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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

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 & 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.  Google Scholar

[2]

J. Lehner, The spectrum of the neutron transport operator for the infinite slab, J. Math. Mech., 11 (1962), 173-181.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

S. Naboko and R. Romanov, Spectral singularities and asymptotics of contractive semigroups. I, Acta Sci. Math. (Szeged), 70 (2004), 379-403.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

L. A.Sahnovič, Dissipative operators with absolutely continuous spectrum, (Russian) Trudy Moskov. Mat. Obšč., 19 (1968), 211-270.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

J. Lehner, The spectrum of the neutron transport operator for the infinite slab, J. Math. Mech., 11 (1962), 173-181.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

S. Naboko and R. Romanov, Spectral singularities and asymptotics of contractive semigroups. I, Acta Sci. Math. (Szeged), 70 (2004), 379-403.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

L. A.Sahnovič, Dissipative operators with absolutely continuous spectrum, (Russian) Trudy Moskov. Mat. Obšč., 19 (1968), 211-270.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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