Abstract
We consider a nonautonomous transport problem, the modelization of the charge exchange dynamics in a monoatomic ionized gas, and apply scattering theory to its dynamics. The free dynamics corresponds to the evolution of the total distribution of particles (neutral plus ionized particles) and the perturbed dynamics corresponds to the evolution of the neutral particles, which is the solution of a nonautonomous transport problem. The existence of the time-dependent wave operators was proved by the first author. In the present paper we follow Howland's formalism in constructing a stationary scattering theory for this nonautonomous transport problem by studying the evolution equation. We prove the existence of the wave operators and by using the smooth perturbation technique we obtain the similarity between perturbed and unperturbed operators.
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REFERENCES
P. Arianfar and H. Emamirad, Relationship between scattering and albedo operators in linear transport theory, Transport Theory Statist. Phys. 23:517–531 (1994).
G. Busoni, Asymptotic behaviour and wave operators in charge exchange, J. Math. Anal. Appl. 212:190–208 (1997).
G. Busoni, Remarks on stability in charge exchange dynamics in a monoatomic ionized gas, Transport Theory Statist. Phys. 17:257–269 (1988).
P. L. Butzer and H. Berens, Semi-Groups of Operators and Approximation (Springer-Verlag, Berlin, New York, 1967).
E. B. Davies, One Parameter Semigroups (Acad. Press, London, New York, 1980).
H. Emamirad, On the Lax and Phillips scattering theory for transport equation, J. Funct. Analysis 62:503–528 (1985).
H. Emamirad, Scattering theory for linearized Boltzmann equation: Survey and new results, Transport Theory Statist. Phys. 16:503–528 (1987).
H. Emamirad, Hypercyclicity in the scattering theory for linear transport equation, Trans. Amer. Math. Soc. 350:3707–3716 (1998).
H. Emamirad and V. Protopopescu, Relationship between the albedo and scattering operators for the Boltzmann equation with semi-transparent boundary conditions, Math. Meth. Appl. Sc. 19:1–13 (1996).
D. E. Evans, Time dependent perturbations and scattering of strongly continuous groups on Banach spaces, Math. Ann. 221:275–290 (1976).
J. Goldstein, Semigroups of Linear Operators and Applications (Oxford University Press, New York, 1985).
J. Hejtmanek, Scattering theory of the linear Boltzmann operator, Comm. Math. Phys. 43:109–120 (1975).
J. S. Howland, Stationary scattering theory for time-dependent Hamiltonians, Math. Ann. 207:315–335 (1974).
V. A. Golant, A. P. Zilinskii, and I. E. Saharov, Foundations of Plasma Physics (Nauka, Moskow, 1983).
Y. Latushkin and S. Montgomery-Smith, Lyapunov theorems for Banach spaces, Bull. Amer. Math. Soc. 31:44–49 (1994).
M. Mokhtar-Kharroubi, Limiting absorption principles and wave operators on L 1(μ) spaces. Application to transport theory, J. Funct. Anal. 115:119–145 (1993).
R. Nagel, Semigroup methods for nonautonomous Cauchy problems, Evolution Equations, G. Ferreyra et al., eds. (Marcel Dekker, 1995), pp. 301–316.
H. Neihardt, On abstract linear evolution equations I, Math. Nachr. 103:283–293 (1981).
G. Nickel, Evolution semigroups solving nonautonomous Cauchy problems, Tübinger Berichte zur Funktionanalysis 5:316–334 (1996).
L. Paquet, Semigroupes généralisés et équations d'évolution, Séminaire de thvorie du Potentiel Paris, Lecture Notes in Math. 713 (Springer-Verlag, 1979), pp. 243–263.
A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations (Springer-Verlag, New York, Berlin, 1983).
V. Protopopescu, On the scattering matrix for the linear Boltzmann equation, Rev. Roum. Phys. 21:991–994 (1976).
R. Rau, Hyperbolic evolution groups and exponentially dichotomic evolution families, J. Dyn. Diff. Eqs. 6:335–350 (1994).
F. Räbiger, R. Rhandi, and R. Schnaubelt, Perturbation and an abstract characterization of evolution semigroups, J. Math. Anal. Appl. 198:516–533 (1996).
F. Räbiger, R. Rhandi, and R. Schnaubelt, Nonautonomous Miyadera perturbations, Ulmer Seminare, Funktionalanalysis und Differentialgleichungen 1:331–357 (1996).
F. Räbiger and R. Schnaubelt, The spectral mapping theorem for evolution semi-groups on spaces of vector-valued functions, Semigroup Forum 52:225–239 (1996).
F. Räbiger and R. Schnaubelt, A spectral characterization of exponentially dichotomic and hyperbolic evolution families, Tübinger Berichte zur Funktion-analysis 3:222–234 (1994).
B. Simon, Existence of the scattering matrix for the linearized Boltzmann equation, Comm. Math. Phys. 41:99–108 (1975).
P. Stefanov, Spectral and scattering theory for the linear Boltzmann equation in the exterieur domain, Math. Nachr. 137:63–77 (1988).
H. Tanabe, Equations of Evolution (Pitman, London, 1979).
T. Umeda, Scattering and spectral theory for the linear Boltzmann equation, J. Math. Kyoto Univ. 24:205–218 (1984).
T. Umeda, Smooth perturbation in ordered Banach spaces and similarity for the linear transport operators, J. Math. Soc. Japan 38:617–625 (1986).
J. Voigt, On the existence of the scattering operators for the linear Boltzmann equation, J. Math. Anal. Appl. 58:541–558 (1977).
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Busoni, G., Emamirad, H. Stationary Scattering Theory for a Charged Particles Transport Problem. Journal of Statistical Physics 96, 377–401 (1999). https://doi.org/10.1023/A:1004536803141
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DOI: https://doi.org/10.1023/A:1004536803141