Marc Quincampoix
email: marc.quincampoix@univ-brest.fr

Jérôme Renault
email: jerome.renault@polytechnique.edu

On the existence of a limit value in some non expansive optimal control problems

limit value in some non expansive optimal control problems

Introduction

We consider the following optimal control denoted Γ t (y 0 ) :

(1)

V t (y 0 ) := inf

u∈U 1 t t s=0
h(y(s, u, y 0 ), u(s))ds, where s → y(s, u, y 0 ) denotes the solution to

(2) y ′ (s) = g(y(s), u(s)), y(0) = y 0 .

Here U is the set of measurable controls from IR + to a given non empty metric space U. Throughout the paper, we will suppose Lipschitz regularity of g : IR d × U → IR d which implies that for a given control u in U and a given initial condition y 0 , equation (2) has a unique absolutely continuous solution.

The main goal of the paper consists in studying the asymptotic behaviour of V t (y 0 ) when t tends to ∞. This problem has been considered in several papers (cf for instance in [START_REF] Arisawa | On ergodic stochastic control[END_REF][START_REF] Bettiol | On ergodic problem for Hamilton-Jacobi-Isaacs equations[END_REF][START_REF] Cardaliaguet | Ergodicity of Hamilton-Jacobi equations with a non coercive non convex Hamiltonian in IR 2 /Z 2 preprint[END_REF]) by approaches ensuring that the limit of V t (y 0 ) is independent of y 0 . In the present paper we exhibit several examples where the limit exists and depends of y 0 . Our aim is to obtain a general result which contains in particular the more easy to state following result, where throughout the paper, < •, • > stands for the canonical scalar product and B is the associated closed unit ball.. Proposition 1.1. Assume that g is Lipschitz, that there exists a compact set N which is -forward -invariant by the control system [START_REF] Arisawa | On ergodic stochastic control[END_REF] and that h is a continuous function which does not depend on u. Assume moreover that :

(3) ∀(y 1 , y 2 ) ∈ N 2 , sup u∈U inf v∈U < y 1y 2 , g(y 1 , u)g(y 2 , v) >≤ 0.

Then problem [START_REF] Arisawa | Ergodic problem for the Hamilton Jacobi Belmann equations II[END_REF] has a value when t converges to +∞ i.e. there exists V (y 0 ) := lim t→+∞ V t (y 0 ). Condition (3) means a non expansive property of the control system, while the condition

∀(y 1 , y 2 ) ∈ N 2 , sup u∈U inf v∈U < y 1 -y 2 , g(y 1 , u) -g(y 2 , v) >≤ -C y 1 -y 2 2
expresses a dissipativity property of the control system. The above dissipativity condition does imply that the limit is independent of y 0 (cf [START_REF] Artstein | The value function of singularly perturbed control systems[END_REF]).

The value function [START_REF] Arisawa | Ergodic problem for the Hamilton Jacobi Belmann equations II[END_REF] can also be characterized through -viscosity -solution of a suitable Hamilton-Jacobi equation. In several articles initiated by the pioneering work [START_REF] Lions | Homogenization of Hamilton-Jacobi Equations[END_REF] the limit of V t (y 0 ) is obtained by "passing to the limit" on the Hamilton-Jacobi equation. This required coercivity properties of the Hamiltonian which could be implied by controlability and/or dissipativity of the control system but which are not valid in the nonexpansive case [START_REF] Artstein | The value function of singularly perturbed control systems[END_REF]. Moreover the PDE approach is out of the scope of the -long enough -present article.

Definition 1.2. The problem Γ(y 0 ) := (Γ t (y 0 )) t>0 has a limit value if lim t→∞ V t (y 0 ) exists. Whenever it exists, we denote this limit by V (y 0 ).

Our main aim consists in giving one sufficient condition ensuring the existence of the limit value. As a particular case of our main result we obtain proposition (1.1).

It is also of interest to know if approximate optimal controls for the value V t (y 0 ) are still approximate optimal controls for the limit value. This leads us to the following definition.

Definition 1.3. The problem Γ(y 0 ) has a uniform value if it has a limit value V (y 0 ) and if :

∀ε > 0, ∃u ∈ U, ∃t 0 , ∀t ≥ t 0 , 1 t t s=0 h(y(s, u, y 0 ), u(s))ds ≤ V (y 0 ) + ε.
Whenever the uniform value exists, the controller can act (approximately) optimally independently of the time horizon. On the contrary, if the limit value exists but the uniform value does not, he really needs to know the time horizon before choosing a control. We will prove that our results do imply the existence of a uniform value. We will be inspired by a recent work in the discrete time case [START_REF] Renault | Uniform value in Dynamic Programming[END_REF].

Let us explain now, how the paper is organized. The second section contains some preliminaries and discussions of limit behaviors in examples. In the third section, we state and prove our main result for the existence of the uniform value.

Preliminaries

We now consider the optimal control problems (Γ t (y 0 )) t described by ( 1) and (2).

Assumptions and Notations

We now describe the assumptions made on g and h.

(4)

      
The function h : IR d × U -→ IR is measurable and bounded The function g :

IR d × U -→ IR d is measurable ∃L ≥ 0, ∀(y, y ′ ) ∈ IR 2d , ∀u ∈ U, g(y, u) -g(y ′ , u) ≤ L y -y ′ ∃a > 0, ∀(y, u) ∈ IR d × U, g(y, u) ≤ a(1 + y )
With these hypotheses, given u in U equation ( 2) has a unique absolutely continuous solution y(•, u, y 0 ) :

IR + → IR d .
Since h is bounded, we will assume without loss of generality from now on that h takes values in [0, 1].

We denote by G(y 0 ) := {y(t, u, y 0 ), t ≥ 0, u ∈ U} the reachable set (i.e. the set of states that can be reached starting from y 0 ).

We denote the average cost induced by u between time 0 and time t by :

γ t (y 0 , u) = 1 t t 0 h(y(s, u, y 0 ), u(s))ds
The corresponding Value function satisfies V t (y 0 ) = inf u∈U γ t (y 0 , u).

Examples

We present here basic examples. In all these examples, the cost h(y, u) only depends on the state y. We will prove later that the uniform value exists in examples 2, 3 and 4.

• Example 1 : here y lies in IR 2 seen as the complex plane, there is no control and the dynamic is given by g(y, u) = i y, where i 2 = -1. We clearly have :

V t (y 0 ) ---→ t→∞ 1 2π|y 0 | |z|=|y 0 | h(z)dz,
and since there is no control, the value is uniform.

• Example 2 : in the complex plane again, but now g(y, u) = i y u, where u ∈ U a given bounded subset of IR, and h is continuous in y.

• Example 3 : g(y, u) = -y + u, where u ∈ U a given bounded subset of IR d , and h is continuous in y.

• Example 4 : in IR 2 . The initial state is y 0 = (0, 0) and the control set is U = [0, 1]. For a state y = (y 1 , y 2 ) and a control u, the dynamic is given by

y ′ (s) = g(y(s), u(s)) = u(s)(1 -y 1 (s)) u 2 (s)(1 -y 1 (s))
, and the cost is h(y) = 1-y 1 (1-y 2 ).

Notice that for any control, y ′ 1 (s) ≥ y ′ 2 (s) ≥ 0, and thus y 2 (t) ≤ y 1 (t) for each t ≥ 0. One can easily observe that G(y 0 ) ⊂ [0, 1] 2 .

If one uses the constant control u = ε > 0, we obtain y 1 (t) = 1exp(-εt) and y 2 (t) = εy 1 (t). So we have V t (y 0 ) ---→ t→∞ 0.

More generally, if the initial state is y = (y 1 , y 2 ) ∈ [0, 1] 2 , by choosing a constant control u = ε > 0 small, one can show that the limit value exists and lim t→∞ V t (y) = y 2 .

Notice that there is no hope here to use an ergodic property, because

{y ∈ [0, 1] 2 , lim t→∞ V t (y) = lim t→∞ V t (y 0 )} = [0, 1] × {0},
and starting from y 0 it is possible to reach no point in (0, 1] × {0}.

• Example 5 : in IR 2 , y 0 = (0, 0), control set U = [0, 1], g(y, u) = (y 2 , u), and

h(y 1 , y 2 ) = 0 if y 1 ∈ [1, 2], = 1 otherwise.
We have u(s) = y ′ 2 (s) = y ′′ 1 (s), hence we may think of the control u as the acceleration, y 2 as the speed and y 1 as the position of some mobile. If u = ε constant, then y 2 (t) = 2εy 1 (t) ∀t ≥ 0.

We have u ≥ 0, hence the speed cannot decrease. Consequently, the time interval where y 1 (t) ∈ [1, 2] cannot be longer than the time interval where y 1 (t) ∈ [0, 1), and we have V T (y 0 ) ≥ 1/2 for each T .

One can prove that V T (y 0 ) ---→

T →∞ 1/2 by considering the following controls : choose t in (0, T ) such that (2/ t) + ( t/2) = T , make a full acceleration up to t and completely stop accelerating after : u(t) = 1 for t < t, and u(t) = 0 for t ≥ t.

Consequently the limit value exists and is 1/2. However, for any control u in U, we either have y(t, u, y 0 ) = y 0 for all t, or y 1 (t, u, y 0 ) ---→ t→∞ +∞. So in any case we have 1 t t 0 h(y(s, u, y 0 ), u(s))ds ---→ t→∞ 1. The uniform value does not exist here, although the dynamic is very regular.

Existence results for the uniform value 3.1 A technical Lemma

Let us define V -(y 0 ) := lim inf t→+∞ V t (y 0 ) and V + (y 0 ) := lim sup t→+∞ V t (y 0 ). Adding a parameter m ≥ 0, we will more generally consider the costs between time m and time m + t :

γ m,t (y 0 , u) = 1 t m+t m h(y(s, u, y 0 ), u(s))ds,
and the value of the problem where the time interval [0, m] can be devoted to reach a good initial state, is denoted by :

V m,t (y 0 ) = inf u∈U γ m,t (y 0 , u).
Of course γ t (y 0 , u) = γ 0,t (y 0 , u) and V t (y 0 ) = V 0,t (y 0 ).

Lemma 3.1. For every m 0 in IR + , we have :

sup t>0 inf m≤m 0 V m,t (y 0 ) ≥ V + (y 0 ) ≥ V -(y 0 ) ≥ sup t>0 inf m≥0 V m,t (y 0 ).
Proof : We first prove sup t>0 inf m≤m 0 V m,t (y 0 ) ≥ V + (y 0 ). Suppose by contradiction that it is false. So there exists ε > 0 such that for any t > 0 we have

inf m≤m 0 V m,t (y 0 ) ≤ V + (y 0 ) -ε . Hence for any t > 0 there exists m ≤ m 0 with V m,t (y 0 ) ≤ V + (y 0 ) -(ε/2). Now observe that V m,t (y 0 ) = inf u 1 t m+t m h(y(s, u, y 0 ), u(s))ds = 1 t inf u { m 0 +t 0 h(y(s, u, y 0 ), u(s))ds - m 0 +t m+t h(y(s, u, y 0 ), u(s))ds - m 0 h(y(s, u, y 0 ), u(s))ds} ≥ m 0 + t t V m 0 +t (y 0 ) -2 m 0 t . Hence m 0 + t t V m 0 +t (y 0 ) -2 m 0 t ≤ V + (y 0 ) -(ε/2).
Passing to the limsup when t goes to +∞ we obtain a contradiction. We now prove V -(y 0 ) ≥ sup t>0 inf m≤0 V m,t (y 0 ). Assume on the contrary that it is false. Then there exists ε > 0 and t > 0 such that V -(y 0 ) + ε ≤ inf m≤0 V m,t (y 0 ). So for any m ≥ 0, we have V -(y 0 ) + ε ≤ V m,t (y 0 ). We will obtain a contradiction by concatenating trajectories. Take T > 0, and write T = lt + r, with l in IN and r in [0, t). For any control u in U, we have : T γ T (y 0 , u) = tγ 0,t (y 0 , u) + tγ t,t (y 0 , u)

+ ... + tγ (l-1)t,t (y 0 , u) + rγ lt,r (y 0 , u) ≥ lt(V -(y 0 ) + ε). Hence γ T (y 0 , u) ≥ T -r T (V -(y 0 ) + ε).
So for T large enough we have V T (y 0 ) ≥ V -(y 0 ) + ε/2, hence a contradiction by taking the liminf when T → ∞ .

Remark : it is also easy to show that for each t 0 ≥ 0, we have inf

m≥0 sup t>t 0 V m,t (y 0 ) ≥ V + (y 0 ).
The following quantity will play a great role in the sequel.

Definition 3.2. V * (y 0 ) = sup t>0 inf m≥0
V m,t (y 0 ).

Main results

Let us state the first version of our main result (which clearly implies Proposition 1.1 stated in the introduction) Proposition 3.3. Assume that (4) holds true and furthermore : (H'1) h(y, u) = h(y) only depends on the state, and is continuous on

IR d . (H'2) G(y 0 ) is bounded, (H'3) ∀(y 1 , y 2 ) ∈ G(y 0 ) 2 , sup u∈U inf v∈U < y 1 -y 2 , g(y 1 , u) -g(y 2 , v) >≤ 0.
Then the problem Γ(y 0 ) has a limit value which is V * (y 0 ), i.e. V t (y 0 ) ----→ t→+∞ V * (y 0 ). The convergence of (V t ) t to V * is uniform over G(y 0 ), and we have

V * (y 0 ) = sup t≥1 inf m≥0 V m,t (y 0 ) = inf m≥0 sup t≥1 V m,t (y 0 ) = lim m→∞,t→∞ V m,t (y 0 ). Moreover the value of Γ(y 0 ) is uniform.
Condition (H'3) can be used to show that (cf Proposition 3.5) :

∀(y 1 , y 2 ) ∈ G(y 0 ) 2 , ∀ε > 0, ∀T ≥ 0, ∀u ∈ U, ∃v ∈ U s.t. : ∀t ∈ [0, T ],
y(t, u, y 1 )y(t, v, y 2 ) ≤ y 1y 2 + ε. Proposition 3.3 can be applied to the previous examples 1, 2 and 3, but not to example 4. Notice that in example 5, we have

V * (y 0 ) = 0 < 1/2 = lim t V t (y 0 ).
We will prove the following generalization of Proposition 3.3. We put Z = G(y 0 ), and denote by Z its closure in IR d . Theorem 3.4. Suppose that (4) holds true and furthermore assume that (H1) h is uniformly continuous in y on Z uniformly in u. And for each y in Z, either h does not depend on u or the set {(g(y, u), h(y, u)) ∈ IR d × [0, 1], u ∈ U} is closed. (H2) : There exist a continuous function ∆ : IR d × IR d -→ IR + , vanishing on the diagonal (∆(y, y) = 0 for each y) and symmetric (∆(y 1 , y 2 ) = ∆(y 2 , y 1 ) for all y 1 and y 2 ), and a function α :

IR + -→ IR + s.t. α(t) --→ t→0 0 satisfying : a) For every sequence (z n ) n with values in Z and every ε > 0, one can find n such that lim inf p ∆(z n , z p ) ≤ ε. b) ∀(y 1 , y 2 ) ∈ Z2 , ∀u ∈ U, ∃v ∈ U such that D ↑ ∆(y 1 , y 2 )(g(y 1 , u), g(y 2 , v)) ≤ 0 and h(y 2 , v) -h(y 1 , u) ≤ α(∆(y 1 , y 2 )).
Then we have the same conclusions as in Proposition 3.3. The problem Γ(y 0 ) has a limit value which is V * (y 0 ). The convergence of V t to V * is uniform over Z, and we have

V * (y 0 ) = sup t≥1 inf m≥0 V m,t (y 0 ) = inf m≥0 sup t≥1 V m,t (y 0 ) = lim m→∞,t→∞ V m,t (y 0 ). Moreover the value of Γ(y 0 ) is uniform.
Remarks :

• Although ∆ may not satisfy the triangular inequality nor the separation property, it may be seen as a "distance" adapted to the problem Γ(y 0 ).

• The assumption : "{(g(y, u), h(y, u)) ∈ IR d × [0, 1], u ∈ U} closed" could be checked for instance if U is compact and if h and g are continuous with respect to (y, u).

• D ↑ is the contingent epi-derivative (cf [START_REF] Aubin | Set-Valued Analysis Birkhaüser[END_REF]) (which reduces to the upper Dini derivative if ∆ is Lipschitz), defined by : D↑∆

(z)(α) = lim inf t→0 + ,α ′ →α 1 t (∆(z + tα ′ )-∆(z)). If ∆ is differentiable, the condition D ↑ ∆(y 1 , y 2 )(g(y 1 , u), g(y 2 , v)) ≤ 0 just reads : < g(y 1 , u), ∂ ∂y 1 ∆(y 1 , y 2 ) > + < g(y 2 , v), ∂ ∂y 2 ∆(y 1 , y 2 ) >≤ 0. • Proposition 3.
3 will be a corollary of Theorem 3.4. It corresponds to the case where : ∆(y 1 , y 2 ) = y 1y 2 2 , G(y 0 ) is bounded, and h(y, u) = h(y) does not depend on u (one can just take α(t) = sup{|h(x)h(y)|, xy 2 ≤ t}).

• H2a) is a precompacity condition. It is satisfied as soon as G(y 0 ) is bounded. It is also satisfied if ∆ satisfies the triangular inequality and the usual precompacity condition : for each ε > 0, there exists a finite subset C of Z s.t. :

∀z ∈ Z, ∃c ∈ C, ∆(z, c) ≤ ε. (see lemma 3.13)
• Notice that H2 is satisfied with ∆ = 0 if we are in the trivial case where inf u h(y, u) is constant.

• Theorem 3.4 can be applied to example 4, with ∆(y 1 , y 2 ) = y 1y 2 1 (L 1norm). In this example, we have for each y 1 , y 2 and u : ∆(y 1 + tg(y 1 , u), y 2 + tg(y 2 , u)) ≤ ∆(y 1 , y 2 ) as soon as t ≥ 0 is small enough.

Proof of Theorem 3.4

We assume in this section that the hypotheses of Theorem 3.4 are satisfied, and we may assume without loss of generality that α is non decreasing and upper semicontinuous (otherwise we replace α(t) by inf ε>0 sup t ′ ∈[0,t+ε] α(t ′ )).

A non expansion property

We start with a proposition expressing the fact that the problem is non expansive with respect to ∆, the idea being that given two initial conditions y 1 and y 2 and a control to be played at y 1 , there exists another control to be played at y 2 such that t → ∆(y(t, u, y 1 ), y(t, v, y 2 )) will not increase. Proposition 3.5. We suppose the hypothesis of Theorem 3.4. Then (5)

       ∀(y 1 , y 2 ) ∈ Z2 , ∀T ≥ 0, ∀ε > 0, ∀u ∈ U, ∃v ∈ U, ∀t ∈ [0, T ], ∆(y(t, u, y 1 ), y(t, v, y 2 )) ≤ ∆(y 1 , y 2 ) + ε, and for almost every t ∈ [0, T ], h(y(t, v, y 2 ), v(t)) -h(y(t, u, y 1 ), u(t)) ≤ α(∆(y(t, u, y 1 ), y(t, v, y 2 ))).
Proof : First fix y 1 , y 2 ε > 0, T > 0 and u. Let us consider the following set-valued map Φ :

IR + × Z × Z × IR → IR d × IR d × IR Φ(t, x, y, l) := co cl{(g(x, u(t)), g(y, v), 0)) | v ∈ U, h(y, v)-h(x, u(t)) ≤ α(∆(x, y))},
where co stands for the convex hull and cl for the closure. Notice that Φ(t, x, y, l) does not depend on l. Using (4), H1) and H2)b), one can check that Φ is a set valued map which is upper semicontinuous in (x, y, l), measurable in t and with compact convex nonempty values [START_REF] Aubin | Set-Valued Analysis Birkhaüser[END_REF][START_REF] Deimling | Multivalued Differential Equations, De gruyter Seris in Nonlinear Analysis and Apllications[END_REF]. We also denote Φ the set valued map defined as Φ but removing the convex hull.

¿From the measurable Viability Theorem [START_REF] Frankowska | Measurable Viability Theorems and Hamilton-Jacobi-Bellman Equation[END_REF] (cf also [START_REF] Carja | Viability, Invariance and Applications[END_REF] section 6.5), condition (H2) b) implies that the epigraph of ∆ (restricted to Z2 × IR) is viable for the differential inclusion [START_REF] Aubin | Set-Valued Analysis Birkhaüser[END_REF] (x ′ (t), y ′ (t), l ′ (t)) ∈ Φ(t, x(t), y(t), l(t)) for a. e. t ≥ 0 So starting from (y 1 , y 2 , ∆(y 1 , y 2 )), there exists a solution (x(•), y(•), l(•)) to ( 6) which stays for any t ≥ 0 in the epigraph of ∆ namely (7) ∆(x(t), y(t)) ≤ l(t) = ∆(y 1 , y 2 ), ∀t ≥ 0, by noticing that l(•) is a constant. ¿From the suppositions made on the dynamics g, the trajectory (x(•), y(•)) remains in a compact set (included in some large enough ball B(0, M)) on the time interval [0, T ]. Because ∆ is uniformly continuous on B(0, M) × B(0, M), there exists η ∈ (0, 1) with

∀(x, x ′ , y, y ′ ) ∈ B(0, M + 1) 4 , x -x ′ + y -y ′ < η =⇒ |∆(x, y) -∆(x ′ , y ′ )| < ε.
Thanks to the Wazewski Relaxation Theorem (cf for instance Th. 10.4.4 in [START_REF] Aubin | Set-Valued Analysis Birkhaüser[END_REF]) applied to Φ, the trajectory (x(•), y(•), l(•)) could be approximated on every compact interval by a trajectory to the differential inclusion defined by Φ. So there exists (y From the choice of η and the very definition of Φ we also have for any

1 (•), y 2 (•), l(•)) satisfying (y ′ 1 (t), y ′ 2 (t), l ′ (t)) ∈ Φ(t,
t ∈ [0, T ] ∆(y 1 (t), y 2 (t)) ≤ ∆(x(t), y(t)) + ε ≤ ∆(y 1 , y 2 ) + ε h(y 2 (t), v(t)) -h(y 1 (t), u(t)) ≤ α(∆(y 1 (t), y 2 (t)))
This completes our proof if, from one hand we observe that y 1 (•) = y(•, u, y 1 ) and from the other hand, we use Filippov's measurable selection Theorem (e.g. Theorem 8.2.10 in [START_REF] Aubin | Set-Valued Analysis Birkhaüser[END_REF]) to Φ for finding a measurable control v ∈ U such that y 2 (•) = y(•, v, y 2 ). QED

The limit value exists

Since α is u.s.c. and non decreasing, we obtain the following consequence of Proposition 3.5.

Corollary 3.6. For every y 1 and y 2 in G(y 0 ), for all T > 0,

|V T (y 1 ) -V T (y 2 )| ≤ α(∆(y 1 , y 2 )).
Define now, for each m ≥ 0, G m (y 0 ) as the set of states which can be reached from x 0 before time m :

G m (y 0 ) = {y(t, u, y 0 ), t ≤ m, u ∈ U}, so that G(y 0 ) = ∪ m≥0 G m (y 0 ).
An immediate consequence of the precompacity hypothesis H2a) is the following Lemma 3.7. For every ε > 0, there exists m 0 in IR + such that :

∀z ∈ G(y 0 ), ∃z ′ ∈ G m 0 (y 0 ) such that ∆(z, z ′ ) ≤ ε.
Proof : Otherwise for each positive integer m one can find z m in G(y 0 ) such that ∆(z m , z) > ε for all z in G m (y 0 ). Use H2a) to find n such that lim inf m ∆(z n , z m ) ≤ ε. Since z n ∈ G(y 0 ), there must exist k such that z n ∈ G k (y 0 ). But for each m ≥ k we have z n ∈ G m (y 0 ), hence ∆(z m , z n ) > ε. We obtain a contradiction.

QED

We can already conclude for the limit value.

Proposition 3.8. V t (y 0 ) ---→ t→∞ V * (y 0 ).
Proof : Because of lemma 3.1, it is sufficient to prove that for every ε > 0, there exists m 0 such that :

sup t>0 inf m≤m 0 V m,t (y 0 ) ≤ sup t>0 inf m≥0 V m,t (y 0 ) + 2ε
Fix ε, and consider η > 0 such that α(t) ≤ ε as soon as t ≤ η. Use lemma 3.7 to find m 0 such that ∀z ∈ G(y 0 ), ∃z ′ ∈ G m 0 (y 0 ) s.t. ∆(z, z ′ ) ≤ η.

Consider any t > 0. We have inf m≥0 V m,t (y 0 ) = inf{V t (z), z ∈ G(y 0 )}, and

inf m≤m 0 V m,t (y 0 ) = inf{V t (z), z ∈ G m 0 (y 0 )}. Let z in G(y 0 ) be such that V t (z) ≤ inf m V m,t (y 0 ) + ε, and consider z ′ ∈ G m 0 (y 0 ) s.t. ∆(z, z ′ ) ≤ η. By corollary 3.6, |V t (z) -V t (z ′ )| ≤ α(∆(z, z ′ )) ≤ ε, so we obtain that inf m≤m 0 V m,t (y 0 ) ≤ V t (z ′ ) ≤ V t (z) + ε ≤ inf m V m,t (y 0 ) + 2ε.
Passing to the supremum on t, this completes the proof. QED Remark 3.9. Observe that for obtaining the existence of the value, we have used a compactness argument (assumption H2)a)) and condition [START_REF] Aubin | Viability Theory Birkhauser[END_REF]. We did not use explicitly assumption H2)b) which is only used for obtaining [START_REF] Aubin | Viability Theory Birkhauser[END_REF].

The rest of the proof is more involved, and is inspired by the proof of Theorem 3.6 in [START_REF] Renault | Uniform value in Dynamic Programming[END_REF].

Auxiliary value functions

The uniform value requires the same control to be good for all time horizons, and we are led to introduce new auxiliary value functions. Given m ≥ 0 and n ≥ 1, for any initial state z in Z = G(y 0 ) and control u in U, we define

ν m,n (z, u) = sup t∈[1,n] γ m,t (z, u), and W m,n (z) = inf u∈U ν m,n (z, u).
W m,n is the value function of the problem where the controller can use the time interval [0, m] to reach a good state, and then his cost is only the supremum for t in [1, n], of the average cost between time m and m + t. Of course, we have W m,n ≥ V m,n . We write ν n for ν 0,n , and W n for W 0,n .

We easily obtain from proposition 3.5, as in corollary 3.6, the following result.

Lemma 3.10. For every z and z ′ in Z, for all m ≥ 0 and n ≥ 1,

|V m,n (z) -V m,n (z ′ )| ≤ α(∆(z, z ′ )). |W m,n (z) -W m,n (z ′ )| ≤ α(∆(z, z ′ )).
The following lemma shows that the quantities W m,n are not that high.

Lemma 3.11. ∀k ≥ 1, ∀n ≥ 1, ∀m ≥ 0, ∀z ∈ Z, V m,n (z) ≥ inf l≥m W l,k (z) - k n .
Proof : Fix k, n, m and z, and put A = inf l≥m W l,k (z). Consider any control u in U. For any i ≥ m, we have

sup t∈[1,k] γ i,t (z, u) = ν i,k (z, u) ≥ W i,k (z) ≥ A.
So we know that for any i ≥ m, there exists

t(i) ∈ [1, k] such that γ i,t(i) (z, u) ≥ A.
Define now by induction i 1 = m, i 2 = i 1 + t(i 1 ),..., i q = i q-1 + t(i q-1 ), where q is such that i q ≤ n + m < i q + t(i q ). We have nγ m,n (z, u)

≥ q-1 p=1 t(i p )A ≥ nA -k, so γ m,n (z, u) ≥ A-k n .
Taking the infimum over all controls, the proof is complete. QED

We know from Proposition 3.8 that the limit value is given by V * . We now give other formulas for this limit. Proposition 3.12. For every state z in Z,

inf m≥0 sup n≥1 W m,n (z) = inf m≥0 sup n≥1 V m,n (z) = V * (z) = sup n≥1 inf m≥0 V m,n (z) = sup n≥1 inf m≥0 W m,n (z).
Proof of proposition 3.12 : Fix an initial state z in Z. We already have

V * (z) = sup t>0 inf m≥0 V m,t (z) ≥ sup t≥1 inf m≥0 V m,t (z). One can easily check that inf m≥0 V m,t (z) ≤ inf m≥0 V m,2t (z) for each positive t. So V * (z) ≥ sup t≥1 inf m≥0 V m,t (z) ≥ sup t≥(1/2) inf m≥0 V m,t (z) ≥ . . . sup t>0 inf m≥0 V m,t (z) = V * (z). Consequently V * (z) = sup t≥1 inf m≥0 V m,t (z). Moreover because V m,t ≤ W m,t we have also V * (z) ≤ sup t≥1 inf m≥0 W m,t (z).
We now claim that V * (z) = sup t≥1 inf m≥0 W m,t (z). It remains to show V * (z) ≥ sup t≥1 inf m≥0 W m,t (z). From Lemma 3.11, we know that for all k ≥ 1, n ≥ 1 and m ≥ 0, we have

V m,nk (z) ≥ inf l≥0 W l,k (z) -1 n , so inf m V m,nk (z) ≥ inf l≥0 W l,k (z) -1 n .
By taking the supremum on n , we obtain

V * (z) = sup n≥1 inf m≥0 V m,n (z) ≥ sup n≥1 inf m≥0 V m,nk (z) ≥ inf l≥0 W l,k (z).
Since k is arbitrary, we have proved our claim. Since the inequalities

inf m≥0 sup n≥1 W m,n (z) ≥ inf m≥0 sup n≥1 V m,n (z) ≥ sup n≥1 inf m≥0 V m,n (z) = V * (z)
are clear, to conclude the proof of the proposition it is enough to show that inf m≥0 sup n≥1 W m,n (z) ≤ V * (z). Fix ε > 0. We have already proved that V * (z) = sup n≥1 inf m≥0 W m,n (z), so for each n ≥ 1 there exists m ≥ 0 such that W m,n (z) ≤ V * (z) + ε. Hence for each n, there exists z ′ n in G(z) such that W 0,n (z ′ n ) ≤ V * (z) + ε. We know from Lemma 3.7 that there exists m 0 ≥ 0 such that : ∀z ′ ∈ G(z), ∃z ′′ ∈ G m 0 (z) s.t. ∆(z ′ , z ′′ ) ≤ ε. Consequently, for each n ≥ 1, there exists z ′′ n in G m 0 (z) such that ∆(z ′ n , z ′′ n ) ≤ ε, and by lemma 3.10 this implies that

W n (z ′′ n ) ≤ W n (z ′ n ) + α(ε) ≤ V * (z) + ε + α(ε).
Up to now, we have proved that for every ε ′ > 0, there exists m 0 such that :

∀n ≥ 1, ∃m ≤ m 0 s.t. W m,n (z) ≤ V * (z) + ε ′ . Since all costs lie in [0, 1], it is easy to check that |W m,n (z) -W m ′ ,n (z)| ≤ |m -m ′ |
for each n, m, m ′ . Hence there exists a finite subset F of [0, m 0 ] such that :

∀n ≥ 1, ∃m ∈ F s.t. W m,n (z) ≤ V * (z) + 2ε ′ .
Considering m in F such that the set {n positive integer, W m,n (z) ≤ V * (z) + 2ε ′ } is infinite, and noticing that W m,n is non decreasing in n, we obtain the existence of a unique m ≥ 0 such that ∀n ≥ 1, W m,n (z) ≤ V * (z) + 2ε ′ . Hence ε ′ being arbitrary, inf m≥0 sup n≥1 W m,n (z) ≤ V * (z), concluding the proof of Proposition 3.12.

QED

We now look for uniform convergence properties. By the precompacity condition H2a), it is easy to obtain that : Lemma 3.13. For each ε > 0, there exists a finite subset C of Z s.t. : ∀z ∈ Z, ∃c ∈ C, ∆(z, c) ≤ ε.

We know that (V n ) n simply converges to V * on Z. Since |V n (z) -V n (z ′ )| ≤ α(∆(z, z ′ )) for all n, z and z ′ , we obtain by lemma 3.13 : Corollary 3.14. The convergence of (V n ) n to V * is uniform on Z.

We can proceed similarly to obtain other uniform properties. We have

V * (z) = sup n≥1 inf m≥0 W m,n (z) = lim n→+∞ inf m≥0 W m,n (z)
since inf m≥0 W m,n (z) is not decreasing in n. Using lemmas 3.10 and 3.13, we obtain that the convergence is uniform, hence we get :

∀ε > 0, ∃n 0 , ∀z ∈ Z, V * (z) -ε ≤ inf m≥0 W m,n 0 (z) ≤ V * (z).
By Lemma 3.11, we obtain :

∀ε > 0, ∃n 0 , ∀z ∈ Z, ∀m ≥ 0, ∀n ≥ 1, V m,n (z) ≥ V * (z) -ε - n 0 n .
Considering n large gives :

(8) ∀ε > 0, ∃K, ∀z ∈ Z, ∀n ≥ K, inf m≥0 V m,n (z) ≥ V * (z) -ε
Write now, for each state z and m ≥ 0 : h m (z) = inf m ′ ≤m sup n≥1 W m ′ ,n (z). (h m ) m converges to V * , and as before, by Lemmas 3.10 and 3.13, we obtain that the convergence is uniform. Consequently, [START_REF] Carja | Viability, Invariance and Applications[END_REF] ∀ε > 0, ∃M ≥ 0, ∀z ∈ Z, ∃m ≤ M, sup n≥1 W m,n (z) ≤ V * (z) + ε.

On the existence of a uniform value

In order to prove that Γ(y 0 ) has a uniform value we have to show that for every ε > 0, there exist a control u and a time n 0 such that for every n ≥ n 0 , γ n (y 0 , u) ≤ V * (y 0 ) + ε. In this subsection we adapt the proofs of Lemma 4.1 and Proposition 4.2. in [START_REF] Renault | Uniform value in Dynamic Programming[END_REF]. We start by constructing, for each n, a control which : 1) gives low average costs if one stops the play at any large time before n, and 2) after time n, leaves the player with a good "target" cost. This explains the importance of the quantities ν m,n . We start with the following Lemma 3.15. ∀ε > 0, ∃M ≥ 0, ∃K ≥ 1, ∀z ∈ Z, ∃m ≤ M, ∀n ≥ K, ∃u ∈ U such that : [START_REF] Deimling | Multivalued Differential Equations, De gruyter Seris in Nonlinear Analysis and Apllications[END_REF] ν m,n (z, u) ≤ V * (z) + ε/2, and V * (y(m + n, u, z)) ≤ V * (z) + ε.

Proof : Fix ε > 0. Take M given by ( 9), so that ∀z ∈ Z, ∃m ≤ M, sup n≥1 W m,n (z) ≤ V * (z) + ε. Take K ≥ 1 given by ( 8) such that :

∀z ∈ Z, ∀n ≥ K, inf m V m,n (z) ≥ V * (z) -ε.
Fix an initial state z in Z. Consider m given by ( 9), and n ≥ K. We have to find u in U satisfying [START_REF] Deimling | Multivalued Differential Equations, De gruyter Seris in Nonlinear Analysis and Apllications[END_REF].

We have W m,n ′ (z) ≤ V * (z) + ε for every n ′ ≥ 1, so W m,2n (z) ≤ V * (z) + ε, and we consider a control u which is ε-optimal for W m,2n (z), in the sense that ν m,2n (z, u) ≤ W m,2n (z) + ε. We have :

ν m,n (z, u) ≤ ν m,2n (z, u) ≤ W m,2n (z) + ε ≤ V * (z) + 2ε. Denoting X = γ m,n (z, u) and Y = γ m+n,n (z, u). time 0 m m + n m + 2n X Y Since ν m,2n (z, u) ≤ V * (z) + 2ε, we have X ≤ V * (z) + 2ε, and (X + Y )/2 = γ m,2n (z, u) ≤ V * (z)+2ε. Since n ≥ K, we also have X ≥ V m,n (z) ≥ V * (z)-ε. And n ≥ K also gives V n (y(m+n, u, z)) ≥ V * (y(m+n, u, z))-ε, so V * (y(m+n, u, z)) ≤ V n (y(m + n, u, z)) + ε ≤ Y + ε. Writing now Y /2 = (X + Y )/2 -X/2 we obtain Y /2 ≤ (V * (z) + 5ε)/2. So Y ≤ V * (z) + 5ε, and finally V * (y(m + n, u, z)) ≤ V * (z) + 6ε.

QED

We can now conclude the proof of theorem 3.4.

Proposition 3.16. For every state z in Z and ε > 0, there exists a control u in U and T 0 such that for every T ≥ T 0 , γ T (z, u) ≤ V * (z) + ε.

Proof : Fix α > 0.

For every positive integer i, put ε i = α 2 i . Define M i = M(ε i ) and K i = K(ε i ) given by lemma 3.15 for ε i . Define also n i = Max{K i , M i+1 α } ≥ 1. We have : ∀i ≥ 1, ∀z ∈ Z, ∃ m(z, i) ≤ M i , ∃u ∈ U, s.t. ν m(z,i),n i (z, u) ≤ V * (z) + α 2 i+1 and V * (y(m(z, i) + n i , u, z)) ≤ V * (z) + α 2 i .

We now fix the initial state z in Z, and for simplicity write v * for V * (z). We define a sequence (z i , m i , u i ) i≥1 by induction :

• first put z 1 = z, m 1 = m(z 1 , 1) ≤ M 1 , and pick u 1 in U such that ν m 1 ,n 1 (z 1 , u 1 ) ≤ V * (z 1 ) + α 2 2 , and V * (y(m 1 + n 1 , u 1 , z 1 )) ≤ V * (z 1 ) + α 2 .

• for i ≥ 2, put z i = y(m i-1 +n i-1 , u i-1 , z i-1 ), m i = m(z i , i) ≤ M i , and pick u i in U such that ν m i ,n i (z i , u i ) ≤ V * (z i )+ α 2 i+1 and V * (y(m i +n i , u i , z i )) ≤ V * (z i )+ α 2 i .

Consider finally u in U defined by concatenation : first u 1 is followed for time t in [0, m 1 + n 1 ), then u 2 is followed for t in [m 1 + n 1 , m 2 + n 2 ), etc... Since z i = y(m i-1 + n i-1 , u i-1 , z i-1 ) for each i, we have y( i-1 j=1 m j + n j , u, z) = z i for each i. For each i we have n i ≥ M i+1 /α ≥ m i+1 /α, so an interval with length n i is much longer than an interval with length m i+1 .

. . . u u 1 u i length m 1 length n 1 length m i length n i

For each i ≥ 1, we have

V * (z i ) ≤ V * (z i-1 ) + α 2 i-1 . So V * (z i ) ≤ + α 2 i-1 + α 2 i-2 ...+ α 2 + V * (z 1 ) ≤ v * + α -α 2 i . So ν m i ,n i (z i , u i ) ≤ v * + α.
Let now T be large.

-First assume that T = m 1 + n 1 + ... + m i-1 + n i-1 + r, for some positive i and r in [0, m i ]. We have : Consequently, here also we have :

γ T (z, u) ≤ v * + 2α + m 1 T .
This concludes the proofs of Proposition 3.16 and consequently, of Theorem 3.4.

  But m j ≤ αn j-1 for each j, soγ T (z, u) ≤ v * + 2α + m 1 T .-Assume now that T = m 1 + n 1 + ... + m i-1 + n i-1 + m i + r, for some positive i and r in [0, n i ]. The previous computation shows that :T -r 0 h(y(s, u, z), u(s))ds ≤ m 1 + (Tr)(v * + 2α). Since ν m i ,n i (z i , u i ) ≤ v * + α, we obtain : T γ T (z, u) = T -r 0 h(y(s, u, z), u(s))ds + T T -r h(y(s, u, z), u(s))ds, ≤ m 1 + (Tr)(v * + 2α) + r(v * + α), ≤ m 1 + T (v * + 2α).

Acknowledgements.

The first author wishes to thank Pierre Cardaliaguet, Catherine Rainer and Vladimir Veliov for stimulating conversations. The second author wishes to thank Patrick Bernard, Pierre Cardaliaguet, Antoine Girard, Filippo Santambrogio and Eric Séré for fruitful discussions.

The work of Jerome Renault was partly supported by the French Agence Nationale de la Recherche (ANR), undergrants ATLAS and Croyances, and the "Chaire de la Fondation du Risque", Dauphine-ENSAE-Groupama : Les particuliers face aux risques.

Références


M Courbage
email: maurice.courbage@univ-paris-diderot.fr

T Durt
email: thomdurt@vub.ac.be

S M Saberi Fathi

A new formalism for the estimation of the CP -violation parameters

Keywords: number:03.65.-w, 13.90.+i, 13.20.Eb, 13.20.He, 13, 20.Jf

 to obtain a phenomenological model of mesons decay. Our approach provides a fairly good estimation of the CP symmetry violation parameter in the case of K, B and D mesons. We also propose a crucial test aimed at discriminating between the standard approach and the time super-operator approach developed throughout the paper.

Introduction

There have been several theoretical approaches to CP violation in kaons (see e.g, the collection of papers edited in [START_REF] Wolfenstein | CP -violation[END_REF]) and the question is partially open today. In this paper, we use a Hamiltonian model, describing a two-level states coupled to a continuum of degrees of freedom, that makes is possible to simulate the phenomenology of neutral kaons. Then, the time superoperator formalism for the decay probability provides new numerical estimate of the parameters of CP violation.

It is well known [START_REF] Perkins | Introduction to High Energy Physics[END_REF] that kaons appears in pair K 0 and K 0 each one being conjugated to each other. The decay processes of K 0 and K 0 correspond to combinations of two orthogonal decaying modes K 1 and K 2 , that are distinguished by their lifetime. The discovery of the small CP -violation effect was also accompanied by the non orthonormality of the short and long lived decay modes, now denoted K S and K L , slightly different from K 1 and K 2 and depending on a CP -violation parameter ǫ. Lee, Oehme and Yang (LOY) [START_REF] Lee | Remarks on possible noninvariance under time reversal and charge conjugation[END_REF] proposed a generalization of the Wigner-Weisskopf theory [START_REF] Weisskopf | Berechnung der naturlichen Linienbreite auf Grund der Diracschen Lichttheorie[END_REF] in order to account the "exponential decay". Later on, L. A. Khalfin [6] has pointed that, for a quantum system with energy spectrum bounded from below, the decay could not be exponential for large times. It was also observed [START_REF] Ghirardi | Decay theory of unstable quantum systems[END_REF] that short-time behavior of decaying systems could not be exponential and this led to the so-called Zeno effect [START_REF] Chiu | Time evolution of unstable quantum states and a resolution of Zeno's paradox[END_REF][START_REF] Misra B Sudarshan E C G | The Zeno's Paradox in Quantum Theory[END_REF]. The departure from the exponential type behavior has been experimentally observed (see references quoted in [START_REF] Chiu | Decay and evolution of the neuteral kaon[END_REF]). L.A. Khalfin also corrected the parameter ǫ at the lowest order of perturbation. His estimation has been presented and reexamined in the reference [START_REF] Chiu | Decay and evolution of the neuteral kaon[END_REF] and applied to other mesons.

We show that our model allows us to obtain a better estimation of the CP -violation parameter for kaon as well as B and D mesons. We also make new predictions that differ from standard predictions and that could be tested experimentally.

This paper is organized as follows. In Section 2, we introduce the time super-operator for decay probability density. Then, in Section 3 we present the 2-levels Friedrichs model. Kaon phenomenology is recalled in Section 4. In Section 5, we present the theory of CP -violation in the Hilbert space and another derivation of the intensity formula for mesons that already has been used in [START_REF] Courbage | A wave-function model for the CPviolation in mesons[END_REF]. Finally, in Section 6, we derive the time super-operator intensity formula and we compute CP -violation parameters for K, B, and D mesons. Then, we compare our results with the experimental data.

2 Decay probability in the time super-operator (T ) approach 2.1 Decay probability in the time operator (T ′ ) approach

In the Wigner-Weisskopf approximation to time evolution of quantum unstable systems, the energy spectrum of the Hamiltonian is extended from -∞ to +∞. In this approximation, a decay time operator T ′ is canonically conjugated to H. That is,

Hψ(ω) = ωψ(ω) (2.1)
T ′ ψ(ω) = -i d dω ψ(ω) (2.2)
so that T ′ satisfies to the commutation relation [H, T ′ ] = iI. The T ′ -representation is obtained by a Fourier transform

ψ(τ ) = 1 √ 2π ∞ -∞
e -iτ ω ψ(ω) dω (2.3) and the unstable states are the those prepared such that the decay occurs in the future, that is, ψ(τ ) = 0 for τ < 0. Any state of the form ψ un (ω) = A/(ω -z 0 ), (z 0 = aib, b > 0), belongs to this space, since,

ψ un (τ ) = iA √ 2πe -iτ z 0 τ ≥ 0, 0 τ < 0 (2.4)
It is clear that these states correspond to a decay probability density:

| ψ un (τ )| 2 = 2π|A| 2 e -bτ (2.5)
This is an exponential distribution of decay times that is very common in particle physics.

Time super-operator (T ) formalism

Rigourously speaking, when the Hamiltonian has a positive spectrum, it is forbidden in principle to define a time operator that satisfies the commutation relation [H, T ′ ] = iI. This argument was elaborated by Pauli who showed that if one could find such an operator T ′ one could use it for generating arbitrary translations in the energy eigenspace so that then the spectrum of Ĥ ought to be unbounded by below, which clearly constitutes a physical impossibility.

In order to escape this contradiction one needs to go to the space of density matrices in order to obtain a time operator that is conjugated to the evolution operator (the Liouville-von Neumann operator) because it is sufficient that the Hamiltonian is not upperly bounded so that the Liouville-von Neumann operator has a spectrum extending from -∞ to ∞. In order to do so, let us consider the Liouville-von Neumann space which is the space of operators ρ on H equipped with the scalar product < ρ, ρ ′ >= T r(ρ * ρ ′ ) for which the time evolution is given by

U t ρ = e -itH ρe itH (2.6)
U t = e -itL is generated by the Liouville von-Neumann operator L given by:

Lρ = Hρ -ρH (2.7)
The time super-operator T is a self-adjoint super-operator on the Liouville-von Neumann space conjugated to L, i.e. [T, L] = iI. This definition is equivalent to the Weyl relation: e itL T e -itL = T + tI.

The average of T in the state ρ is given by4 :

T ρ = ρ, T ρ (2.8)
The time of occurrence of a random event fluctuates and we speak of the probability of its occurrence in a time interval I =]t 1 , t 2 ]. The observable T ′ = -T is associated to such event.

In fact, for a system in the initial state ρ 0 the average time of occurrence T ′ ρ 0 is to be related to the time parameter t and to the average time of occurrence in the state ρ t = e -itL ρ 0 by:

T ′ ρt = T ′ ρ 0 -t (2.9)
This equation follows from the Weyl relation. Let P τ denote the family of spectral projection operators of T :

T = R τ dP τ (2.10)
and let Q τ be the family of spectral projections of T ′ , then, in the state ρ, the probability of occurrence of the event in a time interval I is given, as in the usual formulations, by

P(I, ρ) = Q t 2 ρ 2 -Q t 1 ρ 2 = (Q t 2 -Q t 1 )ρ 2 := Q(I)ρ 2 (2.11)
The unstable "undecayed" states observed at t 0 = 0 are the states ρ such that P(I, ρ) = 0 for any negative time interval I, that is:

Q τ ρ 2 = 0, ∀τ ≤ 0 (2.12)
In other words, these are the states verifying Q 0 ρ = 0. It is straightforwardly checked that the spectral projections Q τ are related to the spectral projections P τ by the following relation:

Q τ = 1 -P -τ (2.13)
Let F τ be the subspace on which P τ projects . Thus, the unstable undecayed states are those states satisfying ρ = P 0 ρ and they coincide with the subspace F 0 5 . For these states, the probability that a system prepared in the undecayed state ρ is found to decay some time during the interval I =]0, t] is Q t ρ 2 = 1 -P -t ρ 2 a monotonically nondecreasing quantity which converges to 1 as t → ∞ while P -t ρ 2 tends monotonically to zero. As noticed by Misra and Sudarshan [START_REF] Misra B Sudarshan E C G | The Zeno's Paradox in Quantum Theory[END_REF], such quantity can not exist in the usual quantum mechanical treatment of the decay processes. It should not be confused with the usual "survival probability of an unstable state χ at time t " defined by |< χ, e -itH χ >| 2 where χ is an eigenstate of the free Hamiltonian.

In fact, the last quantity is interpreted as the probability, at the instant t, for finding the system undecayed when at time 0 it was prepared in the state χ. There is no general reason for this quantity to be monotonically decreasing as should be any genuine probability distribution. This problem does not appear in the time operator approach. Considered so, the time operator approach is non-standard. Actually, the key, non-standard, assumption that underlies the time super-operator formalism is that.

In the Liouville space, given any initial state ρ, its survival probability in the unstable space is given by:

p ρ (t) = P 0 e -itL ρ 2 (2.14)
This is the probability that, for a system initially in the state ρ, no decay is found during [0, t].

Given any initial state ρ, its survival probability in the unstable space is given by [START_REF] Courbage | Semi-Groups and Time operators for quantum unstable systems[END_REF] p ρ (t

) = P 0 e -itL ρ 2 = U -t P 0 U t ρ 2 = P -t ρ 2 (2.15)
Here we used the following relation:

P -t = U -t P 0 U t .
Then, the survival probability is monotonically decreasing to 0 as t → ∞. This survival probability and the probability of finding the system to decay some time during the interval I =]0, t], q ρ (t) = Q ρ (t) 2 are related by:

q ρ (t) = 1 -p ρ (t) (2.16)
Therefore, q ρ (t) → 1 when t → +∞.

The expression of the time operator is given in a spectral representation of H, that is, in the representation in which H is diagonal. As shown in [START_REF] Courbage | On necessary and sufficient conditions for the existence of time and entropy operators in quantum mechanics[END_REF], H should have an unbounded absolutely continuous spectrum. In the simplest case, we shall suppose that H is represented as the multiplication operator on H = L 2 (R + ) :

Hψ(ω) = ωψ(ω).
(2.17)

The Hilbert-Schmidt operators on L 2 (R + ) correspond to the square-integrable functions ρ(ω, ω ′ ) ∈ L 2 (R + × R + ) and the Liouville-von Neumann operator L is given by :

Lρ(ω, ω ′ ) = (ω -ω ′ )ρ(ω, ω ′ ) (2.18)
Then we obtain a spectral representation of L via the change of variables:

ν = ω -ω ′ (2.19) and E = min(ω, ω ′ ) (2.20)
This gives a spectral representation of L:

Lρ(ν, E) = νρ(ν, E), (2.21)
where ρ(ν, E) ∈ L 2 (R × R + ). In this representation, T ρ(ν, E) = i d dν ρ(ν, E) so that the spectral representation, of T is obtained by the inverse Fourier transform:

ρ(τ, E) = 1 √ 2π +∞ -∞ e iτ ν ρ(ν, E)dν = (F * ρ)(τ, E) (2.22)
and T ρ(τ, E) = τ ρ(τ, E).

(2.23)

The spectral projection operators P s of T are given in the (τ, E)-representation by

P s ρ(τ, E) = χ ]-∞,s] (τ )ρ(τ, E) (2.24)
where χ ]-∞,s] is the characteristic function of ]-∞, s]. So, to obtain in the (ν, E)-representation the expression of these spectral projection operators, we use the Fourier transform:

P s ρ(ν, E) = 1 √ 2π s -∞ e -iντ ρ(τ, E) dτ = e -iνs 0 -∞ e -iντ ρ(τ + s, E) dτ. (2.25)
Let g ∈ L 2 (R) and denote its Fourier transform by:

Fg(ν) = 1 √ 2π ∞
-∞ e -iντ g(τ ) dτ . Using the Hilbert transformation:

Hg(x) = 1 π P ∞ -∞ g(t) t -x dt. (2.26)
We have [START_REF] Titchmarsh | Introduction To The Theory of Fourier Integrals[END_REF] the following formula:

1 √ 2π 0 -∞ e -iντ g(τ ) dτ = 1 2 (F(g) -iHF(g)).
(2.27)

Finally, using the well-known property of the translated Fourier transform: σ s g(τ ) = g(τ + s), F(σ s g)(ν) = e iνs F.g(ν)

(2.28)

(2.25) and (2.27) yield:

P s ρ(ν, E) = 1 2 e -iνs [e iνs ρ(ν, E) -iH(e iνs ρ(ν, E))].
(2.29)

Thus: P s ρ(ν, E) = 1 2 [ρ(ν, E) -ie -iνs H(e iνs ρ(ν, E))]. (2.30)
It is to be noted that P s ρ(ν, E) is in the Hardy class H + (i.e. it is the limit as y → 0 + of an analytic function Φ(ν + iy) such that:

∞ -∞ | Φ(ν + iy) | 2 dy < ∞)[14].
3 The two-level Friedrichs model

The Friedrichs interaction Hamiltonian between the two discrete modes and the continuous degree of freedom is given by the operator H on the Hilbert space of the wave functions of the form

| ψ >= {f 1 , f 2 , g(µ)}, f 1 , f 2 ∈ C, g ∈ L 2 (R + ) H = H 0 + λ 1 V 1 + λ 2 V 2 , (3.31)
where λ 1 and λ 2 are the complex coupling constants, and

H 0 | ψ >= {ω 1 f 1 , ω 2 f 2 , µg(µ)}, (ω 1 and ω 2 > 0). (3.32)
The operators V i (i = 1, 2) are given by:

V 1 {f 1 , f 2 , g(µ)} = {< v(µ), g(µ) >, 0, f 1 .v(µ)} V 2 {f 1 , f 2 , g(µ)} = {0, < v(µ), g(µ) >, f 2 .v(µ)} (3.33)
where < v(µ), g(µ) >= dµv * (µ)g(µ), (3.34) is the inner product. Thus H can be represented as a matrix :

H Friedrichs =   ω 1 0 λ * 1 v * (µ) 0 ω 2 λ * 2 v * (µ) λ 1 v(µ) λ 2 v(µ) µ   (3.35)
ω 1,2 represent the energies of the discrete levels, and the factors λ i v(µ) (i = 1, 2) represent the couplings to the continuous degree of freedom. The energies µ of the different modes of the continuum range from -∞ to +∞ when v(µ) = 1, but we are free to tune the coupling v(µ) in order to introduce a selective cut off to extreme energy modes. Let us now solve the Schrödinger equation and trace out the continuum in order to derive the master equation for the two-level system. The two-level Friedrichs model Schrödinger equation with = 1 is formally written as

  ω 1 0 λ * 1 v * (µ) 0 ω 2 λ * 2 v * (µ) λ 1 v(µ) λ 2 v(µ) µ     f 1 f 2 g(µ)   = ω   f 1 f 2 g(µ)   . (3.36)
That is to say:

ω 1 f 1 (ω) + λ * 1 dµv * (µ)g(µ) = ωf 1 (ω), (3.37
)

ω 2 f 2 (ω) + λ * 2 dµv * (µ)g(µ) = ωf 2 (ω), (3.38)
and

λ 1 v(ω)f 1 (ω) + λ 2 v(ω)f 2 (ω) + µg(ω) = ωg(ω). (3.39)
The solution of (5.69), for "outgoing" wave, is:

g(µ) = δ(µ -ω) -lim ǫ→0 λ 1 v(µ)f 1 + λ 2 v(µ)f 2 ω -µ -iǫ . (3.40)
inserting the above equation in the equations(3.37) yields

f 1 (ω) = λ * 1 v * (ω) η + 1 (ω) -λ * 1 λ 2 lim ǫ→0 dµ |v(µ)| 2 µ -ω -iǫ f 2 (ω), (3.41)
where

η + 1 (ω) = ω -ω 1 + |λ 1 | 2 lim ǫ→0 dµ |v(µ)| 2 µ -(ω + iǫ) . (3.42)
We can also obtain the similar relations for f 2 by changing the indices 1 with 2 and vis versa as:

f 2 (ω) = λ * 2 v * (ω) η + 2 (ω) -λ 1 λ * 2 lim ǫ→0 dµ |v(µ)| 2 µ -ω -iǫ f 1 (ω). (3.43)
By substituting f 2 (ω) from the above equation in the equation (3.41) we obtain

f 1 (ω) = 1 1 -λ * 1 λ 2 dµ |v(µ)| 2 µ-ω-i0 2 λ * 1 v * (ω) η + 1 (ω) - λ * 1 |λ 2 | 2 η + 2 (ω) dµ |v(µ)| 2 µ -ω -i0 = 1 1 -O(|λ| 4 ) λ * 1 v * (ω) η + 1 (ω) -O(λ * 1 |λ 2 | 2 ) (3.44)
Thus, to the order two approximation we have

f 1 (ω) ≃ λ * 1 v * (ω) η + 1 (ω) . (3.45)
and the same formula for f 2 as:

f 2 (ω) ≃ λ * 2 v * (ω) η + 2 (ω) . (3.46) Also denote η - i (ω) = η i (ω -iǫ). η ± i (ω), (i = 1, 2)
are complex conjugate of each other, we can see that

η ± i (ω) = ω -ω i + |λ i | 2 P ∞ 0 |v(ω ′ )| 2 ω ′ -ω dω ′ ± iπ|λ i | 2 |v(ω)| 2 , (3.47)
where P indicates the "principal value" and we used the following identity in equation (3.47)

lim ε→0 + 1 x -x 0 ± iε = P 1 x -x 0 ∓ iπδ(x -x 0 ). (3.48) Let |χ = |ǫ 1 f 1 + ǫ 2 f 2 where ǫ i , (i = 1, 2
) is a constant complex number. The physical meaning of such a state is that it corresponds to a coherent superposition of two exponential decay processes. In the following Section we shall compute the projection of |χ χ| on the unstable spaces of time operator and then the survival probability p ρ (t) introduced in the Section 2. We compute its expression for the density matrix |χ χ| in terms of the lifetimes and energies of the (mesonic) resonances. It has been shown [START_REF] Ordonez G Petrosky | Explicit construction of a time superoperator for quantum unstable systems[END_REF] that the average of time operator for the state |χ χ| is equal to the lifetime 1/γ in a first approximation (more precisely in the weak coupling regime that is described in the next section (equation (3.52)). We shall characterize the short time and long time behavior of this survival probability.

Let us now identify the pure state χ with the element ρ = |χ >< χ| of the Liouville space, that is the kernel operator:

ρ = 2 i=1 2 j=1 ρ ij (ω, ω ′ ) = 2 i=1 2 j=1 ǫ i ǫ * j f i (ω)f j (ω ′ ) = 2 i=1 2 j=1 ǫ i ǫ * j F ij . (3.49)
We shall compute the survival probability P -s ρ 2 of the state ρ and show how it reaches the following limit: lim s→∞ P -s ρ 2 → 0.

(3.50)

Weak coupling conditions

As explained above the Liouville operator is given by equation (2.18) and the spectral representation of L is given by the change of variables introduced in (2. [START_REF] Lee | Particle Physics and Introduction to Field Theory[END_REF]) and (2.20). Thus, we obtain for F ij (ν, E), (i, j = 1, 2) :

F ij (ν, E) =        λ i λ * j v(E) η - i (E) v * (E+ν) η + j (E+ν) ν > 0 λ * i λ j v * (E) η + j (E) v(E-ν) η - i (E-ν) ν < 0, (3.51)
Admitting that η + i (ω) in (3.47) in the the O(|λ| 2 ) has one zero in the lower half-plane [START_REF] Marchand | Rigorous results in scattering theory Lectures in Theoretical Physics[END_REF][START_REF] Courbage | Decay probability distribution of quantum-mechanical unstable systems and time operator[END_REF] which approaches ω i for decreasing coupling, we can write:

η + i (ω) = ω -z i . (3.52)
where [START_REF] Courbage | Decay probability distribution of quantum-mechanical unstable systems and time operator[END_REF]. In this article we suppose that ω 1 < ω 2 . Easily, we can verify that

z i = ω i -i γ i 2 where γ i ∼ |λ i | 2 is a real positive constant
η + i (ω) -η - i (ω) = iγ i . (3.53) From (3.47), we have i 2 1 η + i (ω) - 1 η - i (ω) = π|λ i | 2 |v(ω)| 2 |η + i (ω)| 2 .
(3.54)

Consequently, the two above equations yield

π|λ i | 2 |v(ω)| 2 |η + i (ω)| 2 = γ i 2 |η + i (ω)| 2 . (3.55) Therefore, |f i (ω)| 2 ∼ 1 (ω-e ω i ) 2 + γ 2 i 4
which is a Breit-Wigner like distribution. This equation will be used in the next sections.

Phenomenology of kaons

Kaons are bosons that were discovered in the forties during the study of cosmic rays. They are produced by collision processes in nuclear reactions during which the strong interactions dominate. They appear in pairs K 0 , K 0 [START_REF] Perkins | Introduction to High Energy Physics[END_REF][START_REF] Ho-Kim | Elementary Particles and Their Interactions[END_REF].

The K mesons are eigenstates of the parity operator P : P |K 0 = -|K 0 , and

P |K 0 = -|K 0 .
K 0 and K 0 are charge conjugate to each other C|K 0 = |K 0 , and C|K 0 = |K 0 . We get thus

CP |K 0 = -|K 0 , CP |K 0 = -|K 0 . (4.56)
Clearly |K 0 and |K 0 are not CP -eigenstates, but the following combinations

|K 1 = 1 √ 2 |K 0 + |K 0 , |K 2 = 1 √ 2 |K 0 -|K 0 , (4.57) are CP -eigenstates. CP |K 1 = +|K 1 , CP |K 2 = -|K 2 . (4.58)
In the absence of matter, kaons disintegrate through weak interactions [START_REF] Ho-Kim | Elementary Particles and Their Interactions[END_REF]. Actually, K 0 and K 0 are distinguished by their mode of production. K 1 and K 2 are the decay modes of kaons. In absence of CP -violation, the weak disintegration process distinguishes the K 1 states which decay only into "2π" while the K 2 states decay into "3π, πeν, ..." [START_REF] Lee | Particle Physics and Introduction to Field Theory[END_REF]. The lifetime of the K 1 kaon is short (τ S ≈ 8.92× 10 -11 s ), while the lifetime of the K 2 kaon is quite longer (τ L ≈ 5.17× 10 -8 s ).

CP -violation was discovered by Christenson et al. [START_REF] Christenson | Evidence for the 2π decay of the K 2 meson[END_REF]. CP -violation means that the longlived kaon can also decay to "2π". Then, the CP symmetry is slightly violated (by a factor of the order of 10 -3 ) by weak interactions so that the CP eigenstates K 1 and K 2 are not exact eigenstates of the decay interaction. Those exact states are characterized by lifetimes that are in a ratio of the order of 10 -3 , so that they are called the short-lived state (K S ) and long-lived state (K L ). They can be expressed as coherent superpositions of the K 1 and K 2 eigenstates through

|K L = 1 1 + |ǫ| 2 ǫ |K 1 + |K 2 , |K S = 1 1 + |ǫ| 2 |K 1 + ǫ |K 2 , (4.59)
where ǫ is a complex CP -violation parameter, |ǫ| ≪ 1 and ǫ does not have to be real. K L and K S are the eigenstates of the Hamiltonian for the mass-decay matrix [START_REF] Ho-Kim | Elementary Particles and Their Interactions[END_REF][START_REF] Lee | Particle Physics and Introduction to Field Theory[END_REF] which has the following form in the basis |K 0 and |K 0 :

H = M - i 2 Γ ≡ M 11 -i 2 Γ 11 M 12 -i 2 Γ 12 M 21 -i 2 Γ 21 M 22 -i 2 Γ 22 (4.60)
where M and Γ are individually hermitian since they correspond to observables (mass and lifetime). The corresponding eigenvalues of the mass-decay matrix are equal to

m L - i 2 Γ L , m S - i 2 Γ S (4.61)
The CP -violation was established by the observation that K L decays not only via three-pion, which has natural CP parity, but also via the two-pion ("2π") mode with an experimentally observed violation amplitude |ǫ exp. | of the order of 10 -3 , which was truly unexpected at the time.

Let us now reconsider how the simple model (4.59), (4.60) is related to the experimental data.

A series of detections is performed at various distances from the source of a neutral kaon beam in order to estimate the variation of the populations of emitted pion π + , π -pairs in function of the proper time. This is done for times of the order of τ S . The experiment shows that an interference term is present in the expression of the excitation rates of detectors in function of their distance to the source. It follows from (4.59) that the transition amplitude of the K L beam is given by

ψ(t) = A e -i(m S -i 2 Γ S )t + ǫ exp e -i(m L -i 2 Γ L )t (4.62)
with A a global proportionality factor that remains constant in time. Then the intensity I(t) = |ψ(t)| 2 is given by:

I(t) = I 0 e -Γ S t + |ǫ exp | 2 e -Γ L t + |ǫ exp |e -( Γ S +Γ L 2 )t) cos(△mt + arg(ǫ exp )) (4.63)
where

|ǫ exp | = Amplitude (K L → π + , π -) Amplitude (K S → π + , π -) (4.64)
By fitting the expressions (4.63) and (4.64) with the observed data one derives an estimation of the mass difference between the short and long lived state as well as the phase of ǫ exp and its amplitude.

All this leads to an experimental estimation of ǫ exp [START_REF] Yao | CP -violation in K L decays[END_REF] |ǫ exp | = (2.232 ± 0.007) × 10 -3 , arg(ǫ exp ) = (43.5 + 0.7) • . (4.65)

5 The Wigner-Weisskopf type theory of the CP -violation in the Hilbert space

Let us present the fundamental ideas of the theory of spontaneous emission of an atom interacting with the electromagnetic field, given by Wigner and Weisskopf . This treatment aims at obtaining an exponential time dependence for decaying states by integrating over the continuum energy. That is, we assume that the modes of the fields are closely spaced. Then, we have to assume that the variation of v(µ) over µ is negligible with |µ| "uncertainty of the original state energy", i.e. v(µ) ≈ v independent of µ or in the simple case it is taken to obey v(µ) = 1. Also another assumption is that the lower limit of integration over ω is replaced by -∞.

The two-level Friedrichs model time-dependent Schrödinger equation, in the Wigner-Weisskopf regime becomes:

  ω 1 0 λ * 1 0 ω 2 λ * 2 λ 1 λ 2 µ     f 1 (t) f 2 (t) g(µ, t)   = i ∂ ∂t   f 1 (t) f 2 (t) g(µ, t)   .
(5.66) which means:

ω 1 f 1 (t) + λ * 1 ∞ -∞ dµg(µ, t) = i ∂f 1 (t) ∂t , (5.67
)

ω 2 f 2 (t) + λ * 2 ∞ -∞ dµg(µ, t) = i ∂f 2 (t) ∂t , (5.68)
and

λ 1 f 1 (t) + λ 2 f 2 (t) + µg(µ, t) = i ∂g(µ, t) ∂t .
(5.69)

Let us now solve the Schrödinger equation and trace out the continuum in order to derive the master equation for the two-level system. From the equation (5.69) we can obtain g(µ, t), taking g(µ, 0) = 0, as

g(µ, t) = -ie -iωt t 0 dτ λ 1 f 1 (τ ) + λ 2 f 2 (τ ) e iωτ , (5.70)
where t > 0. Then, we substitute g(µ, t) in the equation (5.67) and we obtain

i ∂f 1 (t) ∂t = ω 1 f 1 (t) -iλ * 1 ∞ -∞ dµe -iµt t 0 dτ λ 1 f 1 (τ ) + λ 2 f 2 (τ ) e iµτ , (5.71)
we also obtain the same relation for f 2 (t) from equation(5.68):

i ∂f 2 (t) ∂t = ω 2 f 2 (t) -iλ * 2 ∞ -∞ dµe -iµt t 0 dτ λ 1 f 1 (τ ) + λ 2 f 2 (τ ) e iµτ .
(5.72)

Finally, one obtains the following Markovian form of the reduced Schrödinger equation [START_REF] Courbage | Two-Level Friedrichs model and Kaonic phenomenology[END_REF] i ∂ ∂t

f 1 (t) f 2 (t) = ω 1 -iπ|λ 1 | 2 -iπλ * 1 λ 2 -iπλ 1 λ * 2 ω 2 -iπ|λ 2 | 2 f 1 (t) f 2 (t)
.

(5.73) Thus, we obtain an effective non-Hermitian Hamiltonian evolution,

H eff = M -i γ 2 .
The eigenvalues of the above effective Hamiltonian under the weak coupling constant approximation are:

ω + = ω 1 -iπ|λ 1 | 2 + O(λ 4 ), ω -= ω 2 -iπ|λ 2 | 2 + O(λ 4 ), (5.74)
In a first and very rough approximation, the eigenvectors of the effective Hamiltonian are the same as the postulated kaons states.

|f + = 1 0 = |K 1 and |f -= 0 1 = |K 2 , (5.75)
Phenomenology imposes that the complex Friedrichs energies ω ± coincide with the observed complex energies. The Friedrichs energies depend on the choice of the four parameters ω 1 , ω 2 , λ 1 and λ 2 and the observed complex energies are directly derived from the experimental determination of four other parameters, the masses m S and m L and the lifetimes τ S and τ L . We must thus adjust the theoretical parameters in order that they fit the experimental data. This can be done by comparing the eigenvalue of the effective matrix with the eigenvalue of the mass-decay matrix which is taken in the expression (4.61). Finally, we have

ω 1 = m S , 2π|λ 1 | 2 = Γ S , ω 2 = m L , 2π|λ 2 | 2 = Γ L .
(5.76)

The above identities yield

λ 1 = Γ S 2π e iθ S , λ 2 = Γ L 2π e iθ L (5.77)
where θ S and θ L are real constants. CP T invariance: Let us now discuss the CP T invariance in our model. As mentioned in the texts books like [START_REF] Ho-Kim | Elementary Particles and Their Interactions[END_REF][START_REF] Lee | Particle Physics and Introduction to Field Theory[END_REF], CP T invariance imposes some conditions on the mass-decay matrix, i.e. in the K 0 and K 0 bases. But, we note that our effective Hamiltonian is written in the K 1 and K 2 bases. Thus, we have to rewrite in the K 0 and K 0 bases. Thus, the transformation matrix T from the K 1 and K 2 bases to the K 0 and K 0 bases is obtained as

T = 1 √ 2 1 1 1 -1 = T -1 .
(5.79)

Then, the effective Hamiltonian in the K 0 and K 0 bases, H 00 eff is obtained by

H 00 eff = T H eff T -1 = 1 2 1 1 1 -1 ω 1 -iπ|λ 1 | 2 -iπλ * 1 λ 2 -iπλ 1 λ * 2 ω 2 -iπ|λ 2 | 2 1 1 1 -1 .
(5.80)

we have,

H 00 eff = (m S + m L ) -i 2 Γ S + Γ L + 2 √ Γ S Γ L cos △θ , (m S -m L ) -i 2 Γ S -Γ L + 2i √ Γ S Γ L sin △θ (m S -m L ) -i 2 Γ S -Γ L -2i √ Γ S Γ L sin △θ , (m S + m L ) -i 2 Γ S + Γ L -2 √ Γ S Γ L cos △θ .
(5.81) where △θ = θ L -θ S . CP T invariance conditions in (5.78) impose that

△θ = kπ + π 2 , (k = • • • , -1, 0, 1, • • • ).
(5.82)

Here we choose k = -1, consequently, △θ = -π 2 . Then, we have

M 11 = M 22 = (m S + m L ), Γ 11 = Γ 22 = Γ S + Γ L , M 12 = M * 21 = (m S -m L ), Γ 12 = Γ * 21 = Γ S -Γ L -2i √ Γ S Γ L .
(5.83)

CP -violation: Let us study in this case the CP -violation. The Friedrichs model allows us to estimate the value of ǫ. For this purpose, the effective Hamiltonian (5.73) acts on the |K S vector states (4.59) as an eigenstate corresponding to the eigenvalue

ω + = ω 1 -iπ|λ 1 | 2 = m S -i Γ S
2 , so that we must impose that

H eff |(f + + ǫf -) = H eff 1 ǫ = ω + 1
ǫ , from which we obtain after straightforward calculations that

ǫ = 1 2 √ Γ L Γ S (m L -m S ) -i 2 (Γ L -Γ S )
.

(5.84)

By using the experimental ratio (m L -m S ) -(Γ L -Γ S ) ≈ △mτ S ≈ 0.47 and the above experimental values of Γ L , Γ S , m L , m S , we obtain the following estimated value for ǫ:

ǫ = Γ L Γ S e i(46.77) • = 1.82 × 10 -3 2 e i(46.77) • = 14 ǫ exp .
(5.85)

We see that the ǫ argument is the same as the experimental value but the magnitude of the CP -violation parameter is quite larger than its experimental value. The reason is that, as we have shown in a previous work [START_REF] Courbage | A wave-function model for the CPviolation in mesons[END_REF], we did not normalize correctly the amplitudes associated to the two interfering decay processes (short and long). In that work we solved the problem by developing an analogy between the temporal density of decay and the spatial density of presence (this constitutes the so-called wave function approach). Now we shall derive intensity formula for the meson decay [START_REF] Courbage | A wave-function model for the CPviolation in mesons[END_REF] using the formalism of the time operator (T ′ ) sketched in Section 2. By considering the relations (3.52) and (3.55) and supposing the v(ω) is a real function, we can write the f 1 (ω) and f 2 (ω), the equations (3.45) and (3.46), as:

f i (ω) = γ i 2 e -iθ i ω -ω i + i 2 γ i , (i = 1, 2), (5.86)
where θ i is the phase of the possibly complex coefficients λ i . By using the Fourier transforms, i.e. equation (2.3), for the above equation, (5.86), we obtain for (i = 1, 2)

fi (τ ) = N √ π γ i e -(ie ω i + γ i 2 )τ -iθ i , τ ≥ 0 0, τ < 0 (5.87)
where N is the normalization constant. For s = -τ < 0, we have

fi (s) =    √ πγ i e (ie ω i + γ i 2 )s-iθ i , s ≤ 0 0, s > 0.
(5.88)

Finally, the normalization relation, i.e.

+∞ -∞ ds | fi (s)| 2 = 1, (i = 1, 2) (5.89) yields: fi (s) =    √ γ i e (ie ω i + γ i 2 )s-iθ i , s ≤ 0 0, s > 0.
(5.90)

Here f i (s), (i = 1, 2) is the form of the density of the probability. Thus, the intensity is obtained by

I(s) = |C| 2 |ǫ 1 f 1 (s) + ǫ 2 f 2 (s)| 2 = I 0 e γ 1 s + |ǫ| 2 γ 2 γ 1 e γ 2 s + |ǫ| γ 2 γ 1 e (γ 1 +γ 2 ) 2 s cos(( ω 1 -ω 2 )s + θ 2 -θ 1 + arg(ǫ)) (5.91)
where ǫ = ǫ 2 /ǫ 1 and C and I 0 = |C| 2 ǫ 2 1 γ 1 are the constants. This corresponds to an effective value for ǫ that is no longer 14 times too large as in expression (5.85) because it must be renormalized. Identifying equations (4.63) and ( 5.91) it is easy to show, as we have also done in [START_REF] Courbage | A wave-function model for the CPviolation in mesons[END_REF], that, ǫ th , the theoretical prediction for the experimental CP -violation parameter, obeys

ǫ th = ǫ Γ L Γ S = Γ L Γ S 1 2 △m Γ S -i △γ 2Γ S .
(5.92)

Substituting in the expression (5.92) the physically observed masses and lifetimes of the short and long kaon states we find that ǫ th ≈ 0.6 ǫ exp which constitutes an improvement in comparison to the non-renormalized estimation (5.85). We shall also reconsider similar results in the case of B and D particles in a next section.

In the next coming section, we shall use the time super-operator (T ) formalism as a non Wigner-Weisskopf approximation method to obtain the CP -violation parameter. This formalism also predicts a CP -violation parameter comparable to the experimental value.

Computation of spectral projections of T in a Friedrichs model

In this section, we will compute the survival probability and we obtain the theoretical CPviolation parameters for the mesons K, B and D. Then, we compare our results to the experimental CP -violation parameters. We shall see that our theoretical results provide a good estimation of the experimentally measured quantities. Moreover, a fine structure appears in the case of kaons, which brought us to conceive an experimental test aimed at falsifying the time super-operator approach, that we shall discuss in the conclusion.

By considering v(ω) a real test function and using the equation (3.55) we obtain F ji (ν, E) in the following form:

F ji (ν, E) =        λ j λ * i ν * j (ν+ν i ) ν > 0 λ * j λ i ν i (ν * j -ν) ν < 0. (6.93)
where i, j = 1, 2 and

ν i := a i + ib i := (E -ω i ) + i γ i 2 .
(6.94) For obtaining P s F ij (ν, E) (s < 0), we shall use the formula (2.30). First we compute:

G ji (ν, E) = H(e isν F ji )(ν, E) = 1 π P ∞ -∞ e isx F ji (x, E) x -ν dx (6.95)
Now, we substitute (6.93) in (6.95), so we have

G ji (ν, E) = 1 π P λ i λ * j 0 -∞ e isx ν i (x -ν)(ν * j -x) dx + λ * i λ j +∞ 0 e isx ν * j (x -ν)(ν i + x) dx (6.96)
which for the ν > 0 has the following form:

G ji (ν, E) = 1 π λ i λ * j 0 -∞ e isx ν i (x -ν)(ν * j -x) dx + λ * i λ j P +∞ 0 e isx ν * j (x -ν)(ν i + x)
dx . (6.97)

A complete computation of the G ii (ν, E) is showed in [START_REF] Courbage | Decay probability distribution of quantum-mechanical unstable systems and time operator[END_REF]. Finally, P s F ij (ν, E) is obtained as: for i = j

P s F ii (ν, E) = i|λ i | 2 e -isν -1 2πν i (ν * i -ν) 0 -∞ e -sy y + iν * i dy - 0 -∞ e -sy y + iν dy + 1 2πν * i (ν + ν i ) 0 -∞ e -sy y -iν i dy - 0 -∞ e -sy y + iν dy +      |λ i | 2 e -isν [ e isν * i ν i (ν * i -ν) -e -isν i ν * i (ν i +ν) ], E < ω 1 0, E > ω 1 . (6.98)
and by considering ω i < ω j , F ij for i = j have the following form :

P s F ji (ν, E) = i e -isν -λ i λ * j 2πν i (ν * j -ν) 0 -∞ e -sy y + iν * j dy - 0 -∞ e -sy y + iν dy + λ * i λ j 2πν * j (ν + ν i ) 0 -∞ e -sy y -iν i dy - 0 -∞ e -sy y + iν dy +                  e -isν [ λ i λ * j e isν * j ν i (ν * j -ν) - λ * i λ j e -isν i ν * j (ν i +ν) ], E < ω i λ i λ * j e -isν e isν * i ν i (ν * j -ν) , ω i < E < ω j 0, E > ω j .
(6.99)

In the equations (6.98) and (6.99) the non-integral terms yield the poles and lead to the resonance, and the integral terms yield an algebraical term analog to the background in the Hamiltonian theories [START_REF] Bohm | Time asymmetry and quantum theory of resonances and decay[END_REF]. We can also compute the same result for the case ν < 0. We will neglect the the background (the integrals terms). Then, the above equation is rewritten as:

P s F ii (ν, E) ≃      |λ i | 2 e -isν [ e isν * i ν i (ν * i -ν) -e -isν i ν * i (ν i +ν) ], E ≤ ω 1 0, E > ω 1 .
and for i = j

P s F ij (ν, E) ≃                  e -isν [ λ i λ * j e isν * j ν i (ν * j -ν) - λ * i λ j e -isν i ν * j (ν i +ν) ], E ≤ ω i λ i λ * j e -isν e isν * i ν i (ν * j -ν) , ω i < E ≤ ω j 0, E > ω j .
Now, we would compute the survival probability, i.e.

p ρ (s) = P ρ (s) = |ǫ 1 | 2 P s F 11 (ν, E) + ǫ 1 ǫ * 2 P s F 12 (ν, E) + ǫ 2 ǫ * 1 P s F 21 (ν, E) + |ǫ 2 | 2 P s F 22 (ν, E) (6.100) where • = ∞ 0 dE ∞ -∞ dν | • | 2 .
We see that P ρ (s) can be written as :

P ρ (s) ≃                            e -isν ǫ * 1 λ 1 ν 1 + ǫ * 2 λ 2 ν 2 ǫ 1 λ * 1 e isν * 1 ν * 1 -ν + ǫ 2 λ * 2 e isν * 2 ν * 2 -ν -ǫ 1 λ 1 ν * 1 + ǫ 2 λ 2 ν * 2 ǫ * 1 λ * 1 e -isν 1 ν 1 +ν + ǫ * 2 λ * 2 e -isν 2 ν 2 +ν E ≤ ω 1 , e -isν ǫ * 1 λ 1 ν 1 + ǫ * 2 λ 2 ν 2 ǫ 2 λ * 2 e isν * 2 (ν * 2 -ν) -ǫ 1 λ 1 ν * 1 + ǫ 2 λ 2 ν * 2 ǫ * 2 λ * 2 e -isν 2 (ν 2 +ν) , ω 1 < E ≤ ω 2 0, E > ω 2 (6.101)
Now, by remembering that b i = |λ i | 2 , (i = 1, 2), the square norm of P ρ (s) is obtained as:

|P ρ (s)| 2 ≃                            ǫ 1 λ 1 ν 1 + ǫ 2 λ 2 ν 2 2 |ǫ 1 | 2 |λ 1 | 2 e 2b 1 s |ν * 1 -ν| 2 + |ǫ 2 | 2 |λ 2 | 2 e 2b 2 s |ν * 2 -ν| 2 + |ǫ 1 | 2 |λ 1 | 2 e 2b 1 s |ν 1 +ν| 2 + |ǫ 2 | 2 |λ 2 | 2 e 2b 2 s |ν 2 +ν| 2 + e (b 1 +b 2 )s ǫ 1 ǫ * 2 λ * 1 λ 2 e i(a 1 -a 2 )s (ν * 1 -ν)(ν 2 -ν) + ǫ 1 ǫ * 2 λ * 1 λ 2 e i(a 1 -a 2 )s (ν * 1 +ν)(ν 2 +ν) + C.C. , E ≤ ω 1 ǫ 1 λ 1 ν 1 + ǫ 2 λ 2 ν 2 2 |ǫ 2 | 2 |λ 2 | 2 e 2b 2 s |ν * 2 -ν| 2 + |ǫ 2 | 2 |λ 2 | 2 e 2b 2 s |ν 2 +ν| 2 , ω 1 < E ≤ ω 2 0, E > ω 2 (6.
102) where the terms that oscillate with a frequency equal to the difference of the two masses, i.e. ( ω 2 -ω 1 ) is kept, the other decay terms oscillating with the frequency of one of the masses only are neglected since we have in the weak-coupling regime and the high-mass.

The integral over ν arrives at:

∞ -∞ dν |P ρ (s)| 2 ≃                          2π ǫ 1 λ 1 ν 1 + ǫ 2 λ 2 ν 2 2 |ǫ 1 | 2 e 2b 1 s + |ǫ 2 | 2 e 2b 2 s + 2iǫ * 1 ǫ 2 λ * 1 λ 2 e (b 1 +b 2
)s e i(e ω 1 -e ω 2 )s (e ω 2 -e ω 1 )+i(b

1 +b 2 ) + C.C. , E ≤ ω 1 2π ǫ 1 λ 1 ν 1 + ǫ 2 λ 2 ν 2 2 |ǫ 2 | 2 e 2b 2 s , ω 1 < E ≤ ω 2 0, E > ω 2 (6.103)
Only the terms of the square norm are depended to E and we have

ǫ 1 λ 1 ν 1 + ǫ 2 λ 2 ν 2 2 = ǫ 1 λ 1 E -ω 1 + ib 1 2 + ǫ 2 λ 2 E -ω 2 + ib 2 2 + ǫ * 1 λ * 1 (E -ω 1 + ib 1 ) ǫ 2 λ 2 (E -ω 2 -ib 2 ) + C.C. (6.
104) The integral over E of the above expression is like the following integrals

dE √ b i (E -ω i ) + ib i 2 = arctan E -ω i b i (6.105) and dE λ * 1 λ 2 (x -a 1 + ib 1 )(E -a 2 -ib 2 ) = -λ * 1 λ 2 ( ω 2 -ω 1 ) + i(b 1 + b 2 ) i arctan b 1 E -ω 1 +i arctan b 2 E -ω 2 + log (E -ω 1 ) 2 + b 2 1 -log (E -ω 2 ) 2 + b 2 2 (6.106)
Now, we integrate from equation(6.104) over E from 0 to ∞. Firstly, for the interval E ∈ [0, ω 1 ] we have

I 1 = e ω 1 0 dE ǫ 1 λ 1 ν 1 + ǫ 2 λ 2 ν 2 2 = |ǫ 1 | 2 arctan ω 1 b 1 + |ǫ 2 | 2 arctan ω 2 -ω 1 b 2 + arctan ω 1 b 2 - ǫ * 1 ǫ 2 √ b 1 b 2 ( ω 1 -ω 2 ) + i(b 1 + b 2 ) i( π 2 + arctan b 1 ω 1 + arctan b 2 ω 2 + arctan b 2 ω 2 -ω 1 ) + 1 2 log b 2 1 ( ω 2 2 + b 2 2 ) ( ω 2 1 + b 2 1 )(( ω 2 -ω 1 ) 2 + b 2 2 ) + C.C. . (6.107) For E ∈] ω 1 , ω 2 ]
we have

I 2 = e ω 2 e ω 1 dE ǫ 1 λ 1 ν 1 + ǫ 2 λ 2 ν 2 2 = |ǫ 1 | 2 arctan ω 2 -ω 1 b 1 + |ǫ 2 | 2 arctan ω 2 -ω 1 b 2 - ǫ * 1 ǫ 2 √ b 1 b 2 ( ω 1 -ω 2 ) + i(b 1 + b 2 ) i(arctan b 1 ω 2 -ω 1 -arctan b 2 ω 2 -ω 1 ) + 1 2 log b 2 1 b 2 2 (( ω 2 -ω 1 ) 2 + b 2 1 )(( ω 2 -ω 1 ) 2 + b 2 2 )
+ C.C. . (6.108)

K-meson

For the weak-coupling constants we have b i ≪ ω i , (i = 1, 2) and also by supposing ω 1 ∼ ω 2 , ( ω 2 -ω 1 ) ∼ b 1 and b 2 b 1 ≪ 1, we have

I 1 ≃ π 2 |ǫ 1 | 2 + 2|ǫ 2 | 2 + ǫ * 1 ǫ 2 λ * 1 λ 2 ( ω 1 -ω 2 ) + i(b 1 + b 2 ) + C.C. ≈ π 2 (6.109) I 2 ≃ π 4 |ǫ 1 | 2 + 2|ǫ 2 | 2 + ǫ * 1 ǫ 2 λ * 1 λ 2 ( ω 1 -ω 2 ) + i(b 1 + b 2 ) + C.C. ≈ π 4 (6.110)
where we used the normalization relation, i.e. (|ǫ

1 | 2 + |ǫ 2 | 2 ) = 1.
Finally, we obtain The derivative of the equation (6.111) yields the time super-operator density of the probability or intensity:

p ρ (s) ≃ π 2 |ǫ 1 | 2 e 2b 1 s + 3 2 |ǫ 2 | 2 e 2b 2 s + iǫ * 1 ǫ 2 λ * 1 λ 2 e (b 1 +b 2 )s e i(e ω 1 -e ω 2 )s ( ω 2 -ω 1 ) + i(b 1 + b 2 ) + C.C. ≃ π 2 |ǫ 1 | 2 e 2b 1 s + 3 2 |ǫ| 2 e 2b 2 s + iǫλ * 1 λ 2 e (b 1 +b 2 )s e i(e ω 1 -e ω 2 )s ( ω 2 -ω 1 ) + i(b 1 + b 2 ) + C.C. .
I(s) := dp ρ (s) ds = I 0 e 2b 1 s + |ǫ| 2 3 2 b 2 b 1 e 2b 2 s + 2|ǫ| b 2 b 1 e (b 1 +b 2 )s cos(( ω 1 -ω 2 )s + φ + θ 2 -θ 1 ) .
(6.113) where I 0 = (π|ǫ 1 | 2 b 1 )/2 and λ i = √ b i e iθ i , (i = 1, 2). This expression differs by 3 2 term from the intensity derived previous by [START_REF] Courbage | A wave-function model for the CPviolation in mesons[END_REF] from the integrated probability of decay of two exponentially decay process or relation (5.92) that we obtained in the Hilbert space which we call the time operator prediction.

Let us now evaluate the predictions related to the above equation in the different time intervals and let us compare them with the intensity introduced in the equation (4.63). Firstly, for t = -s ∼ 10 × τ S or t ≫ τ S which that the term effective is: |ǫ| where

ǫ th = ǫ 3 2 b 2 b 1 (6.115)
and the coefficient 3b 2 2b 1 in the above equation is the correction which is obtained by the time operator formalism and by using the condition ( ω 2 -ω 1 ) ∼ b 1 = 0, then I 2 = 0. Secondly, for the time of the order oft τ S (t < 5τ S ) we have I(s) ≈ I 0 e 2b 1 s and I(t) ≈ I 0 e -γ S t , (t < 5τ S ) (6.116)

Finally, for intermediate times (5τ S < t < 10τ S ) we have

I(s) ≈ I 0 |ǫ| 2 b 2 b 1 e 2b 2 s cos(( ω 1 -ω 2 )s + φ + θ 2 -θ 1 ) and
I(t) ≈ I 0 |ǫ exp | 2 e - " Γ S +Γ L 2 " t cos((m L -m L )s + arg(ǫ exp )) (6.117)
The equations (6.114), (6.116), (6.117) and (5.77) yield

b 1 = γ 1 2 = Γ S 2 = 1 2τ S , ω 1 = m S , b 2 = γ 2 2 = Γ L 2 = 1 2τ L , ω 2 = m L , θ 1 = θ S , θ 2 = θ L (6.118)
The ǫ is obtained in (5.85), thus, we have

ǫ th = ǫ 3 2 Γ L Γ S e -i π 2 = 3 2 Γ L Γ S 1 2 △m Γ S -i △γ 2Γ S (6.119)
where △m = (m L -m S ) and △γ = (Γ L -Γ S ). Then, by replacing the experimental data we have ǫ th = 1.62 × 10 -3 e i(46.77 • ) = 0.73 ǫ exp (6.120)

B-meson

It easy to see that the integral I 2 , for the B-mesons and D-mesons, is zero. So the intensity is written as:

I(s) = I 0 e 2b 1 s + |ǫ th 1 | 2 e 2b 2 s + 2|ǫ th 1 | e (b 1 +b 2 )s cos(( ω 1 -ω 2 )s + φ) (6.121)
where

ǫ th = ǫ b 2 b 1 e -i π 2 (6.122)
This expression is the same, in the case of B and D particles in the time operator and in the super-operator approaches, and the theoretically estimated CP -violation parameter obeys the following equation

ǫ th = ǫ Γ L Γ S e -i π 2 = Γ L Γ S 1 2 △m Γ S -i △γ 2Γ S (6.123)
which is not true in the case of K particles. Also for B and D particles the agreement with observations is quite good as we shall now check.

Another example is the CP -violation in the decay of B and the experimental CP -violation parameter of the B meson is [START_REF] Amsler | B 0 -B 0 mixing[END_REF][START_REF] Amsler | CP -violation in meson decays[END_REF]:

A exp SL ≃ 4Re(ǫ exp B ) = (-0.4 ± 5.6) × 10 -3 ⇒ q p exp = 1.0002 ± 0.0028. ( 6

.127)

where

A exp SL 2 ≈ 1 -q p exp
. By replacing in the equation (6.123) we obtain:

ǫ th B = Γ L Γ H 1 2 △m
Γsi △Γs 2Γs = 0.018 + 0.047 × 10 -3 i (6.128) Thus, our theoretical q p th prediction is:

q p th = 1 -ǫ th 1 + ǫ th = 0.96 (6.129)
which is in fairly good agreement with the experimental value.

D-meson

The other example is the CP -violation in the decay of D meson. The experimental values for CP -violation of D 0 → K 0 S π + π -as reported by Belle [START_REF] Amsler | D 0 -D 0 mixing[END_REF] are as follows:

△Γ 2Γ = 0.37 ± 0.25 +0.07+0.07 -0.13-0.08 , (6.130) △m Γ = 0.81 ± 0.30 +0.10+0.09 -0.07-0.16 (6.131) where 1/Γ = τ, ( = 1) is the mean life time

1 Γ = τ = τ D 0 + τ D 0 2 = (410.1 ± 1.5) × 10 -3 ps (6.132)
The CP -violation parameters are experimentally denoted by q p and given by:

q p exp = 1 -ǫ exp 1 + ǫ exp = 0.86

Concluding remarks

About the relevance and novelty of our results

As we can see, the accuracy of the prediction (6.119) is comparable to the one that we derived within the Wigner-Weisskopf approach (5.92) (time operator instead of time super-operator). Now, as we said before, the present results were derived under the assumption that the spectrum of the continuous mode was not bounded by below (no cut-off). In a precedent publication [START_REF] Courbage | Quantum-mechanical decay laws in the neutral Kaons[END_REF], we considered the Friedrichs model with a Gaussian factor form and energy bounded by below (the spectrum of the continuous mode was assumed there to vary from 0 to +∞). We showed that by introducing a cut-off in the coupling between discrete and continuous modes, the estimated value of ǫ slightly differs, depending on the shape that we impose to the cutoff. Therefore a fine tuning of the estimated CP -violation parameter is possible provided that the factor form is chosen conveniently. Considered so, the precision of the agreement with the measured value of the CP -violation parameter is not very convincing by it self (3 times the experimental value of the kaon CP -violation parameter [START_REF] Courbage | Quantum-mechanical decay laws in the neutral Kaons[END_REF]). What is convincing in our approach is that we obtain the right order of magnitude for the K, B and D particles altogether.

A crucial experiment for testing the validity of the Time Super-Operator (T ) Formalism

The most important novelty of the time operator approach is, in our eyes, that it predicts that the distribution in time of the measured populations of pions pairs significantly differs from the predictions that could be made in the standard approach and/or in the Wigner-Weisskopf approach provided we make a fit over the full distribution (which means not only for times larger than the lifetime of the "Short" state but also for times comparable to it). Indeed, taking account of the three contributions of the distribution, which are the purely exponential, "Short" and "Long" contributions, and the oscillating contribution, one sees that the expression (6.113) radically differs from the expressions (4.63) and (5.91). This is due to the presence of the coefficient 3b 2 2b 1 in the above equation which is the correction obtained by the time super-operator formalism and by using the condition ( ω 2 -ω 1 ) ∼ b 1 , that is, (m L -m s ) ∼ Γ S for kaons. Since in the case of the B and D mesons, no such relation exists, the formula obtained here coincide with the one derived using the Wigner-Weisskopf time operator approach [START_REF] Courbage | A wave-function model for the CPviolation in mesons[END_REF].

So, one can conceive crucial experiments that would allow to falsify the time operator approach and do not radically differ from the original Christenson experiment. These experiments require to measure the population of pairs of pions over a large range of times (distances to the source), and to check whether the best fit is provided by the expression (6.115) or by the expressions (4.63) and (5.91).

In principle these effects will be tested on the LHC at CERN in the coming months (years) so that the crucial experiment that we propose here is feasible in the future.

Concluding remark

The formalism of the mass-decay matrix for the kaon decay was introduced by LOY [START_REF] Lee | Remarks on possible noninvariance under time reversal and charge conjugation[END_REF]. Then several other authors [START_REF] Chiu | Time evolution of unstable quantum states and a resolution of Zeno's paradox[END_REF]6,[START_REF] Courbage | Two-Level Friedrichs model and Kaonic phenomenology[END_REF] improved this model. The LOY model requires the Wigner-Weisskopf approximation, i.e. it requires to assume that the energy interval varies from -∞ to +∞ and also that the coupling between discrete and continuous modes is not restricted by a factor form or cut-off.

In [START_REF] Courbage | Two-Level Friedrichs model and Kaonic phenomenology[END_REF], we used the 2-level Friedriche model and the Wigner-Weisskopf approach to obtain a mass-decay matrix. This approach was improved by using a new concept of probability decay density for mesons in [START_REF] Courbage | A wave-function model for the CPviolation in mesons[END_REF]. Beyond the Wigner-Weisskopf approximation, we used the Friedrichs model with a cutoff that amounts to bound from below the energy spectrum of the Hamiltonian [START_REF] Courbage | Quantum-mechanical decay laws in the neutral Kaons[END_REF]. In the present paper, we derived the decay probability density in the formalism of the time super-operator, that also goes beyond the Wigner-Weisskopf approximation.

M 11 = 21 ( 5

 11215 M 22 , Γ 11 = Γ 22 , M 12 = M * 21 and Γ 12 = Γ *

( 6 .

 6 111) where ǫ = |ǫ| e iφ := ǫ 2 ǫ 1 . (6.112)

2 b 1 e 2 b 2 b 1 .

 21221 2b 2 s and comparing for the same time with the equation (4.63) yields the CP -violation parameter is |ǫ| 2 3 Thus, the equations (6.113) and (4.63) for t = -s ≫ τ S can be written approximately asI(s) ≈ I 0 ǫ th 2e 2b 2 s and I(t) ≈ I 0 |ǫ exp | 2 e -γ L t , (-s = t ≫ τ S ) (6.114)

The linearity that usually characterizes the relation between average values of observable A and density matrix M : tr(M A) seems to be violated here, but one should not forget that (a) in the case of pure states the density matrix equals its square and (b) this paradox is easily solved in the case of mixtures by imposing that ρ is the square root of the density matrix M = ρ * ρ, tr(M A) = tr(ρ * Aρ).

Therefore, a subspace Ft 0 is a set of decaying states prepared at time t0. We call it an unstable space of T.


Wassila Riah

Nathalie Cheviron

Jean Trap

Steven Criquet

Clément Peltre

Sabine Houot

Marc Legras

Jean-François Vian

Cécile Antonio Bispo

Riah Wassila

Trap Jean

Criquet Steven

Houot Sabine

Bispo Antonio

Pérès Guénola

Laval Karine

Trinsoutrot-Gattin Isabelle

Mougin Christian

Effect or organic amendments on soil enzymatic activities. Results of the "Bioindicators" French program

teaching and research institutions in France or abroad, or from public or private research centers.

The organic amendments have been applied every two years in September  Lipase, laccase, arylamidase activities may be linked to organic matter biochemical state or the composition of the soil organic carbon

C cycle

Conclusion

Effect of organic amendment on soil enzymatic activities  Organic amendments drive to an increase in soil organic carbon content, as exogenous organic matter added is stable.

 ACP, FDA, DEH, CEL, GAL, XYL and ARYLS activities were not affected by organic amendments, 18 months after the last input.

-

  (1) ESITPA, Unité ARGI'TERR 76134 Mont Saint Aignan, FR (2) INRA, Unité PESSAC 78026 Versailles, FR (3) Aix-Marseille Université, IMBE UMR CNRS 7263 -IRD 237 , 13397 Marseille Cedex 20, FR (4) INRA/AGROPARISTECH, UMR Environnement et Grandes Cultures, 78850 Thiverval-Grignon, FR (5) ADEME, 49004 ANGERS cedex 01, FR (6) Université de Rennes 1, UMR CNRS 6553 ECOBIO, Station Biologique, 35380 Paimpont, FR Effect of organic amendments on soil enzymatic activities. Results of the French "Bioindicator program" What are the needs in agricultural soils ? Tools to measure biological state and soil functions and at least to evaluate agricultural soil ecosystem the bioindicators on the basis of referential values in order to interpret the results ADEME (French Agency for Environment and Energy Management) initiated a national research program to develop such indicators : "Bioindicator Program" (2006-2012). http://ecobiosoil.univ-rennes1.fr/ADEME-Bioindicateur Objectives -to provide relevant biological tools (sensibility, accessibility, cost, ...) depending on the purpose : -soil monitoring -characterization -risk assessment -to transfer the tools to stakeholders (policy makers, farmers, technical advisers…) -Test and comparison of 47 bioindicators (from microflora to fauna and flora) -Study large number of situations 13 situations (47 plots) -Homogeneity of procedures and protocoles -Development of a data base to manage a large amount of data (~200.000) Soil enzymes play an important role in organic matter decomposition and nutrient cycling  Enzymes are recognized as early indicators of soil functioning changes due to modifications of soil management or agricultural practices Origin of Soil Enzyme Activity 1. Soil microorganisms -living and dead 2. Plant roots and crop residues 3. Soil animals Enzymes are proteins with specific catalytic properties  40 to 60% of enzyme activity can come from stabilized enzymes  not necessarily highly correlated with microbial biomass or respiration Objectives Do enzyme activities reveal changes in the biological functioning of soils caused by organic amendment ? Could soil enzyme be indicators of soil organic matter quality or organic products stability ? Located at Feucherolles, in northwestern France (50 km west from Paris) -The climate is oceanic -The soil is cultivated following a wheat-maize rotation -At the date of sampling soils were cultivated with wheat -The last inputs have been made 18 months ago

Lipase

  Fourteen enzyme activities involved in the different nutrient cycling were measured by the spectrophotometric methods Soil enzymes assays are based mostly on the quantification of the product released after adding known amount of substrate to the soil Enzyme assays Enzyme activity expression : nmole of hydrolyzed or oxidazied substrate / min / g of dried soil 1. Homogenizing, sieving and weighing soil 2. Preparation of soil solution 3. Addition of a specific substrate and incubation 4. Stop the enzymatic reaction and measure absorbance with a spectrophotometer 5. Data analysis Organic carbon content  all organic amendments drive to an increase in soil organic carbon content by comparison with the control FUM, BIO and DVB have the highest effect.  OMR is intermediate between control and other organic treatments. Synthesis of enzymatic activities response to organic amendment according to Kruskal-wallis test n=4 NAG and GLU enzyme activities did not reflect the soil organic matter content  The effect of exogenous organic matter (EOM) on GLU and NAG activities was transient These enzyme activities are less suitable for monitoring long-term effect of OM 3. b Variation patterns of enzymatic activities affected by organic amendments Results  Significant increase of URE activity in the soil regardless of addition of compost or manure organic carbon composition?  Lipase, laccase, arylamidase, urease and alkalin phosphatase activities were modified according to the spreading of organic amendments into soil  Organic amendments seem to have transient effect on β-glucosidase, Nacetyl-glucosaminidase. These enzyme activities are less suitable for monitoring long-term effect of OM The municipal solid waste compost (F-OMR) plot that have the lowest OMSI and organic carbon content exhibited in average the highest enzymatic activity values (AKP, LIP, GLU,URE and ARYLN)

Table 1 :

 1 E.C. numbers, specific substrates (concentrations and pH buffers) , references of protocols used to measure enzyme activities

			E.C.
	Enzymes	Codes	number Substrates (

buffer pH and concentration) References Global activities

	Dehydrogenase	DEH	1.1.1.1	2,3,5 triphenyltetrazolium chloride ((pH 7; 0.12M) Schaefer et al, 963
	Fluorescein diacetate	FDA	209.877.6 Fluorescein diacetate (pH 7.6; 1000 µg.mL -1 )	Adam & Duncan, 2001