Appeared in Comunications in Applied Analysis 6 (2002), no. 3, 415-428

SPHERICAL HARMONIC HYPERGROUP IN AN

INVERSE PROBLEM IN TRANSPORT THEORY

H. Emamirad and G.S. Heshmati

Abstract. The inverse problem of recovering the absorption coefficient σ and the collision kernel k from the albedo operator of the boundary of a bounded domain of ℝ^{n} ( n ≥ 3) in a stationary transport problem is discussed by Choulli and Stefanov in [C-S2]. The problem remained open for n = 1, 2. In this paper by using the harmonic analysis techniques we write the integral term generated by the scattering kernel of the transport equation as a generalized convolution product of the spherical harmonic hypergroup. This gives a very simple expression of one-dimensional transport equation, which in turn, by use of the Fredholm alternative gives a negative answer for recovering uniquely the collision kernel k and positive answer for recovering uniquely the absorption coefficient σ.

1. Introduction

Let Ω be an open bounded subset of ℝ^{n} with C^{1} boundary ∂Ω and V = {v ∈ ℝ^{n} ; |v| < 1}. We consider the following boundary value problem for the stationary linear transport equation

(BVP) \{ \begin{matrix}−v.∇_{x}φ − σ(x, v)φ + ∫_{V} k(x, v′, v)φ(x, v′)dv′ = 0 (x, v) ∈ Ω × V \\ φ |_{Γ_{−}}= g_{−} \end{matrix}

Here g_{−} is a given function on Γ_{−} and Γ_{±} = {(x, v) ∈ ∂Ω × V ; ±n(x).v > 0}, where n(x) is the outer normal to ∂Ω at x ∈ ∂Ω. We assume that the absoption cross-section σ and the collision kernel k satisfy the following admissible conditions:

(1) 0 ≤ σ ∈ L^{∞}(Ω × V).

(2) 0 ≤ k(x, v′, .) ∈ L^{1}(V) for a.e; (x, v′) ∈ Ω×V and σ_{p}(x, v) ≡ ∫_{V} k(v, v′)dv′ ∈ L^{∞}(Ω × V).

1991 Mathematics Subject Classification. Primary 82C70, 33C45; Secondary 35Q. Key words and phrases. Spherical harmonic hypergroups, Legendre polynomials, Gegenbauer kernel, Transport equation Albedo operator, Pseudomeasure operator. The authors are supported by Tehran University under contract 611/1/366.

Typeset by AMS-TEX

To guarantee that the boundary value problem (BVP) is uniquely solvable one can assume either that

σ(x, v) − σ_{p}(x, v) ≥ ν > 0 for a.e. (x, v) ∈ Ω × V,  (1.1)

or that

∥τσ∥_{L^{∞}} < ∞,  ∥τσ_{p}∥_{L^{∞}} < 1.  (1.2)

where τ = τ_{−} + τ_{+} and τ_{±}(x, v) = min{t ≥ 0 ; x ± tv ∈ ∂Ω}. The first assumption, which is treated in [D-L, Chapter 20] says that the absorption rate is greater than the production rate. This implies in the time-dependent situation that the corresponding dynamics is subcritical. While the second condition asserts that the mean free path is relatively small(see [C-S2]). Once, by imposing (1.1) or (1.2) to make sure that (BVP) is uniquely solvable, one can define the albedo operator

A : g_{−} ↦ φ |_{Γ_{+}},

that maps the incoming flux on the boundary into the outgoing one. The relationship between the scattering and albedo operators is amply discussed in [A-E],[E-P] and [C-S1]. In [C-S2], Choulli and Stefanov are interested in the following inverse problem:

Does the albedo operator A determine uniquely the absorption and production rates?

In the same paper they have also proved the following theorem:

Theorem 1.1. Let (σ, k) and (σ̂, k̂) be two admissible pairs with σ = σ(x, |v|) and σ̂ = σ̂(x, |v|) which satisfying (1.1) or (1.2). If the corresponding albedo operators A and Â coincide, then

(a) for n ≥ 3, we have σ = σ̂ and k = k̂;

(b) for n = 2, we have σ = σ̂

This Theorem leaves two open problems:

(1) Can we recover uniquely the coefficients σ(x, v) and k(x, v′, v) from the albedo operator when n = 1?

(2) What about the coefficient k(x, v′, v) when n = 2? One of the aim in this paper is to answer to the first open problem. To do this, we use the technique of the harmonic analysis especially the recent development in the theory of hypergroups (see [B-H]). In fact, the use of such theory for such inverse problem is not indispensable. As we will see this can be done simply by use of the Legendre polynomial expansion of the solution. But, our another aim is also to introduce this formalism in the transport theory by puting the integral term of the transport equation in term of a generalized convolution product and to study the harmonic analysis properties of this operator. We begin the section 2 of the paper with a rough introduction on the theory of the hypergroups. The spherical harmonic hypergroup endows the space L^{1}([−1, 1])) with a generalized convolution product ∗_{0}. This convolution product is defined by the Gegenbauer kernel of index zero G_{0}(x, y, z) as

[f ∗_{0} g](x) = ∫∫_{I×I} G_{0}(x, y, z)f(y)g(z)dydz  (1.3)

where I = [−1, 1] and

G_{0}(x, y, z) = Γ(\frac{1}{2})^{−2}(1 − x^{2} − y^{2} − z^{2} + 2xyz)_{+}^{−1/2}.  (1.4)

The resulting convolution will have sufficient resemblance to the classical convolution properties. In particular

{ \atop f ∗_{0} g}(n) = \frac{2}{2n + 1}f̂(n)ĝ(n).  (1.5)

where

f̂(n) = \frac{2n + 1}{2} ∫_{I} f(x)P_{n}(x)dx

is the generalized Fourier coefficient of f. Now, if 𝔞 is the algebra of the continuous fonctions on I of the form

f = ∑_{n≥0} f̂(n)P_{n}

such that ∥f∥ = ∑_{n≥0} |f̂(n)| < ∞, we will see that 𝔞 equipped with the norm ∥ · ∥ is a commutative, semisimple Banach algebra for the simple product of functions and for f, g ∈ 𝔞, one has

(f ∗_{0} g)(x) = ∫_{I} ∫_{I} G_{0}(x, y, z) ∑_{n≥0} f̂(n)P_{n}(y)g(z)dydz

= ∑_{n≥0} f̂(n)P_{n}(x) ∫_{I} P_{n}(z)g(z)dz.  (1.6)

This expression permits us to put the transport equation (BVP) in one-dimensional plane configuration in a very simple expression. Namely, for one-velocity timeindependent neutron transport equation in a slab we have X = [−1, 1] and V = S^{2}. In a homogeneous medium, if we assume that the scattering kernel is degenarate and does not vary with x, then k(x, ω, ω′) = k(ω · ω′). We assume furthermore that the system has azimuthal symmetry – that is, when the dependence of the distribution function φ on ω is only thru μ = ω · e, where e is a fixed polar vector of unit length. In this situation the transport equation reads as

\frac{∂}{∂x}μφ(x, μ) + σ(x, μ)φ(x, μ) = \frac{1}{4π} ∫_{S^{2}} k(ω · ω′)φ(x, ω′ · e)dω′.  (1.7)

Expanding k(ω · ω′) in Legendre polynomials series and using (1.6), the equation (1.7) with the boundary conditions can be written in the following form

(TP)  \{ \begin{matrix}\frac{∂}{∂x}μφ(x, μ) + σ(x, μ)φ(x, μ) = \frac{1}{2}(k ∗_{0} φ)(x, μ) \\ φ(−1, μ) = g_{+}(μ), for 0 ≤ μ ≤ 1, \\ φ(1, μ) = g_{−}(μ), for − 1 ≤ μ ≤ 0. \end{matrix}

With these notations the corresponding albedo operator would be

(AO)  \{ \begin{matrix}Ag_{+}(μ) = φ(−1, μ ≤ 0), \\ Ag_{−}(μ) = φ(1, μ ≥ 0). \end{matrix}

The section 3 is devoted to the harmonic analysis properties of the integral operator

S : f ∈ L^{2}(I) ⟼ k ∗_{0} φ̂ ∈ L^{2}(I).

In this paper we will content ourself with showing only that the operator S is a Hilbert-Schmidt type operator and give a condition under which the equation Sf = f has a no non-zero solution. These properties serve us in the sequel for the inverse problem. In fact, the operator S has also other properties such as asymptotic stability ( see [L-M]) or pseudomeasure character (see [Kah]), which their demonstrations lead us far from our main objective. We have seen that the admissibility conditions on σ and k determines uniquely a solution φ for (TP). This solution provides us a source function S(x, μ) = \frac{1}{2}(k ∗_{0} φ)(x, μ). Replacing the second member of (1.7) by S gives the following boundary value problem

(PAP)  \{ \begin{matrix}\frac{∂}{∂x}μφ(x, μ) + σ(x, μ)φ(x, μ) = S(x, μ) \\ φ(−1, μ) = g_{+}(μ), for 0 ≤ μ ≤ 1, \\ φ(1, μ) = g_{−}(μ), for − 1 ≤ μ ≤ 0. \end{matrix}

An interesting inverse problem in purely absorbing media is the determination of source function from the underling measure space of the solution. This problem is discussed in [Win] and [Mok]. In the section 4, we will supply the following answer to the open problem (1). If k ≡ 0 in (TP) then σ can be recover uniquely from the albedo operator (AO) only if the function σ(x, μ) = σ(μ) does not vary with x. But for recovering the scattering kernel k from the albedo operator we give an negative answer. Namely for σ ≡ 1, if k_{1} and k_{2} are two different scattering kernels in L^{2}(I), then there exist two solutions φ_{1} and φ_{2} such that the corresponding albedo operators coincide.

2. Spherical harmonic hypergroup

In this section we will introduce an intuitive representation of the hypergroups through the spherical harmonics which affects particularly our framework.

Let G be a locally compact topological group and M(G) the space of finite Borel measures on G. then (M(G), ⋆) is a measure algebra endowed with a generalized convolution ⋆. This convolution or convos as is called by Jewett [Jew] is defined by (μ ⋆ ν)(E) = ∫_{G} μ(Ex^{−1})dν(x) satisfies a set of axioms called hypergroup axioms. Now, if H is not a group but it can be endowed with a generalized convolution ⋆ which satisfies the hypergroup axioms then (H, ⋆) is called a hypergroup. Here we are not going to give the definitive set of the hypergroup axioms. The interested reader is directed to [B-H] or [Jew] for a prise statement of these axioms. A typical example of such convolution is given by I. I. Hirschmann, Jr. in [Hi1] and [Hi2]. He describes two measure algebras (M(G), ∗_{α}) and (M(ℕ_{0}), ⋆_{α}) by mean of the ultraspherical polynomials. In the sequel we will define (I, ∗_{α}), while (ℕ_{0}, ⋆_{α}) will be defined later only for α = 0. Let

Q_{n}^{α}(x) = \frac{(−1)^{n}}{2^{n}(α + 1)...(α + n)}(1 − x^{2})^{−α} \frac{d^{n}}{dx^{n}} (1 − x^{2})^{α+n}

be the n^{th} order ultraspherical polynomial of index α > −1. They form a orthogonal system on I = [−1, 1] for the measure dμ_{α}(x) = (1 − x^{2})^{α}dx ( cf. [Erd]), with the relation

∫_{I} Q_{m}^{α}(x)Q_{n}^{α}(x)dμ_{α}(x) = \{ \begin{matrix}0 if n ≠ m \\ (ω_{n}^{α})^{−1} if n = m \end{matrix}

where

ω_{n}^{α} = \frac{(2n + 2α + 1)Γ(n + 2α + 1)}{2^{2α+1}Γ(α + 1)^{2}Γ(n + 1)}.

For α > −\frac{1}{2}, the Gegenbauer’s multiplication formula (cf. [Ga1, Ga2] ):

Q_{n}^{α}(x)Q_{n}^{α}(y) = ∫_{I} G_{α}(x, y, z)Q_{n}^{α}(z)dμ_{α}(z),  (−1 < x, y < 1)  (2.1)

shows that the kernel G_{α}(., ., z) is stochastic (see [L-M]). In fact,

G_{α}(x, y, z) = \frac{Γ(α + 1)}{Γ(\frac{1}{2})Γ(α + \frac{1}{2})} \frac{(1 − x^{2} − y^{2} − z^{2} + 2xyz)_{+}^{α− \frac{1}{2}}}{(1 − z^{2})^{α}(1 − x^{2})^{α}(1 − y^{2})^{α}}

where by f_{+} we designate the positive part of the function f. Hence

G_{α}(x, y, z) ≥ 0,

and by putting n = 0 in (2.1), Q_{0}^{α}(x) = 1 and the symmetry of G_{α} with respect to x, y et z, yield that

∫_{I} G_{α}(x, y, z)dμ_{α}(y) = 1.

For any f, g ∈ L^{1}(dμ_{α}), the generalized convolution product of f by g is defined by

(f ∗_{α} g)(x) = ∫_{I} ∫_{I} G_{α}(x, y, z)f(y)g(z)dμ_{α}(y)dμ_{α}(z).

In the sequel, the norm of f ∈ L^{p}(dμ_{α}) is assigned by

∥f∥_{p} = (∫_{I} |f(x)|^{p}dμ_{α}(x))^{1/p} .

In [Ga1] and [Ga2], G. Gasper discussed this constructions not only for the ultraspherical polynomials P_{n}, but also for the Jacobi polynomials and shows the following Lemma.

Lemma 2.1 [Ga1,Ga2]. (a) L^{1}(dμ_{α}) endowed with the convolution product ∗_{α} is a semisimple commutative Banach algebra.

(b) For 1 ≤ p, q, r ≤ ∞, and \frac{1}{r} = \frac{1}{p} + \frac{1}{q} − 1, if f ∈ L^{p}(dμ_{α}) and g ∈ L^{q}(dμ_{α}), then f ∗_{α} g ∈ L^{r}(dμ_{α}) and one has ∥f ∗_{α} g∥_{r} ≤ ∥f∥_{p}∥g∥_{q}.

(c) For any f ∈ L^{1}(dμ_{α}), we define the generalized Fourier coefficient of f by

f̂(n) = ω_{n}^{α} ∫_{I} fQ_{n}^{α}dμ_{α}.

then for f, g ∈ L^{1}(dμ_{α}),

{ \atop f ∗_{α} g}(n) = (ω_{n}^{α})^{−1}f̂(n)ĝ(n).

It is important to note ∗_{α} is a distinct convolution for each α ≥ −\frac{1}{2}. That is, for α = 0 we deal with spherical harmonics Q_{n}^{0} which are exactly the Legendre polynomials. The convolution ∗_{0} is easly extended to the point masses by defining:

d(δ_{x} ∗_{0} δ_{y})(z) = G_{0}(x, y, z)dz, δ_{x} ∗_{0} δ_{1} = δ_{x}, and δ_{x} ∗_{0} δ_{−1} = δ_{−x}.

Let 𝔞 be the algebra of the continuous fonctions on I of the form

f = ∑_{n≥0} f̂(n)P_{n}

such that ∥f∥ = ∑_{n≥0} |f̂(n)| < ∞.

Lemma 2.2. The algebra 𝔞 equipped with the norm ∥·∥ is a commutative, semisimple Banach algebra for the simple product of functions.

Proof. To see this it suffices to prove that

∥fg∥ ≤ ∥f∥∥g∥.  (2.2)

This relation can be obtained by the second hypergroup (ℕ_{0}, ⋆_{0}) which was introduced by Hirschmann. Based on the formula

P_{m}(x)P_{n}(x) = ∑_{k=0}^{m+n} h_{m,n,k}P_{k}(x),  (2.3)

where the coefficients satisfy

h_{m,n,k} ≥ 0  and  ∑_{k≥0} h_{n,m,k} = 1.  (2.4)

Hence from (2.3), for f = ∑_{n≥0} f̂(n)P_{n} and g = ∑_{n≥0} ĝ(n)P_{n} the orthogonality between P_{ℓ} and P_{k} implies that

f̂g(k) = \frac{2k + 1}{2} ∫_{I} ∑_{n≥0} ∑_{m≥0} f̂(n)P_{n}(x)ĝ(m)P_{m}(x)P_{k}(x)dx

= \frac{2k + 1}{2} ∑_{n≥0} ∑_{m≥0} f̂(n)ĝ(m) ∫_{I} ∑_{ℓ=0}^{n+m} h_{n,m,ℓ}P_{ℓ}(x)P_{k}(x)dx

= ∑_{n≥0} ∑_{m≥0} h_{n,m,k}f̂(n)ĝ(m).

Finally from (2.4),

∥fg∥ = ∑_{k≥0} |f̂g(k)| = | ∑_{n≥0} ∑_{m≥0} f̂(n)ĝ(m) (∑_{k≥0} h_{n,m,k}) |

≤ (∑_{n≥0} |f̂(n)|) ( ∑_{m≥0} |ĝ(n)|)

which implies (2.2).

Lemma 2.3. For any f ∈ 𝔞, there exist g, h ∈ L^{2}(I), such that f = g ∗_{0} h and such that

∥f∥_{α} = ∥g∥_{2}^{2} = ∥h∥_{2}^{2}.  (2.5)

Conversely if g, h ∈ L^{2}(I), then g ∗_{0} h ∈ 𝔞 and

∥g ∗_{0} h∥_{α} ≤ ∥g∥_{2}∥h∥_{2}.  (2.6)

Proof. For any f ∈ 𝔞, we define ĝ(n) = \sqrt{\frac{2n+1}{2}|f̂(n)|},

ĥ(n) = \{ \begin{matrix}0 & if f̂(n) = 0 \\ \sqrt{\frac{2n+1}{2}} \frac{_{f}(n)}{\sqrt{|_{f}(n)|}} & if f̂(n) ≠ 0, \end{matrix}

g = ∑_{n≥0} ĝ(n)P_{n} and h = ∑_{n≥0} ĥ(n)P_{n}.

From the Parseval’s identity and the fact that f̂(n) = \frac{2}{2n+1} ĝ(n)ĥ(n) = { \atop g ∗_{0} h}(n) one gets (2.5) and consequently ∑_{n≥0} |f̂(n)| < ∞.

Conversely for g, h ∈ L^{2}(I), by Cauchy–Schwarz inequality we have

∑_{n≥0} |{ \atop g ∗_{0} h}(n)| = ∑_{n≥0} |\frac{2}{2n + 1}ĝ(n)ĥ(n)|

≤ (∑_{n≥0} \frac{2}{2n + 1}|ĝ(n)|^{2})^{1/2} (∑_{n≥0} \frac{2}{2n + 1}|ĥ(n)|^{2})^{1/2} ,

which implies (2.6).

3. Harmonic analysis properties of the integral operator S

In the transport equation (1.7), the integral term has the following form

\frac{1}{4π} ∫_{S^{2}} k(ω · ω′)φ(x, ω′ · e)dω′

Expanding k(ω · ω′) in Legendre polynomials series this integral term becomes

\frac{1}{4π} ∑_{j=0}^{∞} k̂_{j} ∫_{S^{2}} P_{j}(ω · ω′)φ(x, ω′ · e)dω′

Now we use a well-known spherical harmonic property of the Legendre polynomials(see e.g. [Mül, Lemma 12])

∫_{S^{2}} P_{j}(ω · ω′)f(ω′ · α)dω′ = λ_{j}P_{j}(ω · α)

for all f ∈ 𝔞, where λ_{j} = 2π ∫_{−1}^{1} f(t)P_{j}(t)dt. Hence, the integral term is

\frac{1}{2} ∑_{j=0}^{∞}k̂_{j} ∫_{−1}^{1} p_{j}(μ)P_{j}(μ′)φ(x, μ′)dμ′

= \frac{1}{2} ∑_{j=0}^{∞} k̂_{j} ∫_{−1}^{1} ∫_{−1}^{1} G_{0}(μ, μ′, μ”)P_{j}(μ”)dμ”φ(x, μ′)dμ′

= \frac{1}{2} ∫_{−1}^{1} ∫_{−1}^{1} G_{0}(μ, μ′, μ”)k(μ”)dμ”φ(x, μ′)dμ′

= \frac{1}{2}k ∗_{0} φ = \frac{1}{2}Sφ.

Some properties of the operator S which are useful in the sequel are gathered in the following Propositions Proposition 3.1. The operator S of the form Sf = k ∗_{0} f is a Hilbert-Schmidt operator.

Proof. From the expression of G_{0}(x, y, z)(see (1.4)), it is not hard to infer that

G_{0} ∈ L^{2}(I^{3}).  (3.1)

Now if we write the operator S as an integral operator

[Sf](x) = ∫_{I} s(x, y)f(y)dy,

the kernel s would be

s(x, y) = ∫_{I} G_{0}(x, y, z)k(z)dz.

Thus from (3.1)

∫∫_{I×I} |s(x, y)|^{2}dxdy ≤ ∫∫_{I×I} (∫_{I} |G_{0}(x, y, z)|^{2}dz) (∫_{I} |k(z)|^{2}dz) dxdy
≤ ∥k∥_{2}^{2}∥G_{0}∥_{L^{2}(I^{3})}^{2} < ∞.

This estimation shows that the operator S is of Hilbert-Schmidt type.

Proposition 3.2. If k ∈ L^{2}(I) and ∥k∥_{2} < 1/\sqrt{2}, then the equation Sf = f has no non-zero solution.

Proof. It follows from Lemma 2.3 that if Sf = k ∗_{0} f = f has a non-zero solution then f ∈ 𝔞 and there exists an integer n such that f̂(n) ≠ 0 and from k̂(n)f̂(n) = \frac{2n+1}{2} f̂(n) it follows that

∫_{I} k(x)P_{n}(x)dx = 1.

But

|∫_{I} k(x)P_{n}(x)dx| ≤ \sqrt{2}∥k∥_{2} < 1.

Hence the equation Sf = f has no non-trivial solution.

4. An one-dimensional inverse problem in transport theory

Let us first consider the purely absorbing one-dimensional problem

(PAP)  \{ \begin{matrix}\frac{∂}{∂x}μφ(x, μ) + σ(x, μ)φ(x, μ) = S(x, μ) \\ φ(−1, μ) = g_{+}(μ), for 0 ≤ μ ≤ 1, \\ φ(1, μ) = g_{−}(μ), for − 1 ≤ μ ≤ 0. \end{matrix}

It is not hard to verify that the solution of (PAP) can be expressed explicitly by

φ(x, μ) = \{ \begin{matrix}e^{\frac{−1}{μ} ∫_{−1}^{x} σ(y,μ)dy}g_{+}(μ) + \frac{1}{μ} ∫_{−1}^{x} e^{− \frac{1}{μ} ∫_{x′}^{x} σ(y,μ)dy}S(x′, μ)dx′, \\ for 0 < μ ≤ 1 \\ e^{\frac{1}{μ} ∫_{x}^{1} σ(y,μ)dy}g_{−}(μ) − \frac{1}{μ} ∫_{x}^{1} e^{− \frac{1}{μ} ∫_{x}^{x′} σ(y,μ)dy}S(x′, μ)dx′, \\ for − 1 ≤ μ < 0. \end{matrix}

It is well-known that for S ∈ L^{2}(I × I), g_{−} ∈ L^{2}((−1, 0), μdμ) and g_{+} ∈ L^{2}((0, 1), μdμ) the problem (PAP) has a unique solution in W^{2} = \{φ ∈ L^{2}(I × I) | μ\frac{∂φ}{∂x} ∈ L^{2}(I × I) }(see e.g. [D-L]). Thus, by virtue of Cessenat’s results [Ce1,Ce2], the albedo operator A maps L^{2}(I, μdμ) into itself. Hence, the above expression infers that if S = 0 and if the function σ(x, μ) = σ(μ) do not vary with x, the albedo operator A determines uniquely σ(μ) by

σ(μ) = \{ \begin{matrix}μ ln \frac{g_{+}(μ)}{h_{+}(μ)} for 0 < μ ≤ 1 \\ μ ln \frac{h_{−}(μ)}{g_{−}(μ)} for − 1 ≤ μ < 0, \end{matrix}

where h_{+}(μ) = [Ag_{−}](μ), for 0 < μ ≤ 1 and h_{−}(μ) = [Ag_{+}](μ) for −1 ≤ μ < 0. When k ∈ L^{2}(I) and k ≠ 0, then we loose the uniqueness of this inverse problem. More precisely we have

Theorem 4.1. Let k be a scattering kernels in L^{2}(I), such that ∥k∥_{2} < 1/(2\sqrt{2}). Given g_{+}(μ), h_{+}(μ) defined for 0 ≤ μ ≤ 1 and g_{−}(μ), h_{−}(μ) defined for −1 ≤ μ < 0, the transport equation

\frac{∂}{∂x}μφ(x, μ) + φ(x, μ) = \frac{1}{2}(k ∗_{0} φ)(x, μ) = \frac{1}{2}[Sφ](x, μ), (x, μ) ∈ I × I (4.1)

with boundary conditions

\{ \begin{matrix}φ(−1, μ) = g_{+}(μ), φ(1, μ) = h_{+}(μ) = [Ag_{−}](μ) for 0 ≤ μ ≤ 1, \\ φ(1, μ) = g_{−}(μ), φ(−1, μ) = h_{−}(μ) = [Ag_{+}](μ) for − 1 ≤ μ ≤ 0. \end{matrix} (4.2)

has a solution in the form

φ(x, μ) = ∑_{m=0}^{∞} \frac{2m + 1}{2}φ̃_{m}(μ)P_{m}(x).  (4.3)

where the Fourier coefficients φ̃_{m}(μ) can be obtained by the recursive relations

Sφ̃_{0} − φ̃_{0} = b_{−}  (4.4)
Sφ̃_{1} − φ̃_{1} = b_{+} − μφ̃_{0}  (4.5)
S(φ̃_{m+1} − φ̃_{m−1}) − (φ̃_{m+1} − φ̃_{m−1}) = −(2m + 1)μφ̃_{m}, m ≥ 1  (4.6)

with

b_{−}(μ) = \{ \begin{matrix}μ[h_{+}(μ) − g_{+}(μ)] for 0 < μ ≤ 1 \\ μ[g_{−}(μ) − h_{−}(μ)] for − 1 ≤ μ < 0, \end{matrix}

and

b_{+}(μ) = \{ \begin{matrix}μ[h_{+}(μ) + g_{+}(μ)] for 0 < μ ≤ 1 \\ μ[g_{−}(μ) + h_{+}(μ)] for − 1 ≤ μ < 0, \end{matrix}

Proof. By integrating the transport equation with boundary conditions over [−1, 1], we get (4.4) and (4.3) gives also that

φ̃_{0}(μ) = ∫_{−1}^{1} φ(x, μ)dx

By fixing the albedo operator A one defines a function b_{−} ∈ L^{2}(I). From Proposition 3.1, the operator S is compact, thus we can apply the Fredholm alternative and assert that the equation (4.4) has a unique solution φ̃_{0} ∈ L^{2}(I). In fact the contrary implies that the homogenous equation Sφ̃ = φ̃ has a nontrivial solution which is in contradiction with Proposition 3.2. By the same argument one can show that the equation (4.5) has a unique solution φ̃_{1} ∈ L^{2}(I), where

φ̃_{1}(μ) = ∫_{−1}^{1} φ(x, μ)xdx

This can be obtained by multiplying (4.1) by x and integrating it over [−1, 1]. For obtaining an arbitary φ̃_{m} ∈ L^{2}(I), we multiply (4.1) by P_{m+1}(x) and intergate it over I, which implies that

∫_{−1}^{1} \frac{∂}{∂x}(μφ(x, μ))P_{m+1}(x)dx + φ̃_{m+1}(μ) = Sφ̃_{m+1}(μ),

where

φ̃_{m}(μ) = ∫_{−1}^{1} φ(x, μ)P_{m}(x)dx.

The integration by part gives

μ [φ(1, μ) − (−1)^{m+1}φ(−1, μ)] − ∫_{−1}^{1} μφ(x, μ)P′_{m+1}(x)dx
= Sφ̃_{m+1} − φ̃_{m+1}.  (4.7)

By substracting

μ [φ(1, μ) − (−1)^{m−1}φ(−1, μ)] − ∫_{−1}^{1} μφ(x, μ)P′_{m−1}(x)dx
= Sφ̃_{m−1} − φ̃_{m−1}

from (4.7) and using the identity

P′_{m+1}(x) − P′_{m−1}(x) = (2m + 1)P_{m}(x),

one obtains (4.6) for m ≥ 1. Once more the uniqueness of the solution of the equations

Sφ̃ − φ̃ = −(2m + 1)μφ̃_{m}.

allows us to construct from (4.4),(4.5) and (4.6) by induction the sequences φ̃_{0}, φ̃_{1}, φ̃_{2}, ⋯ and consequently φ(x, μ) from (4.2).

Corollary 4.2. For σ ≡ 1, if k_{1} and k_{2} are two different scattering kernel in L^{2}(I), such that ∥k_{i}∥_{2} < 1/(2\sqrt{2}) for i = 1, 2. Then there exist two solutions φ_{1} and φ_{2} such that the corresponding albedo operators coincide.

Proof. Given the boundary values

L^{2}(I) ∋ φ_{−}(μ) = \{ \begin{matrix}g_{+}(μ), for 0 ≤ μ ≤ 1, \\ g_{−}(μ), for − 1 ≤ μ ≤ 0 \end{matrix}

and

L^{2}(I) ∋ [Aφ_{−}](μ) = \{ \begin{matrix}h_{+}(μ), for 0 ≤ μ ≤ 1, \\ h_{−}(μ), for − 1 ≤ μ ≤ 0, \end{matrix}

for two different scattering kernels k_{1} and k_{2}, one can construct two solutions for the systems (k_{1}, φ_{−}, Aφ_{−}) and (k_{2}, φ_{−}, Aφ_{−}) in the form

φ_{1}(x, μ) = ∑_{m=0}^{∞} \frac{2m + 1}{2}φ_{m}^{1}(μ)P_{m}(x)

and

φ_{2}(x, μ) = ∑_{m=0}^{∞} \frac{2m + 1}{2}φ_{m}^{2}(μ)P_{m}(x)

satisfies (4.1) with boundary condition (4.2), for which the albedo operators coincide. In fact for 0 < μ ≤ 1, adding (4.4), (4.5) and (4.6) for m = 1, 2 ⋯ one retrieve that

0 = 2μh_{+}(μ) − μ ∑_{m=0}^{∞}(2m + 1)μφ̃_{m}(μ).

In the same manner, by multiplying (4.4) by −1, (4.6) by (−1)^{m}, for m = 1, 2, ⋯ and adding with (4.5), one obtains

0 = 2μg_{+}(μ) − μ ∑_{m=0}^{∞}(−1)^{m}(2m + 1)μφ̃_{m}(μ).

Consequently φ_{1}(1, μ) = φ_{2}(1, μ) = h_{+}(μ) and φ_{1}(−1, μ) = φ_{2}(−1, μ) = g_{+}(μ) and the same holds for −1 ≤ μ ≤ 0.

Acknowledgements

We would like to thank Professor Claude Bardos for many helpful conversations we have had.

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SP2MI–L3MA. Université de Poitiers, Teleport 2 - Boulevard Marie et Pierre Curie, BP 30179, 86962 Futuroscope Chasseneuil, Cedex, France

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