Spectral analysis for a class of linear pencils arising in transport theory

Transcription

1 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

2 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

3 References [CZ] K.M. Case, and P.F. Zweifel, Linear Transport Theory, Addison-Wesley, Reading, MA, [Ch] S. Chandrasekhar, Radiative Transfer, New York, [D] B. Davison, Neutron Transport Theory, Oxford, [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, [M] M. V. Maslennikov, The Milne Problem with Anisotropic Scattering, Proc. Steklov Inst. Math. 97, 1968; English transl., Amer. Math. Soc., Providence, R. I Petru A. Cojuhari Spectral analysis for a class of linear pencils 3 / 30

4 1. The time-independent linear transport equation in one dimensional slab configuration with anisotropic scattering has the following form ωω 0 f (x, ω) + f (x, ω) x S 2 k(x, ω, ω )f (x, ω )dω = 0, (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 (selected in the direction of increasing x), by ωω it is denoted the scalar product (defined on R 3 ) of ω, ω S 2 [KLH]. Petru A. Cojuhari Spectral analysis for a class of linear pencils 4 / 30

5 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 k(x, ω, ω ) = g(ωω ), x ; ω, ω S 2, (2) that is, k does not depends on the position variable x /the host medium is homogeneous/ depending only on ωω (the rotational 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

6 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, [K] V.Kourganoff, Basic methods in transfer problems. Radiative equilibrium and neutron diffusion, Clarondon Press, Oxford, [B] I.W.Busbridge, The mathematics of radiative transfer, Cambridge Univ. Press, New York, [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

7 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 = 2π g(µ)p j (µ)dµ (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

8 By seeking a solution in the form u(ω)e λx, in our assumptions, from E q. (1) it follows (3) u(ω) λωω 0 u(ω) = g(ωω )u(ω )dω, ω Ω a.e., 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 Ω g 0 ln 1 + λ 1 λ = 2λ, 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

9 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)) (Cu)(ω) = g(ωω )u(ω )dω, u L 2 (Ω). Then Eq. (3) is written as follows Ω (I λa 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 λa C. Petru A. Cojuhari Spectral analysis for a class of linear pencils 9 / 30

10 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

11 The problem is to study the structure of the spectrum of the operator pencil L(λ) = I λa 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 λa 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

12 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 λa 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

13 2. Next, we consider a linear operator pencil L(λ) = I λa 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 /unperturbed pencil/ L 0 (λ) = I λa. Petru A. Cojuhari Spectral analysis for a class of linear pencils 13 / 30

14 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) = λsa j (I λa) 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

15 Theorem 2. Let L(λ) = I λa C with C 1, and suppose that the assumption (A) is satisfied. Then the spectrum of the operator pencil L(λ) on the interval Λ is finite. Remark. The total number n(λ; L) of the eigenvalues of L(λ) belonging to Λ can be estimated in terms of Schatten s norms T p by supposing, of course, that T B p (H 1 ) (1 p < ). P.A. Cojuhari, J. Math. Anal. Appl. 326 (2007), Petru A. Cojuhari Spectral analysis for a class of linear pencils 15 / 30

16 The following result is also useful for our purposes. Denote by E the spectral measure associated with A, and put L 0 (λ) = 1 λs de(s), W 0 (λ) = sgn(1 λs)de(s) (the integration is taken over the spectrum of A). Theorem 3. Let A and B be self-adjoint operators in H, B = I C, where C B (H) and C 1, and let the interval Λ = (a, b) ( < a < b ) be a spectral gap in the spectrum of the (unperturbed) operator pencil L 0 (λ) = I λa. Furthemore, let λ = a be not a characteristic number of A, i.e., ker(l 0 (a)) = {0}, and suppose that S and T are bounded operators so that C = S TS and L 0 (a) 1/2 S P B (H), where P is an orthogonal projection such that dim(i P)H <. Then the spectrum of the operator pencil L(λ) = B λa on Λ is only discrete for which a is not an accumulation point. P.A. Cojuhari, Methods of Functional Analysis and Topology, vol. 20, no. 1 (2014), Petru A. Cojuhari Spectral analysis for a class of linear pencils 16 / 30

17 3. We apply the abstract results given by Theorems 2 and 3 to our concrete situation of the characteristic equation of radiation energy transfer u(ω) λωω 0 u(ω) = g(ωω )u(ω )dω, ω Ω a.e.. We take Ω H = L 2 (Ω), (Au)(ω) = ωω 0 u(ω), (Cu)(ω) = g(ωω )u(ω )dω, Ω and we have to find conditions on the scattering indicatrix under which the discrete spectrum of the associated operator pencil L(λ) = I λa C to be finite on the interval Λ = ( 1, 1). Petru A. Cojuhari Spectral analysis for a class of linear pencils 17 / 30

18 We assume that g is a continuous function on [ 1, 1], and denote by Φ(t) (0 t 2) the oscillation of g, i.e., Φ(t) = max g(t 1 ) g(t 2 ), where the maximum is taken over t 1 1, t 2 1 and t 1 t 2 t. The discrete spectrum, and so the whole spectrum, of the operator pencil L(λ) is situated symmetrically with respect to the origin λ = 0. So, it is enough to consider the situation on the interval (0, 1). Petru A. Cojuhari Spectral analysis for a class of linear pencils 18 / 30

19 Denote by K 0 the operator of finite rank defined by the degenerate kernel and set k 0 (ω, ω ) = g(ωω 0 ) + g(ω 0 ω ) g(1), h(ω, ω ) = k(ω, ω ) k 0 (ω, ω ) for the kernel of the integral operator C K 0, where K(ω, ω ) is the kernel of C, i.e., k(ω, ω ) = g(ωω ). It follows h(ω, ω ) g(ωω ) g(ω 0 ω ) + g(ωω 0 ) g(ω 0 ω 0 ) and, hence, Φ( ωω ) (ω 0 ω ) + Φ( ωω 0 ) (ω 0 ω 0 ), h(ω, ω ) 2Φ( ω ω 0 ), and / in view of the symmetric nature of the kernel h(ω, ω ) / h(ω, ω ) 2Φ( ω ω 0 ) 1/2 Φ( ω ω 0 ) 1/2, ω, ω Ω. Petru A. Cojuhari Spectral analysis for a class of linear pencils 19 / 30

20 Further, we let S = (I A) 1/2, and observe that the operator (I A) 1/2 (C K 0 )(I A) 1/2 can be extended up to an integral operator T determined by the kernel t(ω, ω ) = h(ω, ω )(1 ωω 0 ) 1/2 (1 ω ω 0 ) 1/2, ωω Ω. The operator T is self-adjoint and, it is an integral operator of the Hilbert-Schmidt type provided that 2 0 Φ(t) dt <. t Petru A. Cojuhari Spectral analysis for a class of linear pencils 20 / 30

21 Theorem 4. Let the scattering indecatrix g be a continuous function on [ 1, 1] satisfying the following condition 2 0 Φ(t) dt <. t Then the spectrum of the operator pencil L(λ) = I λa C on the interval ( 1, 1) is finite. P.A. Cojuhari, J. Math. Anal. Appl. 326 (2007), P.A. Cojuhari, I.A. Feldman, Mat. Issled., 77 (1984), g(t) g(s) C t s δ (δ > 0). P.A. Cojuhari, I.A. Feldman, Mat. Issled., 47 (1978), g L 1 ( 1, 1). I.A. Feldman, Dokl. Akad. Nauk SSSR, no. 6, 214 (1974), g (4) L 1 ( 1, 1). Petru A. Cojuhari Spectral analysis for a class of linear pencils 21 / 30

22 In view of physical interests, a study in more details is required. It will be convenient to pass to other coordinates. Let be the closed rectangle on the plane = {(µ, ϕ) R 2 / 1 µ 1, 0 ϕ 2π}, and consider a function θ : Ω according to the rule θ(µ, ϕ) = 1 µ 2 (cos ϕ)e µ 2 (sin ϕ)e 2 + µe 3 / {e 1, e 2, e 3 } is an arbitrary orthonormal basis in R 3 / In these new coordinates the characteristic equation of radiation energy transfer is written as = 1 1 (4) u(µ, ϕ) λµu(µ, ϕ) = 2π dµ g(µµ + 1 µ 1 2 µ 2 cos(ϕ ϕ ))u(µ, ϕ )dϕ. 0 The Eq. (4) is considered on the space L 2 ( ). Petru A. Cojuhari Spectral analysis for a class of linear pencils 22 / 30

23 As is well-known the / spherical / functions [ 2n + 1 Y nm (µ, ϕ) = 4π (n m )! (n + m )! ( m n; n = 0, 1,...), ] 1/2 P m n (µ)e imϕ where P m n (µ) denote Legendre associated functions Pn m (µ) = (1 µ 2 m /2 d m ) dµ m P n(µ), form an orthonormal basis in L 2 ( ). Petru A. Cojuhari Spectral analysis for a class of linear pencils 23 / 30

24 For any integer m let H m denote the subspace of L 2 ( ) generated by Y nm (µ, ϕ) (n = m, m + 1,...) H m consists of all functions of the form u(µ)e imϕ, u(µ) L 2 ( 1, 1). If u(µ)e imϕ is a solution of the characteristic equation (4), then where k m (µ, µ ) = u(µ) λµu(µ) = 2π k m (µ, µ )u(µ )dµ, g(µµ + 1 µ 2 1 µ 2 cos α) cos mαdα 1 µ, µ 1. Petru A. Cojuhari Spectral analysis for a class of linear pencils 24 / 30

25 Let A denote the operator of multiplication by the argument µ, and C m be the integral operator determined by the kernel k m (µ, µ ). (C m u)(µ) = (Au)(µ) = µu(µ), 1 1 k m (µ, µ )u(µ )dµ. /Now, A and C m are operators in L 2 ( 1, 1)/. It seen is that each of the subspaces H m are invariant with respect to both of operators A and C m. Petru A. Cojuhari Spectral analysis for a class of linear pencils 25 / 30

26 In this way the linear operator pencil L(λ) is decomposed in an orthogonal sum of pencils L m (λ) = I λa C m (m = 0, ±1, ±2,...). / L m (λ) is the restriction of L(λ) on H m / The spectrum σ(l) of L(λ) coincides with the union of the spectra σ(l m ) of the operator pencils L m (λ), so that σ(l) = m σ(l m ). Theorem 5. Assume that the scattering indicatrix g satisfies / as above / the condition 2 0 Φ(t) dt <. t Then the spectrum of each operator pencil L m (λ) on the interval ( 1, 1) is finite. Moreover, the spectrum of L m (λ) for large enough m in the interval ( 1, 1) is empty. Petru A. Cojuhari Spectral analysis for a class of linear pencils 26 / 30

27 The pencil L m (λ) is written in the orthonormal basis Y nm (µ, ϕ) (n = m, n + 1,...) as follows 0 a 1m 0 0 a 1m 0 a 2m 0 0 a 2m 0 a 3m g m g m g m , where [ (n + m ) 2 m 2 ] 1/2 a nm = 4(n + m ) 2, n = 1, 2,... 1 Petru A. Cojuhari Spectral analysis for a class of linear pencils 27 / 30

28 This representation is obtained easily based on the relationships µp m n (µ) = n m + 1 2n + 1 P m n+1(µ) + n + m 2n + 1 Pm n 1(µ), m = 0, 1,..., using the fact that Y nm (µ, ϕ) are eigenfunctions of the operator C corresponding respectively to eigenvalues g n (n = 0, 1,...). It is easily seen that the spectra of operator pencils L m (λ) and L m (λ) coincide, so, it can be considered only L m with nonnegative integers m. Petru A. Cojuhari Spectral analysis for a class of linear pencils 28 / 30

29 Theorem 6. Let the scattering indicatrix g be a summable function on [ 1, 1] satisfying n g n <. n=1 Then the spectrum of each the operator pencil L m (λ) with m 1 on the interval ( 1, 1) is finite. For m = 0 the same is true under the condition (n log n) g n <. n=1 The last result is due to C.G. Lekkerkerker, Three-term recurrence relations in transport theory, Integr. Eq. Oper. Theory, 4/2 (1981), K. M. Case, Scattering theory, orthogonal polynomials, and the transport equation, J. Math. Phys. 15 (1974), The general case is considered in P.A. Cojuhari, Methods of Funct. Anal. and Topology, 20, no. 1 (2014), Finally we note the Petrufollowing A. Cojuhari criterion. Spectral analysis for a class of linear pencils 29 / 30

30 Finally we note the following criterion. Theorem 7. Assume g n = O(n δ ) with some δ > 1. If lim sup n 2 (g n + g n+1 ) < m 2 + 1/4, then the spectrum of L m (λ) on ( 1, 1) is finite, and if lim sup n 2 (g n + g n+1 ) > m 2 + 1/4, then the spectrum of L m (λ) on ( 1, 1) is infinite / consisting only of isolated eigenvalues of finite multiplicity /. Petru A. Cojuhari Spectral analysis for a class of linear pencils 30 / 30