Consider the Boltzmann equation {partial_derivative}/{partial_derivative}{sub t}u(t,x,v)=-v. {del}{sub x}u(t,x,v)-{sigma}{sub a}(x,v)u(t,x,v)+{integral}{sub v}k(x,v{prime},v)u(t,x,v{prime})dv{prime} in R{sup n} x V {such_that} (x,v), V being an open subset of R{sup n}, n {ge} 2. Equation (1.1) describes the dynamics of a flow of particles in R{sup n} under the assumption that the interaction between them is neglectable (no non-linear terms). This is the case for example for a low-density flow of neutrons. The term involving {sigma}{sub a} describes the loss of particles from (x,v) {epsilon} R{sup n} x V due to absorption or scattering into another point (x,v{prime}), while the last term in (1.1) involving k represents the production at x {epsilon} R{sup n} of particles with velocity v form particles with velocity v{prime}. The total rate of this production at (x,v{prime}) is given by {sigma}{sub p}(x,v{prime}) = {integral}{sub V} k (x,v{prime},v)dv. Following [RS] we say that the pair ({sigma}{sub a}, k) is admissible, if (i) O {le} {sigma}{sub a} {epsilon} L{sup {infinity}}(R{sup n} x V), (ii) O {le} k(x,v{prime}, {center_dot}) {epsilon} L{sup 1}(V) for a.e. (x,v{prime}) {epsilon} R{sup n} x V and {sigma}{sub p} {epsilon} L{sup {infinity}}(R{sup n} x V), (iii) There is an open bounded set X {contained_in} R{sup n}, such that k(x,v{prime},v) andmore » {sigma}{sub a}(x,v) vanish if x {epsilon} X. One can define the wave operators associated with T, T{sub O} by W{sub -} = s-lim/t{r_arrow}{infinity} U(t)U{sub O}(-t), W{sub +} = s-lim/t{r_arrow}{infinity} U(O)(-t)U(t). if W{sub -}, W{sub +} exist, then one can define the scattering operator S = W{sub +}W{sub -} as a bounded operator in L{sup 1}(R{sup n} x V). Scattering theory for (1.1) has been developed by other authors and we refer to these papers for sufficient conditions guaranteeing the existence of S. An abstract approach based on the Limiting Absorption Principle has been proposed. 22 refs.« less
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Choulli et al. (1996) studied this question.