UDC 517.947.32

MATHEMATICS

M. V. MASLENNIKOV

THE CHARACTERISTIC EQUATION OF THE THEORY OF RADIATION TRANSFER

(Presented by Academician M. V. Keldysh on 16 IX 1965)

The problem of determining the asymptotics of the spatial-angular distribution of radiation in the depth of a thick plane layer of matter reduces to the study of the equation \((^{1})\)

\[ (1+k\overrightarrow{\omega}\mathbf{n})\Phi(\overrightarrow{\omega})=\hat g\Phi(\overrightarrow{\omega}). \tag{1} \]

Here \(k\) is a parameter, \(\overrightarrow{\omega}\) is a vector running over the unit sphere \(\Omega\) of three-dimensional Euclidean space; \(\mathbf{n}\) is a fixed vector from \(\Omega\); \(\hat g\) is an integral operator:

\[ \hat g f(\overrightarrow{\omega})=\int_{\Omega} g(\overrightarrow{\omega\omega'}) f(\overrightarrow{\omega'})\,d\omega'. \]

The kernel \(g(\overrightarrow{\omega\omega'})\) is determined by specifying the scattering indicatrix \(g(\mu)\), \(\mu\in[-1,1]\), and depends on the scalar product \(\overrightarrow{\omega\omega'}\) of the vectors \(\overrightarrow{\omega}\) and \(\overrightarrow{\omega'}\) from \(\Omega\);

\[ \int_{\Omega}\cdots\,d\omega' \]

is the integral with respect to Lebesgue measure on \(\Omega\).

We shall be interested in the question of for which values of \(k\) there exist, and how the nontrivial solutions of (1) are arranged; and how the solutions of the nonhomogeneous equation corresponding to (1) depend on \(k\).

We shall assume that

\[ 1)\quad g\in L_2(\Omega);\qquad g(\mu)\geqslant 0 \text{ on } [-1,1];\qquad 0<2\pi\int_{-1}^{1} g(\mu)\,d\mu\leqslant 1. \]

This condition makes it possible to expand \(g(\mu)\) in a Legendre-polynomial series convergent in the mean:

\[ g(\mu)=\sum_{n=0}^{\infty}\frac{2n+1}{4\pi}g_nP_n(\mu),\qquad 0<g_0\leqslant 1, \]

\[ |g_n|<g_0 \quad \text{for } n=1,2,\ldots . \]

Let \(L_2(\Omega)\) and \(C(\Omega)\) be the spaces of functions on \(\Omega\) that are respectively square-summable and continuous;

\[ (f_1,f_2)=\int_{\Omega} f_1(\overrightarrow{\omega})\overline{f_2(\overrightarrow{\omega})}\,d\omega \]

is the scalar product in \(L_2(\Omega)\);

\[ \|f\|=(f,f)^{1/2},\quad f\in L_2(\Omega), \]

and

\[ \|f\|_{C(\Omega)}=\sup_{\overrightarrow{\omega}\in\Omega}|f(\overrightarrow{\omega})|,\quad f\in C(\Omega). \]

Denote by \(\vartheta\) the zero of \(L_2(\Omega)\), and by \(Z_0\) the complex \(k\)-plane with cuts along the real axis from \(-\infty\) to \(-1\) and from \(1\) to \(\infty\).

Theorem 1. \(\hat g\) is a linear bounded self-adjoint operator mapping \(L_2(\Omega)\) into itself, with norm \(\|\hat g\|=g_0\). \(\hat g(L_2(\Omega))\subset C(\Omega)\); for \(f\in L_2(\Omega)\)

\[ \|\hat g f\|_{C(\Omega)}\leqslant \|f\|\left(2\pi\int_{-1}^{1} g^2(\mu)\,d\mu\right)^{1/2}; \]

as an operator acting from \(L_2(\Omega)\) to \(C(\Omega)\), \(\hat g\) is completely continuous.

Define, for \(k\in Z_0\),

\[ \hat U(k)f(\overrightarrow{\omega})=(1+k\overrightarrow{\omega}\mathbf{n})^{-1}\hat g f(\overrightarrow{\omega}). \]

\[ \hat U(k): L_2(\Omega)\to \]

\(\to L_2(\Omega)\), \(\hat U(k)\) is completely continuous. Let \(\sigma(\hat U(k))\) be its spectrum. Obviously, (1) is solvable if and only if \(1 \in \sigma(\hat U(k))\).

It can be shown that for \(k \in (-1,1)\) and sufficiently large \(p\)

\[ \inf_{\vec\omega\in\Omega} \hat U^p(k)f(\vec\omega)>0 \]

for every nonnegative \(f(\vec\omega)\). Relying on the theory of cones in Banach spaces \((^2)\), from this we obtain:

Theorem 2. Let \(k\in(-1,1)\), \(M(k)=\sup\{|\lambda|\,|\,\lambda\in\sigma(\hat U(k))\}\). Then:

I. \(M(k)>0,\ M(k)\in\sigma(\hat U(k))\).

II. If \(\lambda\in\sigma(\hat U(k))\), \(\lambda\ne M(k)\), then \(|\lambda|<M(k)\).

III. \(M(k)\) is a simple eigenvalue of \(\hat U(k)\).

IV. There exists exactly one function \(\Phi_k\in L_2(\Omega)\) possessing the following properties:

\[ \hat U(k)\Phi_k=M(k)\Phi_k,\qquad (\Phi_k,1)=4\pi. \]

Moreover,

V.

\[ \inf_{\vec\omega\in\Omega}\Phi_k(\vec\omega)>0. \]

VI. If \(x\ne\vartheta\), \(x\in C(\Omega)\), \(x(\vec\omega)\ge0\) on \(\Omega\), and \(x\) is an eigenfunction of the operator \(\hat U(k)\), then there exists \(a>0\) such that \(x=a\Phi_k\).

In order to prove the existence of nontrivial solutions of (1), it is now sufficient to establish the solvability of the equation \(M(k)=1\) in the interval \(-1<k<1\).

Let us note that \(\hat U(k)\) is an analytic function for \(k\in Z_0\), and for \(k_0\in(-1,1)\), according to Theorem 2, \(M(k_0)\) is a simple isolated eigenvalue of \(\hat U(k_0)\). This permits one to apply to the analysis of the behavior of \(M(k)\) and \(\Phi_k\) in a neighborhood of the point \(k_0\) the theory of analytic perturbations of operators (for example \((^6)\)), which gives:

Theorem 3. There exist an open set \(G_0\subset Z_0\), a numerical function \(\lambda(k)\), and a function \(e(k)\) with values in \(L_2(\Omega)\), defined on \(G_0\), such that:

I. \((-1,1)\subset G_0\).

II. \(\lambda(k)\) and \(e(k)\) are holomorphic in \(G_0\).

III. For \(k\in G_0\), \(\|e(k)\|>0\), \(\lambda(k)\ne0\), \(\hat U(k)e(k)=\lambda(k)e(k)\).

IV. For \(k\in(-1,1)\), \(\lambda(k)=M(k)\), \(e(k)=\Phi_k\).

According to Theorem 3, in a neighborhood of each point \(k_0\in G_0\), \(e(k)\) is expanded in a power series in \((k-k_0)\), convergent in the metric of \(L_2(\Omega)\). It turns out that the coefficients of this series are elements of \(C(\Omega)\), and the series itself, for sufficiently small \((k-k_0)\), converges uniformly with respect to \(\vec\omega\in\Omega\). This, in particular, means that for \(k_1\in(-1,1)\), uniformly with respect to \(\vec\omega\in\Omega\),

\[ \lim_{k=k_1}\Phi_k(\vec\omega)=\Phi_{k_1}(\vec\omega). \]

Using the special form of the kernel of the operator \(\hat g\) and the analyticity of \(M(k)\) and \(\Phi_k\), one can obtain the following results:

Theorem 4. If \(k\in(-1,1)\), then \(M(-k)=M(k)\); \(\Phi_k(-\vec\omega)=\Phi_{-k}(\vec\omega)\) for \(\vec\omega\in\Omega\); \(M(0)=g_0\); \(\Phi_0(\vec\omega)\equiv1\) on \(\Omega\); \(dM(k)/dk>0\) for \(k\in(0,1)\); \(dM(k)/dk|_{k=0}=0\); \(\lim_{k=1-0}M(k)=\infty\). There exists exactly one \(\lambda_0\in[0,1)\) such that \(M(\lambda_0)=1\). If \(g_0=1\), then \(\lambda_0=0\); if \(g_0<1\), then \(\lambda_0>0\).

Theorem 5. Let \(k\in(-1,1)\), \(x\in L_2(\Omega)\setminus\{\vartheta\}\),

\[ (1+k\vec\omega\mathbf n)x(\vec\omega)=\hat g x(\vec\omega). \]

I. If \(x(\vec\omega)\ge0\) on \(\Omega\), then either \(k=\lambda_0\) and \(x=a_1\Phi_{\lambda_0}\), or \(k=-\lambda_0\) and \(x=a_2\Phi_{-\lambda_0}\), \((a_{1,2}=\mathrm{const}>0)\).

II. If \(|k|\ne\lambda_0\) and \(x(\vec\omega)\) is a real function, then \(|k|>\lambda_0\) and \(x(\vec\omega)\) assumes on \(\Omega\) both positive and negative values.

Denote by \(\mathfrak N\) the set of those \(k\in Z_0\) for each of which (1) admits a nontrivial solution \(\Phi\in L_2(\Omega)\). Then \(\mathfrak N=\mathfrak N_0\cup(-\mathfrak N_0)\), \(\mathfrak N_0\subset[\lambda_0,1)\), \(\lambda_0\in\mathfrak N_0\), and \(\mathfrak N_0\) is at most countable. If \(\mathfrak N_0\) is infinite, then \(1\) is the only limit point of \(\mathfrak N_0\). For \(k\in Z_0\) define the linear bounded operator \(\widehat W(k)\), acting in \(L_2(\Omega)\), by the rule:
\[ \widehat W(k)x=(1+k\vec\omega\mathbf n)x-\hat g x,\qquad x\in L_2(\Omega). \]
If \(k\in Z_0\setminus\mathfrak N\), then \(\widehat W(k)\) implements a one-to-one mapping of \(L_2(\Omega)\) onto itself. For \(k\in\mathfrak N\) put
\[ H_k=\{x\mid x\in L_2(\Omega),\ \widehat W(k)x=0\}. \]
Then \(H_k\) is a finite-dimensional subspace of \(L_2(\Omega)\). Let \(p_k\) be the dimension of \(H_k\). Then \(p_{\pm\lambda_0}=1\), and for all \(k\in\mathfrak N\), \(p_k=p_{-k}\geqslant1\). In the subspaces \(H_k\), with \(k\in\mathfrak N\), \(k\ne0\), bases \(\{\psi_{kp}\mid p=1,2,\ldots,p_k\}\) may be chosen in such a way that: a) \(\psi_{kp}(\vec\omega)\) is a real continuous function of \(\vec\omega\in\Omega\); b) \((\vec\omega\mathbf n,\psi_{kp}^{\,2})=-\operatorname{sgn}k\) for all \(p=1,2,\ldots,p_k\), and \(((\vec\omega\mathbf n)\psi_{kp},\psi_{kp'})=0\) for \(1\leqslant p<p'\leqslant p_k\); c) for all \(k\in\mathfrak N\), \(k\ne0\), and \(p=1,2,\ldots,p_k\),
\[ \psi_{(-k)p}(\vec\omega)=\psi_{kp}(-\vec\omega); \]
d) if \(g_0<1\), then
\[ \psi_{\lambda_0 1}=\left|(\vec\omega\mathbf n,\Phi_{\lambda_0}^{\,2})\right|^{-1/2}\Phi_{\lambda_0}. \]

Let us now turn to the inhomogeneous equation corresponding to (1).

Theorem 6. Let \(G\) be an open set, \(G\subset Z_0\), and let \(a(k)\) be a function with values in \(L_2(\Omega)\), holomorphic in \(G\). For each \(k\in G\setminus\mathfrak N\) there exists exactly one solution \(\psi=\psi(k)\in L_2(\Omega)\) of the equation
\[ (1+k\vec\omega\mathbf n)\psi(k)=\hat g\psi(k)+a(k). \]
This solution is \(\psi(k)=W^{-1}(k)a(k)\). It is a single-valued analytic function of \(k\) in \(G\setminus\mathfrak N\). Let \(k_0\in G\cap\mathfrak N\). Then there exists \(r>0\) such that
\[ S(k_0,r)\equiv\{k\mid |k-k_0|<r\}\subset G,\qquad S(k_0,r)\cap\mathfrak N=\{k_0\}. \]
In \(S(k_0,r)\setminus\{k_0\}\) the solution \(\psi(k)\) can be represented as follows:

a) if \(g_0=1\) and \(k_0=0\), then
\[ \psi(k,\vec\omega) = -\frac{3}{4\pi}\frac{1-g_1}{k^2}(a(0),1) +\frac{3}{4\pi k}\bigl[(a(0),1)\vec\omega\mathbf n -(1-g_1)(a'(0),1)+(a(0),\vec\omega\mathbf n)\bigr] +u(k,\vec\omega); \]

b) if \(k_0\ne0\), then
\[ \psi(k,\vec\omega) = -\frac{\operatorname{sgn}k}{k-k_0} \sum_{p=1}^{p_{k_0}}(a(k_0),\psi_{k_0p})\psi_{k_0p}(\vec\omega) +u(k,\vec\omega). \]

In both cases \(u(k)\) is a function of \(k\) with values in \(L_2(\Omega)\), holomorphic in \(S(k_0,r)\).

Let us return to the operator \(\widehat U(k)\). It turns out that separation of the polar and azimuthal coordinates of the vector \(\vec\omega\) occurs in all eigenfunctions of \(\widehat U(k)\). In order to describe this phenomenon accurately, we introduce the following notation. Let
\[ \Delta=[-1,1]\times[0,2\pi] \]
be a rectangle in the \((\mu,\varphi)\)-plane, and let \(\sigma\) be an arbitrary orthonormal basis of three-dimensional space with third vector \(\mathbf n\):
\[ \sigma=\{\mathbf e_1,\mathbf e_2,\mathbf n\}. \]
Let
\[ \vec\psi_\sigma(\mu,\varphi) = \sqrt{1-\mu^2}\cos\varphi\,\mathbf e_1 +\sqrt{1-\mu^2}\sin\varphi\,\mathbf e_2 +\mu\mathbf n, \qquad (\mu,\varphi)\in\Delta, \]
and let \(f\in C(\Omega)\). Fix \(\vec\omega\in\Omega\) and \((\mu,\varphi)\in\vec\psi_\sigma^{-1}(\vec\omega)\). For each integer \(s\), put
\[ \widehat P^{(s)}f(\vec\omega) = \frac{1}{2\pi}e^{is\varphi}\int_0^{2\pi} e^{-is\varphi'}f\bigl(\vec\psi_\sigma(\mu,\varphi')\bigr)\,d\varphi'. \]
It turns out that \(\widehat P^{(s)}(C(\Omega))\subset C(\Omega)\), and moreover the \(\widehat P^{(s)}\) can be extended by continuity to all of \(L_2(\Omega)\). As an operator acting in \(L_2(\Omega)\), each \(\widehat P^{(s)}\) is a projection operator. \(\widehat P^{(s)}\widehat P^{(s')}=\delta_{ss'}\widehat P^{(s)}\) for all integers \(s\) and \(s'\). Finally, each \(\widehat P^{(s)}\) commutes with \(\hat g\).

We now note that the function of three variables \((\mu,\mu',\varphi')\)

\[ g\left(\mu\mu' + \sqrt{1-\mu^2}\sqrt{1-\mu'^2}\cos\varphi'\right) \]

is measurable and square-summable on the parallelepiped \([-1,1]\times[-1,1]\times[0,2\pi]\). This makes it possible to define, for integral \(s\), functions \(g_s(\mu,\mu')\) from \(L_2([-1,1]\times[-1,1])\) by the formula

\[ g_s(\mu,\mu')= \int_0^{2\pi} g\left(\mu\mu' + \sqrt{1-\mu^2}\sqrt{1-\mu'^2}\cos\varphi'\right) e^{is\varphi'}\,d\varphi'. \]

Let us associate with each \(g_s\) the operator \(\hat g_s\) acting in \(L_2(-1,1)\):

\[ \hat g_s f(\mu)=\int_{-1}^{1} g_s(\mu,\mu')\, f(\mu')\,d\mu' \quad\text{for } f\in L_2(-1,1). \]

Theorem 7. Let \(k\in Z_0\) and \(\lambda\in\sigma(\hat U(k))\), \(\lambda\ne 0\). If \(x\in L_2(\Omega)\), \(\hat U(k)x=\lambda x\), then among the functions \(x_s=\hat P^{(s)}x\) there is only a finite number of functions \(x_{s_\nu}\), \(\nu=1,2,\ldots,\nu_0\), distinct from zero. Moreover,

\[ \hat U(k)x_{s_\nu}=\lambda x_{s_\nu} \quad\text{and}\quad x=\sum_{\nu=1}^{\nu_0} x_{s_\nu}. \]

There exist functions \(a_\nu\in L_2(-1,1)\) such that for all \(\vec\omega\in\Omega\)

\[ x_{s_\nu}(\vec\omega)=a_\nu(\mu)e^{is_\nu\varphi}, \quad (\mu,\varphi)\in\vec\psi^{-1}_{\sigma}(\vec\omega). \]

The functions \(a_\nu(\mu)\) satisfy the equations

\[ \lambda(1+k\mu)a_\nu(\mu)=\hat g_{s_\nu}a_\nu(\mu), \quad \nu=1,2,\ldots,\nu_0, \]

and depend continuously on \(\mu\in[-1,1]\).

The converse is also true: if \(a_\nu\) satisfy the equations from Theorem 7, then any linear combination of the corresponding \(x_{s_\nu}\) is an eigenfunction of \(\hat U(k)\) belonging to \(\lambda\).

Theorem 8. Let \(k\in(-1,1)\). For all integral \(s\),

\[ \hat P^{(s)}\Phi_k=\delta_{s0}\Phi_k. \]

There exists exactly one function \(\varphi_k(\mu)\) of the variable \(\mu\in[-1,1]\), \(\varphi_k\in L_2(-1,1)\), such that, for \(\vec\omega\in\Omega\),

\[ \Phi_k(\vec\omega)=\varphi_k(\vec\omega\mathbf n). \]

Moreover,

\[ \varphi_k\in C(-1,1),\qquad M(k)(1+k\mu)x(\mu)=\hat g_0 x(\mu), \qquad \int_{-1}^{1}\varphi_k(\mu)\,d\mu=2. \]

The equations

\[ M(k)(1+k\mu)\varphi_k(\mu)=\hat g_s\varphi_k(\mu) \]

for \(s\ne 0\) admit only the trivial solution \(x(\mu)\equiv 0\).

The connection of \(\varphi_k\) with \(\Phi_k\) makes it possible to elucidate many properties of \(\varphi_k\). Deeper results can be obtained by a direct study of the equation for \(\varphi_k\) \((^3\text{--}^5)\).

Received
14 IX 1965

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