We are now going to get significant values. Let us anticipate and underline that the three types are a priori of equal importance from the theoretical point of view. Indeed, another way of presenting all the resonances is to classify them according to their behavior for κ → 0; this will be done in Section 4.3 and we will then show that slightly changing the physical parameters of a system may make a resonance change from one class to another. This is related to the multivaluedness of the function the zero of which gives the resonances.

"Quarks" and a strong coupling

We take m = 110 MeV, a = 1 Fermi and α = 0.6. We then get λ C /a = 1.79 and E 0,1 = 1.74 GeV, E 1,1 = 3.55 GeV, g nor ≃ 0.45 GeV, κ nor = 0.248, ν = 0.11, χ dist = 2.29. Numerical computations show that there are still two types of nonstandard resonances.

In the first class (first column of Table I which gives the corresponding complex energies) there is one resonance, at ζ = 0.03 -0.11 i.

We put it in a first class because, when ν decreases, it approaches the reals, becomes zero (for ν ≃ 0.034), and then real negative for ν < 0.034. The resonance is nonstandard since one can show that it goes to infinity in the lower half-plane when κ nor tends to 0. When κ nor increases, it approaches the reals, becomes zero for a value close to κ nor = 0.44, and then real negative (i.e. an eigenvalue) for κ nor > 0.44.

In the second class (second column of the Table ), we put an infinite family of resonances. They too seem to go to infinity in the lower half-plane when κ nor tends to 0. But numerical computations show various limit points when κ nor goes to infinity. Let us give three of these resonances:

ζ = 0.81 -0.43 i , ζ = 1.22 -0.63 i, ζ = 1.59 -0.75 i.
Lastly, there is the standard resonance (third column), at ζ = 1.082 -7 10 -5 i.

Coming back to the energies, besides the eigenvalue E 0,1 corresponding to state f with no boson, we get the following resonances Energies of the resonances built with the first two levels l = 0 and l = 1 of a particle in a ball of radius 1 F, coupled to the field (α = 0.6).

The result of this section is that the coupling function's singularity at infinity yields interesting resonances in a case where a/λ C is of the order of unity. For the example a/λ C = 0.56, the singularity at infinity creates a resonance slightly above the fundamental, with a width of 210 MeV, together with an infinite number of larger resonances; the real part of the first one sits between the two levels. In the ν → 0-limit, the two resonances of column 1 and 3 tend to the eigenvalues of the two eigenvectors of the Hamiltonian

E 0,1 | 0 0 | +E 1,1 | 1 1 | ⊗ 1 + g nor a * (G) ⊗ | 0 1 | +a(G) ⊗ | 1 0 | ,
in the space spanned by states | 0 ⊗ | G and | 1 ⊗ | vacuum . The others tend to the fundamental state's energy. (Note that here ν → 0 does not make the width of the continuum zero since it is originally infinite, despite the shape of the curve in Figure 1.) Now that we saw that an appreciable value of χ dist can yield interesting nonstandard resonances, let us go back to the electron's case for two comments. Firstly, to get χ dist of the order of 1 one should have to take dimensions much smaller than the Angström. But we do not see any physical meaning in taking a of the order of 100 Fermi. Secondly, it is possible to find states e and f such that ν e,f be close to 1, i.e. such that the states' extent (in the ordinary sense we consider in this Section III) be of the order of the transition's wavelength. It suffices to take p 2 large for state e, say p 2 = 15. Then ζ 1,p 2 = 48.7 and, with a = 5

• A, we have ν e,f close to 1. But then the coupling strength decreases, and as a consequence χ dist e,f remains small. This is due to the smallness of the electron's Compton wavelength, as formula (4) shows it. Moreover ||G|| ≃ 0.11; hence, with respect to the preceding (p 2 = 1, a = 1

• A)-case, χ dist becomes smaller by a factor of 150. Therefore, having ν e,f close to 1 is not sufficient to ensure that nonstandard resonances are interesting: the coupling strength also has its importance.

We are now going to determine resonances in a case close to the preceding one but in which the states' extent is no longer constant: the extent will now depend on the state's excitation degree.

IV. THE RESONANCES WHEN THE PARTICLE COUPLED TO THE FIELD IS IN A FINITE-DEPTH WELL.

In this case the coupling function has singularities both at infinity and at a finite distance.

A. Setting up and equation for the resonances associated to two levels

We consider a one-dimension (x variable ) square well, of depth -V 0 and width a, centered at the origin. The particle in this well has a mass m and we will express the interesting quantities of the problem as functions of the dimensionless parameters α = λ -1 C a and β = (mc 2 ) -1 V 0 . (α no longer denotes the coupling constants α el or α s of Section III.) For the ball, in the a = 1 Fermi case and with the reduced mass of the same two quarks, these parameters took the values α = 0.558 and β = ∞.

For simplicity purpose, we will only consider even states. We recall the wave functions:

Ψ(x) =λ -1/2 C ψ(λ -1 C x), of energies E = ǫV 0 , with ψ(χ) =    N 1 e χ/ξ if χ < -α/2 N 1 sgn(u)(1 + ǫ) -1/2 e -α/(2ξ) cos χ 2β(1 + ǫ) if -α/2 < χ < α/2 N 1 e -χ/ξ if χ > α/2 , N 1 = 2α -1 (1 + ǫ) e 2γ √ -ǫ γ √ -ǫ (1 + γ √ -ǫ) -1 1/2 , γ = α β 2 , ξ = (-2βǫ) -1/2 , u = cos α 2 -1 β(1 + ǫ) , sgn(u)
is the sign of u, and ǫ is a solution for

| cos α 2 -1 β(1 + ǫ) | = √ 1 + ǫ , tan α 2 -1 β(1 + ǫ) > 0.
Once we have chosen two eigenvectors | 1 and | 2 (E 1 < E 2 ) for the decoupled particle, we assume that the interaction of the particle with the field is described by a Hamiltonian H I whose matrix elements are

g(k) := 1, k | H I | 2 = λ Ψ * 1 (x)e -ikx Ψ 2 (x)dx. ( 8
)
The aim in taking such an interaction Hamiltonian is to get a qualitative approach to the set of resonances one could find in more realistic interactions (see the sixth paragraph of the introduction.) The calculus gives

g(k) = λ G (2β) -1/2 λ C k , with G(q) := C 1 s 12 cos γq -q sin γq q 2 + s 2 12 + C 2 i,j=1,2
sin γ(y 1 + (-1) i+1 y 2 + (-1) j+1 q) y 1 + (-1) i+1 y 2 + (-1) j+1 q , (9)

y i = √ 1 + ǫ i , C 1 = 4s 1 s 2 y 1 y 2 C 2 , C 2 = (ǫ 1 ǫ 2 ) 1/4 s 1 s 2 1 + γ √ -ǫ 1 1 + γ √ -ǫ 2 , s i = sgn[cos γy i ], s 12 = √ -ǫ 1 + √ -ǫ 2 .
The normalized coupling constant is ||g|| 2 = (2β)

z -E 2 - ∞ -∞ g(k) 2 z -E 1 -hc|k| dk = 0.
Through the change of variable ζ = z/V 0 and introducing the dimensionless parameters

µ 1 = 2/β, κ 1 = (hmc 3 ) -1/2 λ, (10)
we get the equation for the resonances:

ζ -ǫ 2 -µ 3 1 κ 2 1 ∞ 0 G(q) 2 ζ -ǫ 1 -µ 1 q dq + 2iπµ 2 1 κ 2 1 G(µ -1 1 (ζ -ǫ 1 )) 2 = 0. (11)
(Note that here µ 1 is not the ratio of the transition wavelength to the (mean) extent of the two wave functions. Also κ 1 is not the ratio of the coupling strength to the levels' energy difference.) The mean extent, in the exponential sense from now on, of the two wave functions appears with the pole of G at q = -i s 12 . It corresponds to a pole of g at k = -i s 12 λ -1 C √ 2β, the inverse of half the harmonic mean of the extents λ C ξ 1 and λ C ξ 2 of the two states. This pole of G and the singularity at infinity coming from the sine functions in (9) are at the origin of the nonstandard resonances.

We are now going to give the results of computer calculations of the resonances in some numerical examples. Our main interest is in examining the dependence of the resonances on the extent in space of the various level pairs. We take µ 1 = 0.68 and γ = 10, corresponding to β = 4.32 and α = 6.8. The particle in this well has four bound states. The reason of the choice for µ will appear in Sections 4.3 and 4.4. The energies of these bound states are (in V 0 = 4.32 mc 2 units):

ǫ (1) = -0.980, ǫ (2) = -0.818, ǫ (3) = -0.5, ǫ (4) = -0.063.
For the coupling to the field, we take κ 1 = 1, so as to have a strong coupling.

As for the ball's case, for each pair of levels, we have an infinity of resonances. To present them, we are going to classify them according to their behavior as κ 1 tends to 0 (whereas the classification in Section III was based on the behavior with respect to ν). Among the nonstandard resonances, two of them tend to the pole of G at a finite distance, and an infinity of others tend to infinity. The diagram of Figure 5 gathers a large part of the results. Resonances are represented there by energy levels with a certain width.

B. The only considered levels are levels 1 and 2. (ǫ 1 = -0.980, ǫ 2 = -0.818)

Let us first consider the pair of the first two levels. The mean extension , in λ C units, is s -1 12 µ 1 = 0.36. The coupling strength ||g|| 2 is (2β) 1/4 κ 1 ||G|| 2 = 0.91, in mc 2 units. (We have ||G|| 2 = 0.53). In Figure 2 we give the graph of ||G|| -1 |G|, to which the graph of Figure 1 can be compared. We recall that q = (2β) -1/2 λ C k. Figure 3b shows the displacement of this resonance as κ 1 grows, starting from small values. It is a standard resonance.

ii) The two nonstandard resonances associated with the pole of G at -i s 12 = -1.89 i, which tend to ǫ 1 -iµ 1 s 12 = ǫ 1 -1.29 i when κ 1 goes to 0, are ζ = -0.491 -0.0178 i, ζ = -0.983 -0.1 i.

iii) Lastly, there is an infinite family of resonances which go to infinity in the complex lower half-plane when κ 1 goes to 0:

-0.345 -0.197 i, -0.138 -0.278 i, +0.069 -0.336 i, • • • .

We do not mention a resonance at ζ = -1.42 -0.08 i; it is a replica of the one at ζ = -0.983 -0.1 i (see Section III.G).

C. The only considered levels are levels 1 and 3. (ǫ 1 = -0.980, ǫ 2 = -0.5) Let us now consider levels 1 and 3. The mean extension is now 0.4 and the coupling strength, 0.90, is approximately the same as before.

i) The upper state is displaced to ζ = -0.264 -5.8 10 -3 i.

There is a difference from the preceding case. It can be seen in Figure 3. The path which this (standard) resonance follows when κ 1 increases from 0 to 1 is given by curve (1) in Figure 3a. (Curve ( 2) is presented later on.) On Figure 3b we showed the displacement of the standard resonance for the level pair (1,2). (It passes below the fundamental for κ 1 > 0.98.) We will see later on that we can change the curve 1-type variation of the resonance into a Figure 3b-type through changing only slightly the well's geometry.

ii) The two resonances associated with the pole of G at -i s 12 = -1.70 i, which tend to ǫ 1 -iµ 1 s 12 when κ 1 goes to 0, are ζ = -1.009 -0.196 i, ζ = -0.830 -0.162 i.

iii) Let us now come to the resonances which go to infinity when κ 1 tends to 0. One of them is the one appearing on curve (2) of Figure 3a. Its value is ζ = -0.734 -4.7 10 -3 i.

Although it is a nonstandard resonance, its width is remarkably small. We have just mentioned an explanation for this: the resonance can easily be changed into a standard one by very little varying the physical parameters of the system. We develop this point in Section IV.D. It is therefore not surprising that its width be of the order of the widths of standard resonances. Others are for instance ζ = -0.151 -0.185 i , 0.060 -0.268 i , • • • .

D. Change of a nonstandard resonance into a standard one through a slight change in the well's geometry

Let us come back to the change of curve (1) in Figure 3a into a Figure 3b-type curve through a slight change in the well's geometry. On Figure 3a we see that there are two resonances around ζ ≃ -0.5 -0.08 i, for a value of κ 1 that a computer calculation puts near 0.25. This computation shows that the relative position of curves ( 1) and ( 2) in the neighborhood of the critical point changes from that of Figure 4a into that of Figure 4b as µ 1 goes from 0.67 to 0.69.

-0.56 -0.54 -0.52 -0.48 -0.46 -0.16 -0.14 -0.12 -0.08

-0.06 -0.04 (a)
(1)

κ 1 decreases↓ (2)
κ 1 increases→ -0.56 -0.54 -0.52 -0.48 -0.46 -0.44 -0.16 -0.14 -0.12 -0.08 -0.06 -0.04

(b) (1) (2)
← κ 1 increases

↓ κ 1 decreases

FIG. 4. Change of a nonstandard resonance into a standard one

Thus the resonance at -0.734 -4.6 10 -3 i for µ 1 = 0.68, which sits on curve (2) of Figure 4a for κ = 1 (not shown on the figure) has its value slightly displaced for µ 1 = 0.69, and now sits on curve (1) of Figure 4b. From a nonstandard resonance it has been changed into a standard resonance through the µ 1 -variation.

E. Comparison of the results for pairs (1,2) and (1,3)

The results concerning these two pairs of levels, with coupling strengths to the field nearly identical, lead to the following comments.

Firstly, it follows from Sections IV.C and IV.D that one cannot deduce the order of magnitude of the imaginary part of a resonance from its standard or nonstandard type, since this type may change under a small variation of the parameters. All the resonances are to be put on the same footing, a priori, even those coming from infinity since we get here an example of a small width resonance although it comes from infinity, for small κ 1 .

The instability of the three classes complicates the study of the effect of the states' extent on the resonances' position, since the extent is clearly related to one class only, the (ii) one, connected to the poles of G.

Nevertheless we can say the following. The pole at a finite distance (in the lower half-plane) of the coupling function approaches the reals when we switch from pair (1,2) to pair (1,3), since the mean extent increases. However, we notice that the imaginary part of the resonance coming from this pole increases, from one pair to the other. Therefore, the position of these poles does not give a precise indication about the position of the resonances which are attached to them, besides the case where the coupling would be small enough for the resonance to be close to the pole. Let us note that this may be a consequence of the existence of singularities at infinity.

Still as regards the resonances coming from the pole at a finite distance, we neverthe-less note that the real part of the energy of the lower resonance decreases as the spatial extent of the pair increases. The numerical values for pair (1,4) will confirm these points.

F. The only considered levels are levels 1 and 4 (ǫ 1 = -0.980, ǫ 2 = -0.063) The mean extent is now 0.55 and the coupling strength, 0.81, has decreased. i) The upper state's energy is displaced at

ζ = +0.04 -1.5 10 -3 i.
a displacement which goes in the same direction as the one found for pair (1,3).

ii) The two nonstandard resonances associated with the pole of G at -i s 12 = -1.24 i, that is tending to ǫ 1 -iµ 1 s 12 = -0.980 -0.844 i when κ 1 tends to 0, are

ζ = -1.013 -0.25 i, ζ = -0.85 -0.24 i.
Here also the lower resonance's energy has decreased with respect to pair (1,3)'s case. We still notice that the imaginary part is greater although the coupling is weaker and the pole of G closer to the reals. Also, with respect to the energy of the upper uncoupled state, -0.818 for pair (1, 2) and -0.063 for pair (1,4), we note that the upper resonance is much lower for pair (1, 4): -0.85, than for pair (1, 2): -0.49, even if we take account of the width. These numbers show that the pole of G at a finite distance does not play a role through its imaginary part only: the real part (e 1 in the three cases) of the pole it induces in integral [START_REF] Cohen-Tannoudji | Processus d'interaction entre photons et atomes[END_REF], pole which is responsible for the existence of these resonances, is also important.

iii) Lastly, as regards the resonances coming from infinity at small coupling, there is still an infinity of them. One the one hand, Representation of some resonances with their widths, for various level pairs. The pairs' mean extent is indicated in λ C units. Resonances represented in dark gray, together with the resonance at ζ = -0.73 -4.6 10 -3 i, are those which come from (i.e. whose λ → 0)-limit is) the singularity of g at infinity. Those in light gray, together with the one at -0.49 -0.02 i, come from the poles at a finite distance. We connected the standard resonances to the corresponding bare levels' energies by a dashed line. For each pair, the level e 1 = -0.98, which remains an eigenvalue has not been drawn.

G. The only considered levels are levels 2 and 3. (e 1 = -0.818, e 2 = -0.5)

The mean extent is 0.42 λ C and the coupling strength 0.9 mc 2 . i) The upper state is displaced by the coupling to the field at ζ = -0.437 -5.7 10 -6 i.

ii) The two nonstandard resonances associated to the pole of G at -i s 12 = -1.61 i, which tend to ǫ 1 -iµs 12 = ǫ 1 -1.1 i when κ 1 tends to 0, are ζ = -0.84 -9.84 10 -2 i, ζ = -0.74 -9.9 10 -3 i.

As for the three preceding cases, the real part of the lower one passes below the lower bare level of the pair. But now its imaginary part, i.e. the width, is smaller than the widths for the other pairs. iii) Resonances which are likely to go to infinity for κ 1 → 0 are ζ = -0.296 -0.057 i, ζ = +0.032 -0.064 i, • • • .

Note that the widths are quite small, as for (ii). Some of the resonances whose real parts are negative are represented in the last column of Figure 5.

There is also

• • • , ζ = -1.35 -0.101 i, ζ = -1.135 -0.062 i.
These last values, together with the last resonances of paragraph (iii) of Section IV.F lead us to the following discussion.

H. DISCUSSION For each pair, we find a great number of resonances, probably because of the infinite order "pole" at infinity. One may ask the question whether some of them could be rejected as having no physical signification. It is an involved problem. Here are some observations.

a | e 2 at its poles. We will not enter into this discussion, partly because of the following point. The second contribution of the above-mentioned complex integration comes from a cut starting at the origin, a cut which we have to chose in the lower half-plane (see [START_REF] Landau | [END_REF], A III .4). Its position determines which resonances are to be taken into account. As a consequence, one should be able to calculate the contributions of the various possible cuts in order to know which resonances play a role. We did not address this aspect of our problem. We limited ourselves to presenting the structure of the whole set of resonances, focusing on the point that the standard resonance, nonstandard resonances coming from a pole at a finite distance and nonstandard resonances coming from a singularity at infinity are a priori to be put on the same footing. As regards the two first types, a convincing example was already [2] the Rabi doublet in the cavity case. As regards resonances coming from a singularity at infinity, Section IV.C gives a physical argument to put one of them on the same footing as a standard resonance. c) Even if the choice of the cut is not taken into account, one has however to note that one resonance may be a kind of replica of another one. Let us show this. In [1] we considered a two level model, the coupling to the field being given by the family of functions g ν = ν -1/2 g(ν -1 p), ν being a parameter which dilates g(p) = 2/π p/(1 + p 2 ). We showed that if ν is smaller than ν c (λ) = 2 ∞ 0 p -1 g(p) 2 dp λ 2 ≃ 0.637 λ 2 then the Hamiltonian has a real negative eigenvalue ζ(ν), which vanishes if ν = ν c (λ). Let ζ - denotes the eigenvalue for ν = 0. When ν increases past ν c , the eigenvalue changes into a resonance in the complex lower half-plane, a zero of the analytic continuation across the cut IR + of the function

z → f (ν, z) = z -1 -2λ 2 ∞ 0 g(p) 2 z -νp dp.
(The energies of the two levels are respectively 0 and To sum up the discussion of this section we could say that the paper presents resonances which are important but that it is not possible at this stage to give a complete list of all those which have a physical meaning.

However, it is perhaps not worth carrying the study too far in the case of the well since the complication coming from the singularity of the coupling function at infinity would perhaps not occur in realistic potentials for strong interactions. We already said that the complication is not present for the 1/r potential. (We would nevertheless have to cope with the multiplicities of the poles.)

V. CONCLUSION Let us recall that in the whole study we had to select two levels of the uncoupled system S, an excited one (e 2 ), and the other one (e 1 ) considered as the fundamental state; we forgot all the others. Due to this bias our results can only be qualitative ones. But we do not have any equation for the resonances if there are more than two levels. Besides, we used models and do not at all pretend that we got a quantitative approach to a description of resonances in strong interaction physics.

We showed that the resonances, poles of the continuation of e 2 | [z -H] -1 | e 2 in the complex lower half-plane of the z variable, are in principle as important as the eigenvalues of H, and that all of them are to be considered. Indeed, the imaginary part of a resonance which disappears when the coupling vanishes may sometimes be so small that the resonance is clearly to be taken into account physically (Section IV.C). In the hydrogen atom case, we had shown [2] that the ratio of the widths of the standard resonances to those of the nonstandard resonances is of the order of 10 -11 , very small as a consequence of the smallness of both the coupling constant and the Compton wavelength of the electron.

For the electron in a ball of radius 1 Angström, coupled to the transverse electromagnetic field, we saw here that this ratio is of the same order, 3 10 -10 . To push the nonstandard resonances closer to the reals, one should diminish a. But does it make sense to consider an electron confined inside a ball whose radius is smaller than its Compton wave length? Considering more massive particles (for example ∼ 100 MeV), with a ball's radius of 1 Fermi and a strong coupling to the field, we showed that the ratio lowers to values of the order of 3 10 -4 . We then get widths of a few hundred MeV, comparable to those found experimentally in strong interaction physics, although they are still large with respect to the width of the excited state.

We were able to diminish this ratio still more drastically through considering a particle in a finite depth well, with a strong coupling to the field. Indeed we showed a case where this ratio can be nearly equal to 1 (Section IV.C). We also showed how we can make nonstandard resonances narrower through choosing more extended states (Section IV.G). This is due to the following fact: in a perturbative approach, that is if one start with the uncoupled system S, the coupling (not necessarily small) of a two-level system with a field is described by a function which depends on the two levels and whose analytic structure depends strongly on the space extent of the wave functions of these levels. Nevertheless, we saw that this spreading-induced narrowing is not automatic. It would be interesting to test it with a greater number of levels or when singularities at infinity are absent since they may perturb the connection between the imaginary parts of the poles of the coupling function and the resonances' widths. In strong interactions less elementary than the ones we considered here the set of resonances could be simpler.

We also found resonances which are narrow for other reasons than the excitation degree of the levels. This is due to the instability of the splitting of the set of resonances into standard and nonstandard resonances (Section IV.C and IV.D). One thus sees the usefulness of considering all the resonances.

APPENDIX: EQUATION FOR THE EIGENVALUES, FOR A PARTICLE COUPLED TO A MASSLESS VECTOR FIELD

We assume that only two states of the particle come into play: an excited state, with wave function ψ l 2 ,m 2 ,p 2 ( r) and a fundamental one with wave function ψ l 1 ,m 1 ,p 1 ( r), toward which the former can go through emitting a massless vector boson with state function s( k). Couples (l 2 , m 2 ), (l 1 , m 1 ) of the particle's angular momenta are assumed to be fixed. We shorten ψ l i ,m i ,p i to ψ i .

The interaction Hamiltonian is

H I = -ih 2 2πm α 1/2 j=1,2 ∞ -∞ 1 √ k c * j ( k)e -i k• r + c j ( k)e i k• r ǫ j ( k) • ∇ d k. (A1)
Through using the spherical harmonics (see page 18 of [START_REF] Landau | [END_REF]):

Y the states of a boson with angular momenta (j, m) and helicity λ are described by s j,m,λ ( k) = s(k) Y λ jm ( k). (The scalar product in the boson-state space is s1 ( k)s 2 ( k) d k.) In these formulas, ∇ k is k times the projection of the gradient ∇ k on k ⊥ , the subspace orthogonal to k. Due to our initial assumption, the only possible states of the electron-boson system are in the subspace E 1 spanned by the ψ 1 ⊗ s 's and ψ 2 ⊗ Ω, where s is a linear combination of the s j,m,λ 's and Ω denotes the vacuum of the radiation. Let P E 1 denotes the orthogonal projection on E 1 . We reduce the original Hamiltonian to H app I := P E 1 H I P E 1 , which acts in E 1 ; it has the same matrix elements as H I . So we neglect matrix elements between states like ψ 1 ⊗ s and ψ α 3 ,l 3 ,m 3 , or between ψ 2 ⊗ s and ψ 1 . We also forget contributions from other angular momenta.

Set F 12 ( k) := ψ1 ( r) e -i k• r ∇ r ψ 2 ( r) d r. With C := -ih 2 2πm α 1/2 , and P k ⊥ denoting the projector on k ⊥ , we have

H app I (ψ 2 ⊗ Ω) = ψ 1 ⊗ { k → Ck -1/2 P k ⊥ F 12 ( k)}, (A2a)
H app I (ψ 1 ⊗ s) = -C IR 3 k -1/2 s( k) • (P k ⊥ ¯ F 12 ( k)) ψ 2 ⊗ Ω. ( A2b
)
Looking for an eigenvector in E 1 of the form ψ 1 ⊗ s + ψ 2 ⊗ Ω, with z its eigenvalue, we find

s( k) = -C k -1/2 (E 1 -z + hck) -1 P k ⊥ F 12 ( k), (A3) with z -E 2 -C 2 IR 3 k -1 (E 1 -z + hck) -1 |P k ⊥ F 12 ( k)| 2 d k = 0, (A4)
which is the equation we looked for. From now on we assume that l 1 = 0. Introducing the radial parts of the wave functions, we have ψ 1 ( r) = (4π) -1/2 R 0,p 1 (r) and ψ 2 ( r) = R l 2 ,p 2 (r)Y l 2 m 2 (Ω r ). Setting I p 1 ,l 2 ,p 2 (k) = ∞ 0 rR ′ 0,p 1 (r)R l 2 ,p 2 (r)j l 2 (kr) dr, (A5) we get

P k ⊥ F 12 ( k) = i l 2 -1 √ 4π I p 1 ,l 2 ,p 2 (k) P k ⊥ ∇ k Y l 2 m 2 ( k) = 2 l 2 (l 2 + 1)π i l 2 -1 k -1 I p 1 ,l 2 ,p 2 (k) Y (e)

FIG. 1 .

 1 FIG.1. Graph of ||G|| -1 |G| at infinity. We get an analogous oscillation on Figure1. We recall that this oscillation disappears in the case of the 1/r potential; only the pole at a finite distance remains (see Section II.B).The resonances are given by the equation

FIG. 2 .

 2 FIG. 2. Graph of ||G|| -1 |G|i) The coupling to the field moves the upper state below the fundamental, at an energy ζ = -0.983.

FIG. 3 .

 3 FIG. 3. Displacement of resonances with respect to the coupling strength for pairs (1,3) (a) and (1,2) (b).

ζ

  = -0.48 -0.045 i, ζ = +0.45 -0.32 i, • • • , one of them having a quite small imaginary part. On the other hand• • • , ζ = -2.20 -0.324 i, • • • , ζ = -1.99 -0.264 i,which come within the discussion in paragraphs (b) and (c) of Section IV.H.Figure5gathers the energies of some of the resonances we obtained, with their widths, in V 0 units. It shows that the real part of the resonances coming from the pole of G at a finite distance decreases as the space extent increases. But the imaginary part increases, contrary to what might have been expected. But comparing pair (1,3) and (2,3), we are now going to see that a somewhat larger extent do lead to a smaller width.

  FIG. 5.Representation of some resonances with their widths, for various level pairs. The pairs' mean extent is indicated in λ C units. Resonances represented in dark gray, together with the resonance at ζ = -0.73 -4.6 10 -3 i, are those which come from (i.e. whose λ → 0)-limit is) the singularity of g at infinity. Those in light gray, together with the one at -0.49 -0.02 i, come from the poles at a finite distance. We connected the standard resonances to the corresponding bare levels' energies by a dashed line. For each pair, the level e 1 = -0.98, which remains an eigenvalue has not been drawn.

  ) The first rejection criterion may be the magnitude of the imaginary parts. The physical importance of the resonances appears in the behaviors in t of e 1 , k | exp -iHt/h | e 2 or of U 2 (t) := e 2 | exp -iHt/h | e 2 . The resonances also manifest themselves in peaks of the function k → lim t→∞ e 1 , k | exp -iHt/h | e 2 . U 2 (t) is computed through a complex integration which has two contributions. The first one comes from the poles of e 2 | [z -H] -1 | e 2 , analytically continued into the complex lower half-plane across IR + . They are the resonances we determined. If their imaginary parts are too large, the contributions of these poles yield fast exponential decreases of U 2 (t) or indiscernible peaks in the graph of function k → lim t→∞ e 1 , k | exp -iHt/h | e 2 . Although the order of magnitude of the widths of the resonances we encountered are various, we do not see any reason to put some resonances aside according to this criterion, except for the widest in the infinite series. b) A second criterion is the magnitude of the residues of e 2 | [z -H] -1

l 2 m 2 ( 2 dΩ k = 1 ,) h4 α πm 2 ∞ 0 k - 1 I

 221201 we get the eigenvalue equationz -E 2 -l 2 (l 2 + 1p 1 ,l 2 ,p 2 (k) 2 z -E 1 -hck dk = 0. (A6) Noticing that I p 1 ,l 2 ,p 2 (k) = a -2 G p 1 ,l 2 ,p 2 (ka), we obtain formulas (2) and (3) of the text, with y = ka.

TABLE I .

 I

	1.80 -0.21 i GeV	3.21 -0.79 i GeV 3.95 -1.14 i GeV ....... 4.62 -1.36 i GeV	3.70 -1.3 10 -4 i GeV

  Restricting ourselves as usual to the Hilbert subspace spanned by | 2 and the | 1, k 's, we get the equation for the eigenvalues:

	1/4 λ of an energy. The normalized coupling function is ||g|| -1 -1/2 C 2 g. (||G|| 2 = ||G|| 2 λ; it has the dimension

∞

-∞ |G(q)| 2 dq).

  1.) For small ν, ζ(ν) is close to ζ -.It is a zero of f (ν, .). But, still for small ν, one can show that the analytic continuation of f (ν, .), which is denoted by f + (ν, .), also has a zero near ζ -, which is complex. Let us denote it by ζ + (ν). If we consider the zeros of the branches f + , f ++ , • • • obtained through turning clockwise once, twice, etc.., round the branch point 0, we get an infinity of points. ζ + (ν, .) is a kind of replica of ζ(ν, .) and cannot be considered as a new resonance. As a consequence, out of all these zeros, only ζ(ν) will be kept in the list of relevant resonances. It may be noted that the above-mentioned cut could be chosen in such a way that the residue at ζ + (ν, .) contributes; however there are other choices for which ζ + (ν) does not play any role.


Mohamed Hadded
email: mohamed.hadded@telecom-sudparis.eu

Paul Muhlethaler
email: paul.muhlethaler@inria.fr

Anis Laouiti
email: anis.laouiti@telecom-sudparis.eu

Leila Azouz Saidane
email: leila.saidane@ensi.rnu.tn

TDMA-aware Routing Protocol for Multi-hop Communications in Vehicular Ad Hoc Networks

Keywords: VANETs, MAC, TDMA, CSMA, multi-hop communications, cross-layer

Vehicular Ad-Hoc Networks (VANETs) have become an emerging technology due to the variety of their applications in Intelligent Transportation Systems (ITS). By creating a vehicular network, each vehicle can exchange information to inform drivers in other vehicles about the current status of the traffic flow or a dangerous situation. Multi-hop communications is an effective method that can be used for information exchange over distances greater than the transmission range of the transmitting vehicle. However, it is a great challenge to ensure a stable multi-hop communication link with a low delivery delay due to the high mobility of the vehicles involved. The goal of this paper is to design a TDMA aware Routing Protocol for Multi-hop wireless vehicular ad hoc networks (TRPM) in order to provide the ability to transmit/receive packets over long distances. The proposed routing scheme is based on a medium access control protocol, in which the intermediate vehicles are selected based on the TDMA scheduling. The simulation results reveal that our routing protocol significantly outperforms other protocols in terms of average end-to-end delay, average number of relay vehicles and the average delivery ratio.

I. INTRODUCTION AND MOTIVATION

VANETs are deployed to make communication between vehicles possible using ad hoc devices. Nowadays, these networks have become an emerging technology due to the variety of their applications in Intelligent Transportation System(ITS). By crating a vehicular network, each vehicle can exchange information to inform drivers in other vehicles about the current status of the traffic follow or the existence of a dangerous situation [START_REF] Hadded | TDMA-based MAC protocols for vehicular ad hoc networks a survey, qualitative analysis and open research issues[END_REF]. They can also be used to improve traffic management conditions such as route optimization, flow congestion control and to provide on-board infotainment such as Internet access. Since safety applications in VANETs have stringent QoS requirements, an efficient Medium Access Control (MAC) protocol that can provide a broadcast service with bounded access delays and minimum transmission collision is required. Recently, MAC protocols, notably those that are based on the TDMA technique, have been used to enable multiple vehicles to use the same frequency channel without interfering with other vehicles' transmissions [START_REF] Hadded | An infrastructure-free slot assignment algorithm for reliable broadcast of periodic messages in vehicular ad hoc networks[END_REF]. Moreover, a VANET is characterized by its lack of a central coordinator, and thus a data or safety message may pass through multiple intermediate vehicles during its transmission from the source vehicle to the destination vehicle. However, the nodes in VANETs are characterized by their high mobility, so the network topology can change quickly and frequently [START_REF] Hadded | TDMA-based MAC protocols for vehicular ad hoc networks a survey, qualitative analysis and open research issues[END_REF]. These conditions create further difficulty to building and maintaining a multi-hop routing path between the source and destination. Generally, the routing protocols which are proposed for VANETs are designed to find the best path for endto-end packet delivery, which can satisfy QoS requirements by considering the number of relay nodes and link lifetime. Although these protocols can achieve good performance in terms of the metrics studied, they are not simultaneously optimized to maximize the overall network performance [START_REF] Laouil | Mac-aware routing in wireless sensor networks[END_REF]. In Figure 1, we show an example of a situation where unsuitable routing decisions lead to a large end-to-end delay. The presented VANET scenario consists of 7 vehicles identified by letters (A to G), using a random TDMA scheduling [START_REF] Rhee | DRAND: Distributed randomized tdma scheduling for wireless ad hoc networks[END_REF] represented by vectors of length equal to 6. Each element of a vector represents one time slot that can be used by only one vehicle to send messages. The shortest path in terms of the number of hops provided by the routing protocol does not always ensure the shortest end-to-end delay. For example, when considering vehicle G as the destination vehicle that will broadcast a message collected from vehicle A, the path A-B-D-G is the shortest, in terms of the number of hops, but it produces a delay of 16 time slots (4 time slots to reach slot 𝑡4 which is the transmission slot for vehicle A, then 4 time slots between 𝑡4 and 𝑡2 as 𝑡2 is the transmission slot for vehicle B, then 5 time slots between 𝑡4 and 𝑡1 as 𝑡1 is the transmission slot for vehicle D and finally 3 time slots between 𝑡1 and 𝑡5 as 𝑡5 is the transmission slot in which vehicle G will broadcast the message received from vehicle A). This delay is greater than the delay of the path A-B-C-E-G which uses 3 relay nodes and requires 11 time slots (4 time slots to reach slot 𝑡4, 4 time slots between 𝑡4 and 𝑡2, then 1 slot between 𝑡2 and 𝑡3, 1 slot between 𝑡3 and 𝑡4 and 1 slot between 𝑡4 and 𝑡5). In this paper, we present a new TDMA-aware Routing Protocol for Multihop communications, called TRPM that can ensure coherent decisions between the MAC and routing layers by selecting the next relay node based on the DTMAC scheduling scheme [START_REF] Hadded | An infrastructure-free slot assignment algorithm for reliable broadcast of periodic messages in vehicular ad hoc networks[END_REF]. The proposed cross-layer is evaluated in a highway scenario by studying the end-to-end delay required to deliver a packet from a source to destination.

The rest of the paper is organized as follows. In Section 2, we review related work. Section 3 describes the system models and briefly presents the DTMAC protocol. Section 4 describes our routing protocol, called TRPM for TDMAaware Routing Protocol for Multi-hop communications in VANETs. In this section, we describe how the message is delivered between a source vehicle and a destination vehicle through multiple relay nodes. Section 5 presents the simulation results and the performance evaluation. Finally, conclusions and future work are reported in Section 6.

II. RELATED WORK

The simultaneous transmissions in VANETs due to multiple concurrent vehicles, lead to an increase in the collisions rate which can degrade the network performance in terms of packet delivery ratio and delay. The relevance of this issue has been confirmed by the development of a specific IEEE standard to support VANETs. The IEEE 802.11p [START_REF]IEEE standard for information technology -Telecommunications and information exchange between systems -local and metropolitan area networks -specific requirements part 11 : Wireless LAN medium access control (MAC) and physical layer (PHY) and physical layer (PHY) specifications amendment 6 Std[END_REF] which is the emerging standard deployed to enable vehicular communication, is a contention-based MAC protocol, using a priority based access scheme that employs both Enhanced Distributed Channel Access (EDCA) and Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) mechanisms [START_REF] Uzcátegui | WAVE: A tutorial[END_REF]. This standard is a contentionbased MAC method that cannot ensure a reliable broadcast mechanism with bounded access delays. This disadvantage is particularly detrimental in VANETs which are specially designed to improve road safety. Despite that, several multihop routing protocols use this approach to transmit data. For instance, in [START_REF] Blaszczyszyn | Opportunistic broadcast in VANETs (OB-VAN) using active signaling for relays selection[END_REF] the authors have proposed an opportunistic broadcast protocol named OB-VAN to overcome the problem of packet delivery in VANETs. OB-VAN uses a modified 802.11 MAC layer using an active signaling mechanism to select a suitable next-hop relay from all the candidate vehicles that have correctly received the packet. Since OB-VAN uses a contention-free MAC schemes, it is possible that it cannot operate well in sparse or dense mobility scenarios.

Many alternatives exist to mitigate collision between vehicle transmissions by using a contention-free protocols at the medium access layer. For instance, the authors in [START_REF] Omar | On multihop communications for in-vehicle internet access based on a TDMA MAC protocol[END_REF] have proposed a cross-layer MAC and routing scheme based on contention-free MAC protocol called VeMAC [START_REF] Zhuang | VeMAC: A novel multichannel MAC protocol for vehicular ad hoc networks[END_REF] for multihop in-vehicle Internet access. The goal of this work is to propose a routing protocol which allows a vehicle to discover the existence of a gateway connected to the Internet and exchange packets with it through multi-hop communications. The proposed routing protocol consists of two main phases: gateway discovery and packet forwarding. The first phase defines how a vehicle can discover the existence of a gateway installed along the highway, while the second one defines how a packet is transmitted via multi-hop communications from a vehicle to a gateway and vice versa. However, this multi-hop communication scheme is limited only to infotainment applications (i.e. Internet access) and does not support safety applications, which makes it unsuitable for VANETs which are specially designed to improve road safety. In this paper, we focus on this category of approach and we propose a new TDMA-aware Routing protocol for Multihop Communications in VANETs, in which the next hop decisions are based on the TDMA scheduling at the MAC layer. Unlike [START_REF] Omar | On multihop communications for in-vehicle internet access based on a TDMA MAC protocol[END_REF], our protocol allows a vehicle to send packets over a large distance using V2V communication.

III. SYSTEM MODEL AND DTMAC PROTOCOL

In this paper, we suppose that each vehicle in a VANET is equipped with a GPS (Global Positioning System) that can allow it to obtain an accurate real-time three-dimensional geographic position (latitude, longitude and altitude), speed and exact time. Moreover, synchronization between vehicles may be performed by using GPS timing information. Each road is divided into 𝑁 small fixed areas, denoted by 𝑥 𝑖 , 𝑖 = 1, . . . , 𝑁 (see Figure 2). Note that the area size depends on the transmission range of the vehicles. We assume that the vehicles are equipped with digital maps to determine which area they are in. Moreover, the vehicles employ multi-hop communications to send their packets over a large distance. As shown in Figure 2, a source vehicle denoted by A, transmits its messages to a destination vehicle denoted by G through some relay vehicles. In the following, we will briefly describe the TDMA scheduling principle of DTMAC protocol [START_REF] Hadded | An infrastructure-free slot assignment algorithm for reliable broadcast of periodic messages in vehicular ad hoc networks[END_REF]. The latter uses the vehicle location and slot reuse concept to ensure that vehicles in adjacent areas have collision-free schedules. The channel time is partitioned into frames and each frame is further partitioned into three sets of time slots 𝑆 0 , 𝑆 1 , and 𝑆 2 of size equal to 𝑛 0 , 𝑛 1 and 𝑛 2 , respectively. These sets are associated with vehicles moving in the areas 𝑥 𝑖 , 𝑥 𝑖+1 , and 𝑥 𝑖+2 , respectively. Moreover, the three subsets of time slots are reused between neighboring areas in such a way that no vehicles in different adjacent areas can access the channel at the same time, and thus no interference can occur. When an active vehicle 𝑣 moving within the area 𝑥 𝑖 needs to acquire a time slot on the transmission channel, it starts listening to the channel during the set of time slots reserved for the area in which it is traveling, let 𝑆 𝑗 (𝑣), where 𝑗 = (𝑖 + 2) 𝑚𝑜𝑑 3. At the end of the frame the vehicle 𝑣 can determine the set of its neighbors 𝑁 (𝑣) and the set of busy slots in 𝑆 𝑗 (𝑣), denoted by 𝐵(𝑣). In order to avoid any collision problem, this set of time slots can not be used by any neighboring vehicles 1 . Therefore, vehicle 𝑣 can determine the set of available time slots 𝐹 (𝑣) and then attempt to select one of them at random.

IV. CONTRIBUTION : TDMA-AWARE ROUTING FOR MULTI-HOP COMMUNICATIONS IN VANETS

In this paper we present a new TDMA-aware Routing Protocol for Multi-hop communications in VANETs, named TRPM. The main idea of this protocol is to select the next hop using the vehicle position and the time slot information from the TDMA scheduling. Like the GPSR protocol [START_REF] Karp | GPSR: greedy perimeter stateless routing for wireless networks[END_REF], we assume that each transmitting vehicle knows the position of the packet's destination. In TRPM, the TDMA scheduling information and the position of a packet's destination are sufficient to make correct forwarding decisions at each transmitting vehicle. Specifically, if a source vehicle is moving in area 𝑥 𝑖 , the locally optimal choice of next hop is the neighbor geographically located in area 𝑥 𝑖+1 or 𝑥 𝑖-1 according to the position of the packet's destination. As a result, the TDMA slot scheduling obtained by DTMAC can be used to determine the set of next hops that are geographically closest to the destination. In fact, each vehicle that is moving in the area 𝑥 𝑖 can know the locally optimal set of next hops that are located in adjacent areas 𝑥 𝑖+1 or 𝑥 𝑖-1 by observing the set of time slots 𝑆 (𝑖+3)%3 or 𝑆 (𝑖+1)%3 , respectively. We consider the same example presented above when vehicle G as the destination vehicle that will broadcast a message received from vehicle A. As shown in Figure 2, only two relay vehicles are needed to ensure a multi-hop path between vehicle A and G (one relay node in the area 𝑥 2 and another one in the area 𝑥 3 ). In the following, we suppose that the DTMAC protocol has been used by the vehicles to organize the channel access. The TDMA slot scheduling obtained by DTMAC is illustrated in Figure 2. Firstly, vehicle A forwards a packet to B, as vehicle A uses its frame information to choose a vehicle that is accessing the channel during the set 𝑆 1 . Upon receiving the packet for forwarding, vehicle B will choose by using its frame information a vehicle that's accessing the channel during the set of time slots 𝑆 2 (say vehicle D). Then, vehicle D will forward the packet to G, as G is moving in area 𝑥 4 (accessing the channel during the set 𝑆 0 ) and it is the direct neighbor of vehicle D. By using DTMAC as the MAC layer, we can note that the path A-B-D-G is the shortest, in terms of the number of hops as well as the end-to-end delay which is equal to 6 time slots (2 time slots between 𝑡 0 and 𝑡 2 as 𝑡 2 is the transmission slot for vehicle B, then 2 time slots between 𝑡 2 and 𝑡 4 as 𝑡 4 is the transmission slot for vehicle D and finally 2 time slots between 𝑡 4 and 𝑡 0 as 𝑡 0 is the transmission slot in which vehicle G will broadcast the message received from vehicle A).

The idea of TRPM is the following. Whenever a vehicle 𝑖 accessing the channel during the set 𝑆 𝑘 wants to send/forward a message, it constructs two sets of candidate forwarders based on its scheduling table as follows, where 𝑇 𝑆(𝑗) indicates the time slot reserved by vehicle 𝑗.

• 𝐴 𝑖 = {𝑗 ∈ 𝑁 (𝑖) | 𝑇 𝑆(𝑗) ∈ 𝑆 (𝑘+1)%3 } //
The set of vehicles that are moving in the adjacent right-hand area.

• 𝐵 𝑖 = {𝑗 ∈ 𝑁 (𝑖) | 𝑇 𝑆(𝑗) ∈ 𝑆 (𝑘+2)%3 } //
The set of vehicles that are moving in the adjacent left-hand area.

Each source vehicle uses the position of a packet's destination and the TDMA scheduling information to make packet forwarding decisions. In fact, when a source vehicle 𝑖 is moving behind the destination vehicle, it will select a next hop relay that belongs to set 𝐵 𝑖 ; when the transmitter is moving in front of the destination vehicle, it will select a forwarder vehicle from those in set 𝐴 𝑖 . Algorithm 1 outlines the behavior of our scheme during the procedure for forwarding messages. For each vehicle 𝑖 that will forward a message, we define the normalized weight function WHS (Weighted next-Hop Selection) which depends on the delay and the distance between each neighboring vehicle 𝑗. WHS is calculated as follows:

𝑊 𝐻𝑆 𝑖,𝑗 = 𝛼 * Δ𝑡𝑖,𝑗 𝜏 -(1 -𝛼) * 𝑑𝑖,𝑗 𝑅 (1)
Where:

• 𝜏 is the length of the TDMA frame (in number of time slots).

• 𝑗 is one of the neighbors of vehicle 𝑖, which represents the potential next hop that will relay the message received from vehicle 𝑖.

• Δ𝑡 𝑖,𝑗 is the gap between the sending slot of vehicle 𝑖 and the sending slot of vehicle 𝑗.

• 𝑑 𝑖,𝑗 is the distance between the two vehicles 𝑖 and 𝑗, and 𝑅 is the communication range.

• 𝛼 is a weighted value in the interval [0, 1] that gives more weight to either distance or delay. When 𝛼 is high, more weight is given to the delay. Otherwise, when 𝛼 is small, more weight is given to the distance.

When a vehicle receives a message (as shown in Algorithm 1), it checks whether it is the destination of the packet (line 1), and if it is, it passes the packet to the upper layer (line 2). However, if the packet is destined for another vehicle, the receiver will check if the destination is moving in the same area (line 4), and if it is, the message will be transmitted immediately to its final destination (line 5). Otherwise, if the packet's destination is moving in another area, the receiver will calculate the next hop vehicle towards the destination (lines 7-11). If a relay node is found, the message will be forwarded (line 15), otherwise the message will be queued (line 17). Each forwarding vehicle includes its area ID in the relayed message. These steps are repeated by each relay vehicle until the packet is received by its final destination vehicle. To deliver a packet from a source to a destination, each vehicle 𝑖 receiving a message will use the weight function WSH to select a forwarding vehicle in the next area from those listed in the set 𝐴 𝑖 or 𝐵 𝑖 . By subtracting the area ID contained in the received message, the vehicle 𝑖 can determine the appropriate set of potential relays. For instance, in the situation depicted in Figure 3, vehicle 𝑇 𝑋 will send a message to vehicle 𝑅𝑋. Since, the vehicle 𝑇 𝑋 is moving ahead of vehicle 𝑅𝑋, it will forward the message to vehicle 𝐹 1 that is moving in the area 𝑥 2 and accessing the channel during the set of time slots 𝑆 1 . Vehicle 𝐹 1 needs to wait until its slot to forward the packet (i.e. it needs to wait for 𝑇 𝑆(𝐹 1) -𝑇 𝑆(𝑇 𝑋) slots). As vehicle 𝐹 1 has received the packet from vehicle 𝑇 𝑋 which is moving in the area 𝑥 1 , vehicle 𝐹 1 will immediately select a forwarding vehicle from those located in the area 𝑥 3 which are accessing the channel during the set of time slots 𝑆 2 . Then, assuming that vehicle 𝐹 1 decides to choose vehicle 𝐹 2 as the next hop to relay the packet, once the slot starts, the vehicle 𝐹 1 will retransmit the message to vehicle 𝐹 2 which in turn will forward the packet directly to its final destination 𝑅𝑋. 𝑠𝑒𝑛𝑑 𝑚𝑒𝑠𝑠𝑎𝑔𝑒(𝑚𝑠𝑔.𝑠𝑟𝑐, 𝑚𝑠𝑔.𝑑𝑠𝑡, " ")

𝑞𝑢𝑒𝑢𝑒 𝑚𝑒𝑠𝑠𝑎𝑔𝑒(𝑚𝑠𝑔) 𝑎𝑛𝑑 𝑔𝑜 𝑡𝑜 1 18: end if

As shown in Figure 3, one frame is sufficient to deliver a message from TX to RX, because this message is forwarded three times (i.e. 𝑛 0 + 𝑛 1 + 𝑛 2 = 𝜏 slots). Based on this example, we can theoretically estimate the End-to-End Delay (EED) needed to deliver a message from a source vehicle 𝑖 to a destination vehicle 𝑗. EED is estimated as follows, where 𝑆 𝑑 is the slot duration which is fixed to 0.001s.

𝐸𝐸𝐷 𝑖,𝑗 ≤ ⌈ 𝑑𝑖𝑠𝑡𝑖,𝑗 3 * 𝑅 ⌉ * 𝜏 * 𝑆 𝑑 (2)

V. SIMULATION RESULTS AND PERFORMANCE EVALUATION

A. Simulation Scenarios

We generated a realistic VANET environment by selecting a real highway area from a digital map which took into account lane directions. Figure 4 shows a metropolitan area from the Map of San Jose (California) of size 3000𝑚×100𝑚 exported from OpenStreetMap (OSM) and edited using Java OpenStreetMap Editor (JOSM). Then MOVE and SUMO [START_REF] Karnadi | Rapid generation of realistic mobility models for VANET[END_REF] were used respectively to generate vehicular traffic scenarios and to simulate the area with vehicular traffic. Then, the traffic traces generated by SUMO were used in the 𝑛𝑠2.34 simulator. The simulation parameters used in our experiments are summarized in Table I. Each simulation run lasts for 120 seconds. After the first 2 seconds of simulation, the source vehicle starts to transmit a message 50 bytes in size. The message is transmitted to only one destination vehicle through multiple relay nodes and is repeated periodically after one second. We simulated several scenarios by varying the vehicle density between 40 and 330 vehicles in the whole network.

B. Performance evaluations

We compared the proposed TRPM with two multi-hop communication protocols that have the same underlying principle (i.e. MAC-aware routing protocol). The first one is the Random TDMA aware routing protocol, called RTDMA [START_REF] Rhee | DRAND: Distributed randomized tdma scheduling for wireless ad hoc networks[END_REF]. In this protocol the time slots are allocated to vehicles randomly and the next hop decision for unicast traffic forwarding is based on the vehicles' positions and the time slot information from the random TDMA scheduling. The second protocol is the Contention aware routing protocol, called CRP [START_REF] Ko | Flooding-based geocasting protocols for mobile ad hoc networks[END_REF] which is based on classical flooding, where every vehicle relays each packet received to all its one-hop neighbors at least once until the packet has been received by its final destination vehicle. These protocols are evaluated by varying the Source-to-Destination Distance (SDD) between 550𝑚 and 2550𝑚. Moreover, we evaluated these protocols on the same network scenarios in terms of the average EED, average number of hops and average delivery ratio. The performance of TRPM depends on the value of 𝛼, which determines its behavior. In fact, when 𝛼 is high, more weight is given to the delay. Otherwise, when 𝛼 is small, more weight is given to the distance and thus to the number of hops. For this, we evaluated several values of 𝛼 in different scenarios to find the optimal value of this parameter. Figure 5-a shows the variation of the average EED with the change in 𝛼 values. We can see from this figure that the EED is reduced to less than 150𝑚𝑠 by choosing values of 𝛼 between 0.3 and 0.6 under a high traffic condition scenario. However, when 𝛼 < 0.3 or 𝛼 > 0.6 the average EED is high, the reason being that when 𝛼 is small, more weight is given to the distance, hence the selected relay vehicles between the source and the destination generate more delay due to the high gaps between their sending slots. On the other hand, higher values of 𝛼 give more weight to the delay than to the distance which provides routes that have a good delay but a greater number of relay vehicles as we can see in Figure 5-b. In the following, we present the simulation results and we analyze the performance of our proposed protocol. For these results, the weight factor 𝛼 was fixed to 0.4.

Figure 6-a shows the average EED for all the multihop communication protocols under consideration. We can note from this figure that the TRPM protocol performs very well compared to RTDMA and CRP, especially as the distance between the source and destination increases. For instance, when 𝑆𝐷𝐷 = 2295𝑚, the TRPM protocol achieves an average EED of 234.16𝑚𝑠 while RTDMA and CRP show an average delay of 520.33𝑚𝑠 and 626.3𝑚𝑠, respectively (i.e. approximately 122.2% and 167.47% higher than TRPM). This is mainly because, as discussed in Section 4, the proposed protocol can reduce the gap between the sending slots of relay vehicles that are moving in adjacent areas by dividing the frame into three sets of time slots. Moreover, we can observe that RTDMA performs much better than CRP. These results can be explained by the fact that, in CRP, all candidate relay nodes are considered without taking into account any criteria. This figure also compares the theoretical values of average EED with those obtained by simulation. The theoretical values are close to the simulated values for all shown SDDs. In Figure 6-b, we show the relationship between the average number of relay nodes and the SDD. It is clear from this figure that the number of relay vehicles increases as the distance increases. We can note that TRPM can significantly reduce the number of relay vehicles required to deliver a message compared to RTDMA and CRP protocols. This is due to the fact that TRPM always selects only one relay vehicle for each area (i.e. one relay node for each 310𝑚), in contrast to the RTDMA and CRP in which two or more relay vehicles can be successively selected within the same area. Unlike CRP and RTDMA, these two figures clearly show that TRPM achieves better performances in terms of both average EED and average number of hops, since it uses a next-hop selection function that can balance the two metrics studied.

In order to assess the effect of collision in the performance of these protocols, we evaluate then on the following scenario where there is a background traffic that consists of a periodic message broadcasted by each vehicle every 100𝑚𝑠. Figure 7 shows the average number of relay vehicles for TRPM, RTDMA and CRP protocols when varying vehicle density. Unlike RTDMA and CRP, the average number of hops in TRPM is still constant as vehicle density increases. This is mainly due to the forwarding concept in TRPM which always selects only one forwarding vehicle in each area so that the number of relay vehicles always remains constant. In order to validate the previous results, we evaluate the performance of these protocols in terms of delivery reliability. Figure 8 shows the average delivery ratio of the three protocols under consideration when varying vehicle density. As shown in this figure, TRPM achieves a considerably higher delivery rate of packets than RTDMA and CPR. For instance for a high density (in the case of 280 vehicles), the TRPM protocol achieves an average delivery ratio of 98.4%, in contrast to RTDMA and CRP which show a rate of 75.91% and 65.52%, respectively. We can note that TRPM maintains almost an average delivery ratio close to the ideal rate (i.e. 100%) for all VANET scenarios. This is because TRPM implements an optimized relay vehicle selection mechanism that can avoid redundant transmissions. Moreover, TRPM is a contention-free based protocol that can reduce packet collisions in the presence of background traffic. We can also see that the CRP and RTDMA protocols have very poor performances. These results might well be expected for CRP since it is a flooding based routing protocol in which each vehicle retransmits the message received to all its neighboring vehicles without using any selection mechanism.

VI. CONCLUSION

The stringent requirements of VANET safety applications mean that their messages need to be delivered quickly and with a high degree of reliability. However, designing an efficient multi-hop communication protocol for packet delivery is a major challenge in VANETs due to the rapid changes in network topology and the lack of infrastructure. In this paper, we propose a novel TDMA aware routing protocol to allow a vehicle to send a packet over a long distance through multiple relay vehicles. The packet is delivered from a source vehicle to a destination vehicle using the geographic positions and the time slot information from the TDMA scheduling. Moreover, our protocol takes into account the relay selection efficiency by using a weighted next-hop selection function in order to make coherent next hop decision in terms of both number of relay vehicles and end-to-end delay (EED).

The simulation results show that, compared to two other protocols, our proposed cross layer protocol provides better performances in terms of average end-to-end delay, average number of hops and average delivery ratio.

In future work, we will explore how our TDMA aware routing protocol can be adapted to support multichannel operation and to provide a reliable delivery of messages on both the control and service channels. Moreover, we plan to carry out extensive simulations to compare it with a geographic routing protocol, such as the Greedy Perimeter Stateless Routing (GPSR) protocol.

Fig. 1 .

 1 Fig. 1. VANET network using random TDMA scheduling scheme.

Fig. 2 .

 2 Fig. 2. VANET network using DTMAC scheduling scheme.

Fig. 3 .Algorithm 1 4 :

 314 Fig. 3. Message propagation based on TDMA slot information.

Fig. 4 .

 4 Fig. 4. San Jose (California) urban area captured from Google Maps (left) and exported to a VANET network topology by using MOVE/SUMO (right).

Fig. 7 .

 7 Fig. 7. The average number of relay vehicles vs vehicle density.

Fig. 8 .

 8 Fig. 8. The average delivery ratio vs vehicle density.

Fig. 5 .

 5 Fig. 5. Effect of changing 𝛼 on average EED (left) and on average number of hops (right).

Fig. 6 .

 6 Fig. 6. The average EED (left) and the average number of relay vehicles (right) vs SDD.

  𝑓𝑟𝑤𝑑𝑖={𝑘 ∈ 𝐴𝑖 | 𝑊 𝐻𝑆 𝑖,𝑘 = 𝑚𝑖𝑛 (𝑊 𝐻𝑆 𝑖,𝑙 ∀ 𝑙 ∈ 𝐴𝑖)} 𝑓𝑟𝑤𝑑𝑖={𝑘 ∈ 𝐵𝑖 | 𝑊 𝐻𝑆 𝑖,𝑘 = 𝑚𝑖𝑛 (𝑊 𝐻𝑆 𝑖,𝑙 ∀ 𝑙 ∈ 𝐵𝑖)}

	6:	else
	7:	if 𝑚𝑠𝑔.𝐴𝑟𝑒𝑎 𝐼𝐷 ≺ 𝑥𝑗 then
	8:
	9:	else
	10:
	11:	end if
	12:	end if
	13: end if
	14: if 𝑓𝑟𝑤𝑑𝑖 ≥ 0 then
	15:	𝑠𝑒𝑛𝑑 𝑚𝑠𝑔(𝑚𝑠𝑔.𝑠𝑟𝑐, 𝑚𝑠𝑔.𝑑𝑠𝑡, 𝑚𝑠𝑔.𝑓 𝑟𝑤𝑑𝑖)
	16: else
	17:

TABLE I .

 I Simulation parameters

	Parameter	Value
	Highway length	3 𝑘𝑚
	Vehicle speed	120 𝑘𝑚/ℎ
	Speed standard deviation (𝜎)	30 𝑘𝑚/ℎ
	Transmission range	310 𝑚
	Slots/frame	100
	Slot duration	0.001 𝑠
	Simulation time	120 𝑠

The set of neighbors is the set of vehicles that are moving within the same area.