Estimates of solutions of linear neutron transport equation at large time and spectral singularities

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March 2012, 5(1): 113-128. doi: 10.3934/krm.2012.5.113

Estimates of solutions of linear neutron transport equation at large time and spectral singularities

Roman Romanov ^1,

1.	Laboratory of Quantum Networks and Department of Mathematical Physic, Faculty of Physics, St.Petersburg State University, 198504, Saint Petersburg, Russian Federation

Received August 2010 Revised August 2011 Published January 2012

Abstract
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The spectral analysis of a dissipative linear transport operator with a polynomial collision integral by the Szőkefalvi-Nagy - Foiaş functional model is given. An exact estimate for the remainder in the asymptotic of the corresponding evolution semigroup is proved in the isotropic case. In the general case, it is shown that the operator has at most finitely many eigenvalues and spectral singularities and an absolutely continuous essential spectrum. An upper estimate for the remainder is established.

Keywords: Transport equation, spectral singularities, asymptotics of semigroups..

Mathematics Subject Classification: Primary: 47A45; Secondary: 35Q2.

Citation: Roman Romanov. Estimates of solutions of linear neutron transport equation at large time and spectral singularities. Kinetic and Related Models, 2012, 5 (1) : 113-128. doi: 10.3934/krm.2012.5.113

References:

[1]	B. Sz.-Nagy and C. Foiaş, "Analyse Harmonique des Opérateurs de l'Espase de Hilbert," (French) [Harmonic Analysis of Operators in Hilbert Space], Masson et Cie, Paris, Académiai Kiadó, Budapest, 1967.
[2]	J. Lehner, The spectrum of the neutron transport operator for the infinite slab, J. Math. Mech., 11 (1962), 173-181.
[3]	J. Lehner and G. Wing, On the spectrum of an unsymmetric operator arising in the transport theory of neutrons, Comm. Pure Appl. Math., 8 (1955), 217-234. doi: 10.1002/cpa.3160080202.
[4]	J. Lehner and G. Wing, Solution of the linearized Boltzmann transport equation for the slab geometry, Duke Math. J., 23 (1956), 125-142. doi: 10.1215/S0012-7094-56-02312-2.
[5]	Yu. Kuperin, S. Naboko and R. Romanov, Spectral analysis of the transport operator: A functional model approach, Indiana Univ. Math. J., 51 (2002), 1389-1425. doi: 10.1512/iumj.2002.51.2180.
[6]	B. S. Pavlov, On separation conditions for the spectral components of a dissipative operator, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 123-148, 240.
[7]	M. A. Naĭmark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis, (Russian) Trudy Mosk. Mat. Obšč., 3 (1954), 181-270; English transl., Amer. Math. Soc. Transl. (2), 16 (1960), 103-193.
[8]	B. S. Pavlov, Spectral analysis of a dissipative singular Schrödinger operator in terms of a functional model, in "Partial Differential Equations, VIII" (ed. M. Shubin), Encyclopaedia Math. Sci., 65, Springer, Berlin, (1996), 87-153.
[9]	S. N. Naboko, Functional model of perturbation theory and its applications to scattering theory, (Russian) Boundary value problems of mathematical physics, 10, Trudy Mat. Inst. Steklov, 147 (1980), 86-114, 203; English transl., Proc. Steklov Inst. Math., 147 (1981), 85-116.
[10]	S. N. Naboko, On the conditions for existence of wave operators in the nonselfadjoint case, (Russian) in "Wave Propagation. Scattering Theory" (ed. M. Birman), LGU, Leningrad, (1987), 132-155; English transl., Amer. Math. Soc. Transl. Ser. 2, 157, AMS, Providence, RI, (1993), 127-149.
[11]	S. Naboko and R. Romanov, Spectral singularities and asymptotics of contractive semigroups. I, Acta Sci. Math. (Szeged), 70 (2004), 379-403.
[12]	N. K. Nikol'skiĭ , "Treatise on the Shift Operator. Spectral Function Theory," With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller, Translated from the Russian by Jaak Peetre, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 273, Springer-Verlag, Berlin, 1986.
[13]	R. Romanov, A remark on equivalence of weak and strong definitions of the absolutely continuous subspace for nonself-adjoint operators, in "Spectral Methods for Operators of Mathematical Physics" (Bedlewo, 2002), Oper. Theory Adv. Appl., 154, Birkhäuser, Basel, (2004), 179-184.
[14]	L. A.Sahnovič, Dissipative operators with absolutely continuous spectrum, (Russian) Trudy Moskov. Mat. Obšč., 19 (1968), 211-270.
[15]	A. S. Tikhonov, Functional model and duality of spectral components for operators with continuous spectrum on a curve, (Russian) Algebra i Analiz, 14 (2002), 158-195; English transl., St. Petersburg Math. J., 14 (2003), 655-682.
[16]	R. Romanov and M. Tihomirov, On the selfadjoint subspace of the one-velocity transport operator, Math. Notes, 89 (2011), 106-116. doi: 10.1134/S0001434611010111.

show all references

References:

[1]	B. Sz.-Nagy and C. Foiaş, "Analyse Harmonique des Opérateurs de l'Espase de Hilbert," (French) [Harmonic Analysis of Operators in Hilbert Space], Masson et Cie, Paris, Académiai Kiadó, Budapest, 1967.
[2]	J. Lehner, The spectrum of the neutron transport operator for the infinite slab, J. Math. Mech., 11 (1962), 173-181.
[3]	J. Lehner and G. Wing, On the spectrum of an unsymmetric operator arising in the transport theory of neutrons, Comm. Pure Appl. Math., 8 (1955), 217-234. doi: 10.1002/cpa.3160080202.
[4]	J. Lehner and G. Wing, Solution of the linearized Boltzmann transport equation for the slab geometry, Duke Math. J., 23 (1956), 125-142. doi: 10.1215/S0012-7094-56-02312-2.
[5]	Yu. Kuperin, S. Naboko and R. Romanov, Spectral analysis of the transport operator: A functional model approach, Indiana Univ. Math. J., 51 (2002), 1389-1425. doi: 10.1512/iumj.2002.51.2180.
[6]	B. S. Pavlov, On separation conditions for the spectral components of a dissipative operator, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 123-148, 240.
[7]	M. A. Naĭmark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis, (Russian) Trudy Mosk. Mat. Obšč., 3 (1954), 181-270; English transl., Amer. Math. Soc. Transl. (2), 16 (1960), 103-193.
[8]	B. S. Pavlov, Spectral analysis of a dissipative singular Schrödinger operator in terms of a functional model, in "Partial Differential Equations, VIII" (ed. M. Shubin), Encyclopaedia Math. Sci., 65, Springer, Berlin, (1996), 87-153.
[9]	S. N. Naboko, Functional model of perturbation theory and its applications to scattering theory, (Russian) Boundary value problems of mathematical physics, 10, Trudy Mat. Inst. Steklov, 147 (1980), 86-114, 203; English transl., Proc. Steklov Inst. Math., 147 (1981), 85-116.
[10]	S. N. Naboko, On the conditions for existence of wave operators in the nonselfadjoint case, (Russian) in "Wave Propagation. Scattering Theory" (ed. M. Birman), LGU, Leningrad, (1987), 132-155; English transl., Amer. Math. Soc. Transl. Ser. 2, 157, AMS, Providence, RI, (1993), 127-149.
[11]	S. Naboko and R. Romanov, Spectral singularities and asymptotics of contractive semigroups. I, Acta Sci. Math. (Szeged), 70 (2004), 379-403.
[12]	N. K. Nikol'skiĭ , "Treatise on the Shift Operator. Spectral Function Theory," With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller, Translated from the Russian by Jaak Peetre, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 273, Springer-Verlag, Berlin, 1986.
[13]	R. Romanov, A remark on equivalence of weak and strong definitions of the absolutely continuous subspace for nonself-adjoint operators, in "Spectral Methods for Operators of Mathematical Physics" (Bedlewo, 2002), Oper. Theory Adv. Appl., 154, Birkhäuser, Basel, (2004), 179-184.
[14]	L. A.Sahnovič, Dissipative operators with absolutely continuous spectrum, (Russian) Trudy Moskov. Mat. Obšč., 19 (1968), 211-270.
[15]	A. S. Tikhonov, Functional model and duality of spectral components for operators with continuous spectrum on a curve, (Russian) Algebra i Analiz, 14 (2002), 158-195; English transl., St. Petersburg Math. J., 14 (2003), 655-682.
[16]	R. Romanov and M. Tihomirov, On the selfadjoint subspace of the one-velocity transport operator, Math. Notes, 89 (2011), 106-116. doi: 10.1134/S0001434611010111.

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