Spectral analysis for a class of linear pencils arising in transport - - PowerPoint PPT Presentation

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

• perators 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 ωω0 ∂ ∂x f (x, ω) + f (x, ω) −

• S2 k(x, ω, ω

′)f (x, ω ′)dω ′ = 0,

(1) where f is the distribution /of particles/ function defined on ∆ × S2 (the phase space), ∆ is an open interval on the real axis R, S2 denotes the unit sphere in R3, ω0 is a fixed unit vector (selected in the direction of increasing x), by ωω

′ it is denoted the

scalar product (defined on R3) of ω, ω

′ ∈ S2 [KLH]. 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

• ther words, the dependence on ω is only through the variable

µ = ωω0, −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

′ ∈ S2,

(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

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 ∈ L1(−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 < g0 ≤ 1, |gj| < g0 (j = 1, 2, ...), where gj = 2π 1

−1

g(µ)Pj(µ)dµ (j = 0, 1, ...), and Pj(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−λx, in our assumptions, from Eq. (1) it follows (3) u(ω) − λωω0u(ω) =

• 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 g0 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

• Eq. (3) is considered in the space L2(Ω).

Denote by A the multiplication operator by ωω0 defined on L2(Ω), i.e., (Au)(ω) = ωω0u(ω), u ∈ L2(Ω), and let C be the integral operator (also on L2(Ω)) (Cu)(ω) =

• Ω

g(ωω

′)u(ω ′)dω ′,

u ∈ L2(Ω). Then Eq. (3) is written as follows (I − λA − C)u = 0, u ∈ L2(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

• A is a self-adjoint operator in L2(Ω).
• σ(A) = [−1, 1].
• C is a compact operator in L2(Ω).
• gj

(j = 0, 1, 2, ...) are the eigenvalues of C, the corresponding eigenfunctions are the spherical functions Ynm(m = 0, ±1, ..., ±n; n = 0, 1, ...). /Funk-Hecke Theorem/

• C = g0 ≤ 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

• perator 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 L0(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

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

• f 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

• 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 L0(λ) = I − λA. /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

• perator 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 H1, T is a compact self-adjoint operator on H1, and the

• perator-valued functions

Qj(λ) = λSAj(I − λA)−1S∗ (j = 0, 1, 2; λ ∈ ρ(L0)) are uniformly bounded on Λ, i.e., there exists a constant c, c > 0, such that Qj(λ) ≤ c, (j = 0, 1, 2; λ ∈ Λ).

Petru A. Cojuhari Spectral analysis for a class of linear pencils 14 / 30

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 Tp by supposing, of course, that T ∈ Bp(H1) (1 ≤ p < ∞). P.A. Cojuhari, J. Math. Anal. Appl. 326 (2007), 1394 - 1409.

Petru A. Cojuhari Spectral analysis for a class of linear pencils 15 / 30

The following result is also useful for our purposes. Denote by E the spectral measure associated with A, and put | L0(λ) | =

• | 1 − λs | dE(s), W0(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 L0(λ) = I − λA. Furthemore, let λ = a be not a characteristic number of A, i.e., ker(L0(a)) = {0}, and suppose that S and T are bounded operators so that C = S∗TS and |L0(a)|−1/2S∗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

• nly discrete for which a is not an accumulation point.

P.A. Cojuhari, Methods of Functional Analysis and Topology,

• vol. 20, no. 1 (2014), 10 - 26.

Petru A. Cojuhari Spectral analysis for a class of linear pencils 16 / 30

• 3. We apply the abstract results given by Theorems 2 and 3 to
• ur concrete situation of the characteristic equation of radiation

energy transfer u(ω) − λωω0u(ω) =

• Ω

g(ωω

′)u(ω ′)dω ′,

ω ∈ Ω a.e.. We take H = L2(Ω), (Au)(ω) = ωω0u(ω), (Cu)(ω) =

• Ω

g(ωω

′)u(ω ′)dω ′,

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

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(t1) − g(t2)|, where the maximum is taken over |t1| ≤ 1, |t2| ≤ 1 and |t1 − t2| ≤ t. The discrete spectrum, and so the whole spectrum, of the

• perator pencil L(λ) is situated symmetrically with respect to the
• rigin 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

Denote by K0 the operator of finite rank defined by the degenerate kernel k0(ω, ω

′) = g(ωω0) + g(ω0ω ′) − g(1),

and set h(ω, ω

′) = k(ω, ω ′) − k0(ω, ω ′)

for the kernel of the integral operator C − K0, where K(ω, ω

′) is

the kernel of C, i.e., k(ω, ω

′) = g(ωω ′). It follows

|h(ω, ω

′)| ≤ |g(ωω ′) − g(ω0ω ′)| + |g(ωω0) − g(ω0ω0)| ≤

≤ Φ(|ωω

′) − (ω0ω ′|) + Φ(|ωω0) − (ω0ω0|),

and, hence, |h(ω, ω

′)| ≤ 2Φ(|ω − ω0|),

and / in view of the symmetric nature of the kernel h(ω, ω

′) /

|h(ω, ω

′)| ≤ 2Φ(|ω − ω0|)1/2Φ(|ω − ω0|)1/2,

ω, ω

′ ∈ Ω. Petru A. Cojuhari Spectral analysis for a class of linear pencils 19 / 30

Further, we let S = (I − A)1/2, and observe that the operator (I − A)−1/2(C − K0)(I − A)−1/2 can be extended up to an integral operator T determined by the kernel t(ω, ω

′) = h(ω, ω ′)(1 − ωω0)−1/2(1 − ω ′ω0)−1/2,

ωω

′ ∈ Ω.

The operator T is self-adjoint and, it is an integral operator of the Hilbert-Schmidt type provided that 2 Φ(t) t dt < ∞.

Petru A. Cojuhari Spectral analysis for a class of linear pencils 20 / 30

Theorem 4. Let the scattering indecatrix g be a continuous function on [−1, 1] satisfying the following condition 2 Φ(t) t dt < ∞. 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), 1394 - 1409. P.A. Cojuhari, I.A. Feldman, Mat. Issled., 77 (1984), 98 - 103. |g(t) − g(s)| ≤ C|t − s|δ (δ > 0). P.A. Cojuhari, I.A. Feldman, Mat. Issled., 47 (1978), 35 - 40. g

′ ∈ L1(−1, 1).

I.A. Feldman, Dokl. Akad. Nauk SSSR, no. 6, 214 (1974), 1280 - 1283. g(4) ∈ L1(−1, 1).

Petru A. Cojuhari Spectral analysis for a class of linear pencils 21 / 30

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) ∈ R2/ − 1 ≤ µ ≤ 1, 0 ≤ ϕ ≤ 2π}, and consider a function θ : ∆ → Ω according to the rule θ(µ, ϕ) =

• 1 − µ2(cos ϕ)e1 +
• 1 − µ2(sin ϕ)e2 + µe3

/ {e1, e2, e3} is an arbitrary orthonormal basis in R3 / In these new coordinates the characteristic equation of radiation energy transfer is written as (4) u(µ, ϕ) − λµu(µ, ϕ) = = 1

−1

dµ

′ 2π

g(µµ

′+

• 1 − µ2
• 1 − µ

′2 cos(ϕ−ϕ ′))u(µ ′, ϕ ′)dϕ ′.

The Eq. (4) is considered on the space L2(∆).

Petru A. Cojuhari Spectral analysis for a class of linear pencils 22 / 30

As is well-known the / spherical / functions Ynm(µ, ϕ) = 2n + 1 4π (n − |m|)! (n + |m|)! 1/2 Pm

n (µ)eimϕ

(|m| ≤ n; n = 0, 1, ...), where Pm

n (µ) denote Legendre associated functions

Pm

n (µ) = (1 − µ2)|m|/2 d|m|

dµ|m| Pn(µ), form an orthonormal basis in L2(∆).

Petru A. Cojuhari Spectral analysis for a class of linear pencils 23 / 30

For any integer m let Hm denote the subspace of L2(∆) generated by Ynm(µ, ϕ) (n = |m|, |m| + 1, ...) Hm consists of all functions of the form u(µ)eimϕ, u(µ) ∈ L2(−1, 1). If u(µ)eimϕ is a solution of the characteristic equation (4), then u(µ) − λµu(µ) = 1

−1

km(µ, µ

′)u(µ ′)dµ ′,

where km(µ, µ

′) =

2π g(µµ

′ +

• 1 − µ2

1 − µ

′2 cos α) cos mαdα

−1 ≤ µ, µ

′ ≤ 1. Petru A. Cojuhari Spectral analysis for a class of linear pencils 24 / 30

Let A denote the operator of multiplication by the argument µ, and Cm be the integral operator determined by the kernel km(µ, µ

′).

(Au)(µ) = µu(µ), (Cmu)(µ) = 1

−1

km(µ, µ

′)u(µ ′)dµ ′.

/Now, A and Cm are operators in L2(−1, 1)/. It seen is that each of the subspaces Hm are invariant with respect to both of operators A and Cm.

Petru A. Cojuhari Spectral analysis for a class of linear pencils 25 / 30

In this way the linear operator pencil L(λ) is decomposed in an

• rthogonal sum of pencils

Lm(λ) = I − λA − Cm (m = 0, ±1, ±2, ...). / Lm(λ) is the restriction of L(λ) on Hm / The spectrum σ(L) of L(λ) coincides with the union of the spectra σ(Lm) of the operator pencils Lm(λ), so that σ(L) = ∪mσ(Lm). Theorem 5. Assume that the scattering indicatrix g satisfies / as above / the condition 2 Φ(t) t dt < ∞. Then the spectrum of each operator pencil Lm(λ) on the interval (−1, 1) is finite. Moreover, the spectrum of Lm(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

The pencil Lm(λ) is written in the orthonormal basis Ynm(µ, ϕ) (n = |m|, |n| + 1, ...) as follows

     a1m · · · a1m a2m · · · a2m a3m · · · . . . . . . . . . ...      +      1 − g|m| · · · 1 − g|m|+1 · · · 1 − g|m|+2 · · · . . . . . . . . . ...      ,

where anm = (n + |m|)2 − m2 4(n + |m|)2 − 1 1/2 , n = 1, 2, ....

Petru A. Cojuhari Spectral analysis for a class of linear pencils 27 / 30

This representation is obtained easily based on the relationships µPm

n (µ) = n − m + 1

2n + 1 Pm

n+1(µ) + n + m

2n + 1 Pm

n−1(µ),

m = 0, 1, ..., using the fact that Ynm(µ, ϕ) are eigenfunctions of the operator C corresponding respectively to eigenvalues gn (n = 0, 1, ...). It is easily seen that the spectra of operator pencils Lm(λ) and L−m(λ) coincide, so, it can be considered only Lm with nonnegative integers m.

Petru A. Cojuhari Spectral analysis for a class of linear pencils 28 / 30

Theorem 6. Let the scattering indicatrix g be a summable function on [−1, 1] satisfying

∞

• n=1

n|gn| < ∞. Then the spectrum of each the operator pencil Lm(λ) with m ≥ 1 on the interval (−1, 1) is finite. For m = 0 the same is true under the condition

∞

• n=1

(n log n)|gn| < ∞. The last result is due to C.G. Lekkerkerker, Three-term recurrence relations in transport theory, Integr. Eq. Oper. Theory, 4/2 (1981), 245 - 274.

• K. M. Case, Scattering theory, orthogonal polynomials, and the

transport equation, J. Math. Phys. 15 (1974), 974 - 983. The general case is considered in P.A. Cojuhari, Methods of Funct. Anal. and Topology, 20, no. 1 (2014), 10 - 26. Finally we note the following criterion.

Petru A. Cojuhari Spectral analysis for a class of linear pencils 29 / 30

Finally we note the following criterion. Theorem 7. Assume gn = O(n−δ) with some δ > 1. If lim sup n2(gn + gn+1) < m2 + 1/4, then the spectrum of Lm(λ) on (−1, 1) is finite, and if lim sup n2(gn + gn+1) > m2 + 1/4, then the spectrum of Lm(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