HYPERCYCLICITY IN THE SCATTERING

THEORY FOR LINEAR TRANSPORT EQUATION

H. Emamirad

Abstract. We show how the hypercyclicity of the transport semigroup can intervene in the scattering theory to characterize the density property of the Lax and Phillips representation theorem and conversely, how the existence of the wave operators of the scattering theory can be used for recovering the hypercyclicity of the absorbing transport group in some weighted L^{1} spaces.

1. Introduction

The study of the scattering theory for transport phenomena was initiated by Lax and Phillips [L-P1] for streaming free group

U_{0}(t)f(x, v) = f(x − tv, v),  (1.1)

in the context of the Hilbert space X = L^{2}(ℝ^{3} × S^{2}).

From a compact convex subset Ω of ℝ^{3} they defined the incoming and outgoing subspaces D_{−} and D_{+} as follows : Let ρ be the radius of a ball B_{ρ} around the origine which contains all points of Ω. Let E_{+} = {(x, v) | x·v ≥ ρ} be the free forward and E_{−} = {(x, v) | x · v ≤ −ρ} be the free backward point sets of ℝ^{3} × S^{2}. Corresponding to these sets they defined

D_{±} = {f ∈ L^{2}(ℝ^{3} × S^{2}) | suppf ⊂ E_{±}},

and they have proved that for the dynamics U(t) = U_{0}(t), the following representation theorem holds (see also [L-P2]).

Theorem 1.1. The subspaces D_{+} and D_{−} satisfy the following properties: (i)_{+} U(t)D_{+} ⊂ D_{+} for t ≥ 0, (i)_{−} U(t)D_{−} ⊂ D_{−} for t ≤ 0, (ii) ⋂_{t∈ℝ} U(t)D_{±} = {0}, (iii) ⋃_{t∈ℝ} U(t)D_{±} is dense in X.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 47D05 ; Secondary 82A70.

Key words and phrases. Hypercyclic semigroup, Lax and Phillips scattering theory, two similar semigroups.

In [Em1] and [Em2], the author generalized the above representation theorem in the Banach space X = L^{1}(ℝ^{n} × V) for the transport semigroup U(t), which is governed by the transport equation

\frac{∂u}{∂t} = −v · ∇_{x}u − σ_{a}(x, v)u + ∫_{V} k(x, v′, v)u(x, v′, t)dv′,  (1.2)

where

V = {v ∈ ℝ^{n} , | 0 < v_{min} ≤ |v| ≤ 1 },  (1.3)

is the velocity space, the absorbing cross section σ_{a}(x, v) and producing source function k(x, v, v′) are the positive functions in L^{∞}(ℝ^{n} ×V) and L^{∞}(ℝ^{n} ×V × V) having their supports as functions of x in a compact convex subset Ω of ℝ^{n}. By assuming the transparent boundary condition on (1.2) U(t) becomes a strongtly continuous group for which the Theorem 1.1 holds.

In all the applications of the Lax and Phillips representation theorem the most difficult point is the density property (iii). Since for t ≥ 0, U(t)D_{+} ⊂ D_{+}, thus

\overline{⋃_{t∈ℝ} U(t)D_{+}} = \overline{⋃_{t≤0} U(t)D_{+}} = X.  (1.4_{+})

Similarly,

\overline{⋃_{t∈ℝ} U(t)D_{−}} = \overline{⋃_{t≥0} U(t)D_{−}} = X.  (1.4_{−})

Here we arise a crucial question: is it possible to characterize a subset of the incoming or outgoing subspaces D_{±} which generates this dense subset? It seems that this question is closely related to the hypercyclic character of the semigroups V_{±}(t) which are the restrictions of U(t) to D_{±} for ∓t ≥ 0.

A systematic study of a hypercyclic strongly continuous ( C_{0}- ) semigroup is recently initiated in [D-S-W]. A C_{0}-semigroup of bounded linear oprators {T (t)}_{t≥0} is called hypercyclic provided that there exists x ∈ X such that {T (t)x | t ≥ 0} is dense in X.

In the next section we will prove that on X = L_{ω_{±}}^{1}(E_{±}), where ω_{±}(x, v) > 0 is an adequate weight function, the semigroups V_{0}^{±}(t) = χ_{±}U_{0}(∓t), where χ_{±} are the characteristic functions of E_{±} and U_{0}(t) is defined in (1.1) are hypercyclic. By mean of this result we retrieve the Lax and Phillips representation theorem in L_{ω}^{1}(ℝ^{n} × V) by proving that the dense subset in (iii) can be generated by a single element of X. Then, we extend the weight functions ω_{±} on whole ℝ^{n} × V in such a manner that by imposing the classical assumptions on the pair (k, σ_{a}) appearing in (1.2), U(t) become a C_{0}-group on L_{ω}^{1}(ℝ^{n} × V). By defining V^{±}(t) = χ_{±}U(∓t), we prove that V^{±}(t) = V_{0}^{±}(t), which in turn establishes the Lax and Phillips representation theorem for U(t). This result is already announced in [Em3]. In the section 3, we prove that if {S(t)}_{t≥0} and {T (t)}_{t≥0} are two similar C_{0}-semigroups and if one of them is hypercyclic then the other one is also hypercyclic. In the case when {S(t)}_{t∈ℝ} and {T (t)}_{t∈ℝ} are two C_{0}-groups we show that how the existence of the wave operators in scattering theory can help us to establishing the similarity between these groups.

In the fourth section we will prove the hypercyclicity of the streaming free group U_{0}(t) in some L_{ω}^{1}(ℝ^{n} × V). We show that for ant α > 0, there exist a weight function ω such that

∥U_{0}(t)∥_{L^{1}(ℝ^{n}×V)} ≤ e^{α|t|}  for all t ∈ ℝ.  (1.5)

The construction of this weight function is similar to those of [D-S-W] for translation groups.

Unfortunately with an estimation like (1.5) we are not able to use the existence theorems (as Cook’s Lemma [R-S] or Theorem 1 of [Um]) of the wave operators in order to obtain the hypercyclicity of the transport group U(t). In fact, in any abstract result for the existence of wave operators the free dynamic should be uniformly bounded in the underlying Banach space.

However, if the kernel k is identically zero, the equation (1.2) generates a C_{0}-group U_{1}(t) which is called absorbing transport group. In this case the wave operators have an explicit form and we don’t need to use an abstract theorem for their existence. By the way, a particular case of this equation (when σ_{a}(x, v) = σ(x)) and the corresponding scattering operator are used in the inverse problems in computerized tomography (see [Em-A]).

2. Hypercyclicity of the transport semigroup

Let V be the velocity space defined in (1.3) and E be a subset of ℝ^{n} × V. Definition 2.1. By an admissible weight function on E we mean a measurable positive function ω satisfying

ω(x − tv, v) ≤ r(t)ω(x, v)

for all (x, v) ∈ E, where r is a positive function such that lim inf_{t→∞} r(t) = 0.

A prototype example of an admissible weight function is

ω(x, v) = exp(x · v).

Let us denote by L_{ω}^{1}(E) the space of measurable functions f : E ⟼ ℂ such that

∥f∥_{E} = ∫_{E} |f(x, v)|ω(x, v)dxdv < ∞.

Let ω_{+} and ω_{−} be two admissible weight functions defined respectively on E_{+} and E_{−}. Let ω be a positive extension of ω_{±} on ℝ^{n} × V. On L_{ω}^{1}(ℝ^{n} × V) we will define the streaming free group {U_{0}(t)}_{t∈ℝ} as in (1.1), which in turn defines the dynamics V_{0}^{±}(t) = χ_{±}U_{0}(∓t) on L_{ω_{±}}^{1}(E_{±}).

Theorem 2.2. {V_{0}^{±}(t)}_{t≥0} are strongly continuous hypercyclic semigroups on L_{ω_{±}}^{1}(E_{±}).

For the proof of this theorem we need the following lemma.

Lemma 2.3 [D-S-W, Theorem 2.3]. Let {T (t)}_{t≥0} be a C_{0}-semigroup on a Banach space X. Let X_{0} = {x ∈ X | lim_{t→∞} T (t)x = 0} and X_{∞} = {x ∈ X | ∀ϵ > 0, ∃ u ∈ X and t > 0 such that ∥u∥ < ϵ and ∥T (t)u − x∥ < ϵ}. If both X_{0} and X_{∞} are dense subset of X, then {T (t)}_{t≥0} is hypercyclic.

Proof of Theorem 2.2. The fact that V_{0}^{±}(t) are two C_{0}-semigroups can be proved by the standard argument of translation semigroup. Here we will prove the hypercyclicity of V_{0}^{−}(t), the same argument will work for V_{0}^{+}(t).

Let f be an element of L_{ω_{−}}^{1}(E_{−}) with compact support in E_{−}. Thus, there is some R > ρ such that for any (x, v) ∈ ℝ^{n} × V with −R ≥ x · v, f(x, v) = 0. Therefore for any (x, v) ∈ E_{−} and t ≥ (R − ρ)/v_{min}^{2} one has x · v − t|v|^{2} ≤ −ρ − tv_{min} ≤ −R and f(x − tv, v) = 0. This implies that X_{0} contains all functions of compact support in L_{ω_{−}}^{1}(E_{−}), so that X_{0} is dense.

For the density of X_{∞}, we choose an arbitrary function f of compact support in E_{−} and we define u_{t}(x, v) = f(τ_{t}(x, v), v), where τ_{t}(x, v) = (x + tv, v). For any t ≥ 0 and u_{t} ∈ L_{ω_{−}}^{1}(E_{−}), we have

∥u_{t}∥_{E_{−}} = ∫_{E_{−}} |f(x + tv, v)|ω_{−}(x, v)dxdv
= ∫_{E_{−}} |f(x, v)|ω_{−}(x − tv, v)dxdv
≤ r_{−}(t)∥f∥_{E_{−}}.

Hence, from vanishing property of r_{−}(t), for any ϵ > 0, we have ∥u_{t}∥_{E_{−}} < ϵ for some t enough large, while V_{0}^{−}(t)u_{t} = f. Consequently f ∈ X_{∞}. Since f has been picked from a dense subset, X_{∞} is dense in L_{ω_{−}}^{1}(E_{−}) and Lemma 2.3 concludes. □

Corollary 2.4. The Lax and Phillips representation Theorem holds for streaming free group U_{0}(t) in L_{ω}^{1}(ℝ^{n} × V).

Proof. The properties (i) and (ii) can be obtained readly as in [Em1]. But for (iii) we have even more; in fact, we will prove that there exists φ_{+} ∈ L_{ω_{+}}^{1}(E_{+}) such that instead of (1.4^{+}) we have

\overline{⋃_{t≤0} U_{0}(t)D_{+}} = \overline{⋃_{t≤0,s≥0} U_{0}(t)V_{0}^{+}(s)\{φ_{+}\}} = X.  (2.1^{+})

In fact for any ϵ > 0 and any f ∈ L_{ω}^{1}(ℝ^{n} × V) there exists g ∈ C_{0}^{∞}(ℝ^{n} × V) such that ∥f − g∥_{ℝ^{n}×V} < \frac{ϵ}{2}. Let t be enough large such that h = U_{0}(t)g ∈ L_{ω_{+}}^{1}(E_{+}) = D_{+}. From the hypercyclicity of V_{0}^{+}(s) it follows that there exists s ≥ 0 such that ∥V_{0}^{+}(s)φ_{+} − h∥_{E_{+}} < ϵ(r_{+}(t) + r_{−}(t))^{−1}/2. Hence,

∥f − U_{0}(−t)V_{0}^{+}(s)φ_{+}∥_{ℝ^{n}×V} ≤ \frac{ϵ}{2} + ∥g − U_{0}(−t)V_{0}^{+}(s)φ_{+}∥_{ℝ^{n}×V}

≤ \frac{ϵ}{2} + (r_{+}(t) + r_{−}(t))^{−1}∥U_{0}(t)g − V_{0}^{+}(s)φ_{+}∥_{ℝ^{n}×V} ≤ ϵ

The same argument holds for E_{−} and we can write

\overline{⋃_{t≥0} U_{0}(t)D_{−}} = \overline{⋃_{t≥0,s≥0} U_{0}(t)V_{0}^{−}(s)\{φ_{−}\}} = X.  (2.1^{−})

□

We will now describe some condition on the pair (k, σ_{a}), appearing in (1.2), for the strong continuity of transport semigroup. We assume that a finite convex scattering body, Ω is surrounded by vacuum and we define the incoming and outgoing spaces as before. Let ω_{±} be an admissible weight function on E_{±}. Define

ω(x, v) = \{ \begin{matrix}ω_{+}(x, v)  if (x, v) ∈ E_{+} \\ ω_{−}(x, v)  if (x, v) ∈ E_{−} \\ 1  elsewhere \end{matrix}

As in [Si], we assume that the pair (k, σ_{a}), satisfies the following hypotheses:

(H1) k(x, v, v′) is a nonnegative measurable function on ℝ^{n} × V × V for Lebesgue measure dxdvdv′ and σ_{a}(x,v) is a nonnegative measurable function on ℝ^{n} × V for Lebesgue measure dxdv;

(H2) For each (x, v) , k(x, v,.) is in L^{1}(V); (H3) σ_{a} and σ_{p} ≡ ∫_{V} k(x, v, v′)dv′ are uniformly bounded on ℝ^{n} × V. (H4) k(x, v, v′) and σ_{a}(x, v) vanish for x ∉ Ω.

We consider the following Cauchy problem in L_{ω}^{1}(ℝ^{3} × V):

(P)  \{ \begin{matrix}\frac{∂u}{∂t} = T u \\ u(x, v, 0) = f(x, v), \end{matrix}

where T = T_{0} + A = T_{0} + A_{a} + A_{p} with T_{0}f ≡ −v · ∇_{x}f, A_{a}f ≡ −σ_{a}(x, v)f and [A_{p}f](x, v) ≡ ∫_{V} k(x, v′, v)f(x, v′) dv′.

Theorem 2.5. Under hypotheses (H1) to (H4), the operator T generates a C_{0}–group on L_{ω}^{1}(ℝ^{3} × V).

Proof. Knowing that T_{0} generates a C_{0}-group U_{0}(t), it is enough to show that A is a bounded perturbation of T_{0}. According to (H3), the uniform boundedness of σ_{a} on ℝ^{n} × V implies the boundedness of the operator A_{a}. While for the boundedness of A_{p}, we have to prove that

∫_{V} k(x, v′, v)ω(x, v)dv ≤ Cω(x, v′)  for all (x, v′) ∈ ℝ^{n} × V. (2.2)

Since for (x, v) ∈ E_{±} we have |x| ≥ |x||v| ≥ ±x · v ≥ ρ, thus (H4) implies that for x ∉ Ω, the left-hand side of (2.2) is identically zero and for x ∈ Ω, ω(x, v) = 1, hence (2.2) follows from (H2). □

Let us denote by U(t) the C_{0}–group generated by T. Define V^{±}(t) = χ_{±}U(∓t) on L_{ω_{±}}^{1}(E_{±}).

Theorem 2.6. V^{±}(t) = V_{0}^{±}(t) on L_{ω_{±}}^{1}(E_{±}).

Proof. According to Duhamel’s formula

U(−t) = U_{0}(−t) − ∫_{−t}^{0} U_{0}(−t − s)AU(s)ds

or

V^{+}(t) = V_{0}^{+}(t) − χ_{+} ∫_{0}^{t} U_{0}(−τ)AU(τ − t)dτ.  (2.3)

Taking (H4) into account, for any f, the function g(x, v) = [AU(τ − t)f](x, v) vanishes for x ∉ Ω. Since for any (x, v) ∈ E_{+} one has (x + τv, v) ∈ E_{+}, thus χ_{+}(x, v)U_{0}(−τ)g(x, v) = 0 for all τ ≥ 0. This together with (2.3) imply that V^{+}(t) = V_{0}^{+}(t) and the same argument holds for V^{−}(t) = V_{0}^{−}(t). □

Analogously to Corollary 2.4, we can show that

Corollary 2.7. The Lax and Phillips representation Theorem holds for transport group U(t) in L_{ω}^{1}(ℝ^{n} × V).

Proof. By virtue of Corollary 2.4, it is enough to remark that U(t)D_{±} ⊂ D_{±} for ±t ≥ 0. In fact, U(t)f = U_{0}(t)f + ∫_{0}^{t} U(t − s)AU_{0}(s)fds. For f ∈ D_{+}, U_{0}(s)f ∈ D_{+} and AU_{0}(s)f is identically zero. Moreover, we have

\overline{⋃_{±t≥0} U(t)D_{∓}} = \overline{⋃_{±t≥0,s≥0} U(t)V^{∓}(s)\{φ_{∓}\}} = X.

□

3. The similarity between two C_{0}-semigroups

In this section we introduce the notion of similarity between two C_{0}-semi-groups and we give a sufficient condition to realize this similarity for two C_{0}-group.

Definition 3.1. Let {S(t)}_{t≥0} and {T (t)}_{t≥0} be two C_{0}- semigroups on a Banach space X. S(t) and T (t) are called similar if there exists an isomorphism P on X such that

T (t) = P^{−1}S(t)P for all t ≥ 0.

Theorem 3.2. Let S(t) and T (t) be two similar C_{0}-semigroups. If one of them is hypercyclic, then the other one is also hypercyclic.

Proof. Let S(t) be a hypercyclic C_{0}-semigroup and y an arbitrary element of X. For x = P y and ϵ > 0, there exists x_{0} ∈ X and t ≥ 0 such that ∥S(t)x_{0} − x∥ ≤ ϵM^{−1}, where M = ∥P^{−1}∥. Hence, for y_{0} = P^{−1}x_{0} we have

∥T (t)y_{0} − y∥ ≤ M∥S(t)P y_{0} − P y∥ < ϵ,

which proves the hypercyclicity of T (t). □

One of the essential aims in the scattering theory is to establish the similarity for a pair of linear operators. Consider a pair of linear operators T and T_{0}, where T is a bounded perturbation of T_{0} (T = T_{0} + A with A ∈ L(X)). Assume that both T and T_{0} generate the bounded C_{0}-groups e^{tT} and e^{tT_{0}} on X. Then the existence of the wave operators (see [Em2, Um])

W_{±}(T, T_{0}) = s − lim_{t→±∞} e^{−tT} e^{tT_{0}} and W_{±}(T_{0}, T) = s − lim_{t→±∞} e^{−tT_{0}}e^{tT} ,

implies that

W_{±}(T, T_{0})W_{±}(T_{0}, T)f = W_{±}(T_{0}, T)W_{±}(T, T_{0})f = f for all f ∈ X,

and by letting s → ±∞ in

e^{tT} e^{−sT} e^{sT_{0}} = e^{(t−s)T} e^{(s−t)T_{0}}e^{tT_{0}}

we obtain the following intertwining identity

e^{tT} W_{±}(T, T_{0})f = W_{±}(T, T_{0})e^{tT_{0}}f, for all f ∈ X.

This proves that the C_{0}-groups e^{tT} and e^{tT_{0}} (consequently T and T_{0}) are mutually similar.

4. Hypercyclicity of the transport absorbing group

For the characterization of hypercyclic streaming free group U_{0}(t)f(x, v) = f(x − tv, v), we need the following Lemme.

Lemma 4.1 [D-S-W, Theorem 2.2 and Theorem 2.5]. Let {T (t)}_{t≥0} be a C_{0}-group on a Banach space X, then the following assertions are equivalent:

(i) The semigroup {T (t) | t ≥ 0} is hypercyclic.

(ii) The semigroup {T (−t) | t ≥ 0} is hypercyclic.

(iii) For all ϵ > 0 there exists a dense subset D ⊂ X such that for all f ∈ D there exists a dense subset D′ ⊂ X such that for all g ∈ D′ there exist h ∈ X and t > 0 such that ∥g − h∥ < ϵ and ∥f − T (t)h∥ < ϵ.

Throughout this section, we designate by ∥.∥_{ω} the norm of L_{ω}^{1}(ℝ^{n} × V), where ω is an admissible weight function.

Theorem 4.2. For any α > 0 there exists an admissible weight function ω(x, v) on ℝ^{n} × V such that the steaming free group {U_{0}(t) | t ∈ ℝ} is hypercyclic in L_{ω}^{1}(ℝ^{n} × V) and satisfies

∥U_{0}(t)f∥_{ω} ≤ e^{α|t|}∥f∥_{ω}  for all t ∈ ℝ and f ∈ L_{ω}^{1}(ℝ^{n} × V). (4.1)

For the proof of this Theorem, we need to construct a weight function ω such that it were admissible in some sense for t → ±∞. We proceed this construction in the following lemma.

Lemma 4.3. For any α > 0 there exists a weight function ω such that for any compact set K ⊂ ℝ^{n} there exists a positive function r such that lim inf_{t→±∞} r(t) = 0 and

ω(x − tv, v) ≤ r(t)ω(x, v) for all (x, v) ∈ K × V and all t ∈ ℝ. (4.2)

Furthermore

\frac{ω(x + tv, v)}{ω(x, v)} ≤ e^{α|t|} for all (x, v) ∈ ℝ^{n} × V and all t ∈ ℝ.  (4.3)

Proof. For any α > 0, we can choose an even real-valued function ϕ : ℝ ⟼ ℝ such that ϕ is continuously differentiable, its derivative is bounded by α and ϕ(t) = 0 for all t ∈ [−ρ, ρ] and

lim sup_{s→∞} ϕ(s) = lim sup_{s→−∞} ϕ(s) = ∞
lim inf_{s→∞} ϕ(s) = lim inf_{s→−∞} ϕ(s) = −∞

For example the following characterization of its derivative can determine entirely this function: Take an even function ϕ ∈ C^{1}(ℝ) such that |ϕ′| ≤ α and

ϕ′(t) = \{ \begin{matrix}0  if & t ∈ [0, ρ] \\ α  if & t ∈ [ρ + 2n(n − 1) + ϵ_{0}, ρ + 2n(n − 1) + n − ϵ_{0}]∪ \\ & [ρ + 2n(n − 1) + 3n + ϵ_{0}, ρ + 2n(n − 1) + 4n − ϵ_{0}] \\ −α  if & t ∈ [ρ + 2n(n − 1) + n + ϵ_{0}, ρ + 2n(n − 1) + 3n − ϵ_{0}] \end{matrix}

for n = 1, 2, ⋯ and some ϵ_{0} > 0 small enough.

Let us define the weight function ω(x, v) = e^{ϕ(x·v)}. Then

\frac{ω(x − tv, v)}{ω(x, v)} ≤ exp \{− ∫_{0}^{t} ϕ′(x · v − s|v|^{2})|v|^{2}ds\} .  (4.4)

Since |v| ≤ 1 and bounded away from zero, for any (x, v) ∈ K × V we have |x · v| ≤ M and there exists a ∈ ℝ such that exp{− ∫_{0}^{t} ϕ′(x · v − s|v|^{2})|v|^{2}ds} ≤ exp{αn + ∫_{a}^{a−t|v|^{2}} ϕ′(s)ds} = r(t), where n is the largest integer such that ρ + 2n(n − 1) ≤ M. The fact that for any a ∈ ℝ

lim sup_{t→±∞} ∫_{a}^{t} ϕ′(s)ds = +∞  and  lim inf_{t→±∞} ∫_{a}^{t} ϕ′(s)ds = −∞

implies (4.2) and by (4.4) we infer easily (4.3). □

Proof of Theorem 4.2. Let ω be the weight function constructed in the previous Lemma. Let f and g be two functions in L_{ω}^{1}(ℝ^{n} × V) with compact support, say K × V. We will show that Condition (iii) of Lemma 4.1 holds with f and g. Since these functions are taken from a dense set, this implies that {U_{0}(t) | t ≥ 0} is hypercyclic. Since the weight function is even we conclude that {U_{0}(t) t ≤ 0} is also a hypercyclic semigroup on L_{ω}^{1}(ℝ^{n} × V) which in turn proves the Theorem.

By retaking the notation in the proof of the Theorem 2.2, τ_{t}(x, v) = (x + tv, v), we can choose t enough large so that τ_{t}(K × V) ∩ K × V = ∅ and define

h(x, v) = \{ \begin{matrix}g(x, v) & if  (x, v) ∈ K × V \\ f(x + tv, v) & if  (x + tv, v) ∈ K × V \\ 0 & elsewhere \end{matrix}

It is not hard to see that

∥g − h∥_{ω} = ∫_{τ_{t}(K×V)} f(x + tv, v)ω(x, v)dxdv
= ∫_{K×V} f(x, v)ω(x − tv, v)dxdv
≤ r(t)∥f∥_{ω}

which is sufficiently small for enough large t and the same is true for U_{0}(t)h−f, since ∥U_{0}(t)h − g∥_{ω} ≤ r(−t)∥g∥_{ω}. □

Let us denote U_{1}(t) the C_{0}-group generated by the penetration operator T_{1} = T_{0}+A_{a}, where the absortion operator A_{a} is defined by A_{a}f = −σ_{a}(x, v)f. The C_{0}-group U_{1}(t) has an explicit form given by

[U_{1}(t)f](x, v) = exp {− ∫_{0}^{t} σ_{a}(x − sv, v)ds} f(x − tv, v)

In this case the wave operators exist and have also the explicit forms

W_{±}(T_{0}, T_{1}) = lim_{t→±∞} e^{− ∫_{0}^{t} σ_{a}(x−(s−t)v,v)ds}f(x, v)
= e^{− ∫_{∓∞}^{0} σ_{a}(x−τv,v)dτ} f(x, v)

and

W_{±}(T_{1}, T_{0}) = lim_{t→±∞} e^{− ∫_{0}^{−t} σ_{a}(x−τv,v)dτ} f(x, v)
= e^{− ∫_{0}^{∓∞} σ_{a}(x−τv,v)dτ} f(x, v).

Since σ_{a} has a compact support with respect to x in Ω,

∫_{−∞}^{∞} σ_{a}(x − tv, v)dt ≤ v_{min}^{−1}M_{a}[diam Ω],  (4.5)

where M_{a} = ∥v^{−1}σ_{a}∥_{∞} denotes the mean free path for σ_{a}. Thus, these wave operators are bounded in L_{ω}^{1}(ℝ^{n} × V) for any admissible weight function ω. Here we assume the following

(H5)  inf_{(x,v)∈ℝ^{n}×V} σ_{a}(x, v) = σ > 0.

Theorem 4.4. Assume (H5), then there exist a weight function ω satisfying the assumptions of Lemma 4.3 such that in L_{ω}^{1}(ℝ^{n} × V), the C_{0}-group U_{1}(t) is hypercyclic and satisfies

∥U_{1}(t)∥_{ω} ≤ 1 for t ≥ 0.

Proof. The hypercyclicity of U_{1}(t) follows from the existence of the wave operators and Theorem 3.2. For α ≤ σ, we construct a weight function according to Lemma 4.3, then

∥U_{1}(t)f∥_{ω} = ∫_{ℝ^{n}×V} e^{− ∫_{0}^{t} σ_{a}(x−(s−t)v,v)ds}|f(x, v)|ω(x + tv, v)dxdv
≤ e^{(α−σ)t}∥f∥_{ω} ≤ ∥f∥_{ω} for any t > 0.
□

In spite of the contraction of U_{1}(t) for t ≥ 0, this group is not uniformly bounded for t ≤ 0, hence we can not use the Cook’s Lemma (see [R-S]) for the existence of W_{±}(T, T_{1}) and W_{±}(T_{1}, T).

References

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Université de Poitiers. Département de Mathématiques. 40, ave. du Recteur Pineau. 86022 Poitiers Cedex, France

   E-mail address: emamirad@matpts.univ-poitiers.fr