Spectral analysis for a class of linear pencils arising in transport theory Petru A. Cojuhari AGH University of Science and Technology Kraków, Poland Vienna, December 17 – 20, 2016 Petru A. Cojuhari Spectral analysis for a class of linear pencils 1 / 30
Our intention is to discuss certain spectral aspects of linear operators pencils which occur naturally in modeling transport phenomena in matter. The phenomena relate mainly to neutron transport in a nuclear scattering experiment as, for instance, in a nuclear reactor, or in radiative transfer of energy, and also other similar processes. In each case, the transport mechanism involves the migration of particles /neutrons, photons, etc./ through a host medium. We shall concern ourselves exclusively with transport phenomena involving neutral particles and, for the sake of simplicity, the effects of external forces /of fields/ will be ignored. In other words, we deal with the situation in which - the motion of particles are affected only collisions with the atomic nuclei of the host medium; - the collisions are well-defined events and take place locally and instantaneously; The number of particles is not necessarily however conserved in a collision /some particles may disappear (absorption) and also change their velocity (scattering)/. Petru A. Cojuhari Spectral analysis for a class of linear pencils 2 / 30
References [CZ] K.M. Case, and P.F. Zweifel, Linear Transport Theory, Addison-Wesley, Reading, MA, 1967. [Ch] S. Chandrasekhar, Radiative Transfer, New York, 1960. [D] B. Davison, Neutron Transport Theory, Oxford, 1957. [KLH] H. G. Kaper, C. G. Lekkerkerker, and J. Hejtmanek, Spectral Methods in Linear Transport Theory, Oper. Theory Adv. Appl., vol. 5, Birkhäuser Verlag, Basel, 1982. [M] M. V. Maslennikov, The Milne Problem with Anisotropic Scattering, Proc. Steklov Inst. Math. 97, 1968; English transl., Amer. Math. Soc., Providence, R. I. 1969. Petru A. Cojuhari Spectral analysis for a class of linear pencils 3 / 30
1. The time-independent linear transport equation in one dimensional slab configuration with anisotropic scattering has the following form ∂ � ′ = 0 , ′ ) f ( x , ω ′ ) d ω ωω 0 ∂ x f ( x , ω ) + f ( x , ω ) − S 2 k ( x , ω, ω (1) where f is the distribution /of particles/ function defined on ∆ × S 2 (the phase space), ∆ is an open interval on the real axis R , S 2 denotes the unit sphere in R 3 , ω 0 is a fixed unit vector ′ it is denoted the (selected in the direction of increasing x ), by ωω ′ ∈ S 2 [KLH]. scalar product (defined on R 3 ) of ω, ω Petru A. Cojuhari Spectral analysis for a class of linear pencils 4 / 30
We consider the situation of azimuthal symmetry that means that the distribution function is independent of the azimuth, in other words, the dependence on ω is only through the variable µ = ωω 0 , − 1 ≤ µ ≤ 1 . In addition, we assume that the scattering kernel k is of the form ′ ∈ S 2 , ′ ) = g ( ωω ′ ) , k ( x , ω, ω x ∈ ∆; ω, ω (2) that is, k does not depends on the position variable x /the host ′ (the rotational medium is homogeneous/ depending only on ωω invariance property). The function g determined k as in (2) is called the scattering function or, in other terminology especially in the theory of radiative transfer, the scattering indicatrix [M]. Petru A. Cojuhari Spectral analysis for a class of linear pencils 5 / 30
The problem of finding non-trivial solutions of equation (1) is known as the Milne problem. The Milne problem has been studied extensively by many authors. Besides the already mentioned monographs [CZ] [Ch] [D] [KLH], [M], the following works [H] E.Hopf, Mathematical problems of radiative equilibrium, Cambridge Tracts in Math. and Math. Phys., no. 31, Cambridge Univ. Press, New York, 1934. [K] V.Kourganoff, Basic methods in transfer problems. Radiative equilibrium and neutron diffusion, Clarondon Press, Oxford, 1952. [B] I.W.Busbridge, The mathematics of radiative transfer, Cambridge Univ. Press, New York, 1961. [M] J.R.Mika, Neutron transport with anisotropic scattering, Nuclear Sci. Eng. 11, 1961 should be mentioned due to of which a rigorous mathematical study was initiated, although, those refer to particular cases, as, for example, in [H], [K] or [B] the simplest case of isotropic scattering ( g ( µ ) ≡ const ) is considered. Petru A. Cojuhari Spectral analysis for a class of linear pencils 6 / 30
For our purposes we study the problem under the following assumptions concerning the scattering indicatrix (a) g is a nonnegative summable function on [ − 1 , 1 ] , i.e., g ≥ 0 and g ∈ L 1 ( − 1 , 1 ); (b) the probability of survival of a particle in a single event of interaction with the material is positive; this is equivalent to the following 0 < g 0 ≤ 1 , | g j | < g 0 ( j = 1 , 2 , ... ) , where � 1 g j = 2 π g ( µ ) P j ( µ ) d µ ( j = 0 , 1 , ... ) , − 1 and P j ( µ ) are the Legendre polynomials. Petru A. Cojuhari Spectral analysis for a class of linear pencils 7 / 30
By seeking a solution in the form u ( ω ) e − λ x , in our assumptions, from E q . (1) it follows � ′ ) u ( ω ′ ) d ω ′ , ω ∈ Ω a . e ., ( 3 ) u ( ω ) − λωω 0 u ( ω ) = g ( ωω Ω Here and in what follows Ω denotes the unit sphere in R . If g ( µ ) = const , then the spectral parameter λ , as is seen, satisfies the transcentental equation g 0 ln 1 + λ 1 − λ = 2 λ, that in known in the Hopf-Chandrasekhar theory as the characteristic equation of radiation energy transfer. This term is also applied to the general case of integral equation (3). Petru A. Cojuhari Spectral analysis for a class of linear pencils 8 / 30
Eq. (3) is considered in the space L 2 (Ω) . Denote by A the multiplication operator by ωω 0 defined on L 2 (Ω) , i.e., ( Au )( ω ) = ωω 0 u ( ω ) , u ∈ L 2 (Ω) , and let C be the integral operator (also on L 2 (Ω) ) � ′ ) u ( ω ′ ) d ω ′ , ( Cu )( ω ) = g ( ωω u ∈ L 2 (Ω) . Ω Then Eq. (3) is written as follows ( I − λ A − C ) u = 0 , u ∈ L 2 (Ω) , and in this way the problem is reduced to the study of the corresponding linear operator pencil L ( λ ) = I − λ A − C . Petru A. Cojuhari Spectral analysis for a class of linear pencils 9 / 30
• A is a self-adjoint operator in L 2 (Ω) . • σ ( A ) = [ − 1 , 1 ] . • C is a compact operator in L 2 (Ω) . • g j ( j = 0 , 1 , 2 , ... ) are the eigenvalues of C , the corresponding eigenfunctions are the spherical functions Y nm ( m = 0 , ± 1 , ..., ± n ; n = 0 , 1 , ... ) . / Funk-Hecke Theorem / • � C � = g 0 ≤ 1 . Petru A. Cojuhari Spectral analysis for a class of linear pencils 10 / 30
The problem is to study the structure of the spectrum of the operator pencil L ( λ ) = I − λ A − C . Our approach is based on the technique of perturbation theory for linear operators. We consider the operator pencil as a perturbation of the pencil L 0 ( λ ) = I − λ A by the operator C . It seems that the situation is the same as in ordinary case when a given operator is perturbed by another operator. But, it is not the case. The situation with operator pencils is much more complicated then of ordinary case. In spite of the fact that both operators A and B = I − C are self-adjoint the spectrum of the operator pencil L ( λ ) can contain complex (non-real) points / such a situation can be realized by simple example even for finite-dimensional case./ Petru A. Cojuhari Spectral analysis for a class of linear pencils 11 / 30
Proposition 1. Under above assumptions suppose that there exists a regular point of the operator pencil L ( λ ) . Then the spectrum of L ( λ ) outside of the real line can be only discrete. In the particular case of C ∈ B ∞ ( H ) and � C � ≤ 1 the non-real spectrum of L ( λ ) is empty, i.e., σ ( L ) ⊂ R . Remarks 1. If kerA = { 0 } and C ∈ B ∞ ( H ) , then the existence of a regular point of the pencil L ( λ ) is ensured. 2. If the operator B is definite / either B > 0 or B < 0 /, then the spectrum of L ( λ ) = B − λ A lies on the real axis and, moreover, the eigenvalues of L ( λ ) if there exist are semi-simple, i.e., there are no associated eigenvectors for L ( λ ) . Petru A. Cojuhari Spectral analysis for a class of linear pencils 12 / 30
2. Next, we consider a linear operator pencil L ( λ ) = I − λ A − C , in which A and C are self-adjoint operators in a Hilbert space H , � C � < 1 . Suppose that Λ = ( a , b ) ⊂ R is a spectral gap of L 0 ( λ ) = I − λ A . /unperturbed pencil/ Petru A. Cojuhari Spectral analysis for a class of linear pencils 13 / 30
We need the following assumption. (A) There exists an operator of finite rank K such that the operator C − K admits a factorization of the form C − K = S ∗ TS , where S is a bounded linear operator from H into another Hilbert space H 1 , T is a compact self-adjoint operator on H 1 , and the operator-valued functions Q j ( λ ) = λ SA j ( I − λ A ) − 1 S ∗ ( j = 0 , 1 , 2 ; λ ∈ ρ ( L 0 )) are uniformly bounded on Λ , i.e., there exists a constant c , c > 0 , such that � Q j ( λ ) � ≤ c , ( j = 0 , 1 , 2 ; λ ∈ Λ) . Petru A. Cojuhari Spectral analysis for a class of linear pencils 14 / 30
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