Fine Spectral Analysis of Isotropic Partly Elastic Neutron Transport Operators

Abstract

This paper deals with spectral theory of a new class of neutron transport operators involving collision operators of the form $K=K_{i}+K_{e}$ where $K_{i}$ (resp. $K_{e}$) describes the inelastic (resp. elastic) collisions of neutrons with the host material. We give a fine analysis of their asymptotic point spectra for isotropic space homogeneous cross sections in bounded geometries. We provide a new formalism relying on spectral analysis of some non compact symmetrizable operators arising in transport theory. Additional results on essential spectra are also given.

Appendix

For reader’s convenience, we recall here some functional analytic results from [12, 13] on symmetrizable operators on Hilbert spaces (see also [7, 17, 18] for classical results). Let $H$ be a complex Hilbert space with scalar product $(.,.)$ and norm

$$ \vert x \vert :=\sqrt{(x,x)}. $$

For any bounded linear operator $O\in \mathcal{L}(H)$, we denote its operator norm by $\Vert O \Vert _{\mathcal{L}(H)}$ or more simply by $\Vert O \Vert$ for the sake of brevity. Let $S\in \mathcal{L}(H)$ be a nonnegative (not necessarily injective) self-adjoint operator on $H$. An operator $G\in \mathcal{L}(H)$ is said to be (left)-symmetrizable by $S$ if $SG$ is self-adjoint. For brevity, we say that $G$ is symmetrizable by $S$. If in addition

$$ Gx=\lambda x \quad (\lambda \neq 0\ \text{and}\ x\neq 0)\Rightarrow Sx \neq 0 $$

then we say that $G$ is fully symmetrizable by $S$; in this case, it is easy to see that the eigenvalues of $G$ must be real since $Sx\neq 0$ amounts to $(Sx,x)>0$ ($S$ and $\sqrt{S}$ have the same kernel). A symmetrizable operator $G$ is strongly symmetrizable by $S$ if

$$ Ker(S)\subset Ker(G); $$

in this case, it is also easy to see that $G$ is necessarily fully symmetrizable by $S$. A sufficient condition of strong symmetrizability is e.g. the existence of a positive constant $c$ such that

$$ \vert Gx \vert \leq c \vert \sqrt{S}x \vert ,\quad x \in H; $$

(37)

this assumption is satisfied in the context of neutron transport theory (see (10)). We point out that under (37) the spectrum of $G$ is real (see [12, Lemma 3 and Theorem 3(ii)]) (in general, the spectrum of a symmetrizable operator need not be real, see e.g. [17]); if in addition $H$ is ordered by a generating cone $H_{+}$ and if $G$ is positive (in the sense that $G$ leaves invariant the positive cone $H_{+}$) then

$$ r_{\sigma }(G)=\sup_{ \{ x\in H;\ (Sx,x)=1 \} }(SGx,x) $$

(38)

(see [12, Corollary 5]).

Definition 37

For any $G\in \mathcal{L}(H)$ symmetrizable by $S$, we define

$$ \alpha^{+}(G):=\inf \bigl\{ \lambda >0;\ \sigma (G)\cap ( \lambda ,+ \infty ) \text{ consists at most of discrete spectrum} \bigr\} $$

and

$$ \alpha^{-}(G):=\sup \bigl\{ \lambda < 0;\ \sigma (G)\cap ( - \infty , \lambda ) \text{ consists at most of discrete spectrum} \bigr\} $$

where “discrete spectrum” refers to isolated eigenvalues with finite algebraic multiplicity.

In general

$$ \max \bigl\{ \alpha^{+}(G),-\alpha^{-}(G) \bigr\} \leq r_{ess}(G) $$

where $r_{ess}(G)$ is the essential spectral radius of $G$ (i.e. the radius of the smallest closed disc with center 0 containing the essential spectrum of $G$). However, under (37) the spectrum of $G$ is real and

$$ \max \bigl\{ \alpha^{+}(G),-\alpha^{-}(G) \bigr\} =r_{ess}(G). $$

We recall W.T. Reid’s inequality:

Lemma 38

([18, Theorem 2.1 and Corollary 1])

Let $G\in \mathcal{L}(H)$ be symmetrizable by $S$. Then, for all $x,y\in H$,

$$ \bigl\vert (x,SGy) \bigr\vert \leq \Vert G \Vert \sqrt{(x,Sx)} \sqrt{(y,Sy)}. $$

(39)

In particular

$$ \bigl\vert (x,SGx) \bigr\vert \leq \Vert G \Vert (x,Sx). $$

(40)

By restricting ourselves to the positive spectrum of $G$ we have:

Theorem 39

([13, Theorem 9])

Let $H$ be a complex Hilbert space and let $G\in \mathcal{L}(H)$ be fully symmetrizable by a nonnegative self-adjoint operator $S\in \mathcal{L}(H)$. We assume that:

$$ \beta_{1}^{+}:=\sup_{ \{ x\in H;\ (Sx,x)=1 \} }(SGx,x)> \alpha^{+}(G). $$

Then:

• (i) $\beta_{1}^{+}$ is an isolated eigenvalue of $G$ associated to an eigenvector $e_{1}^{+}$ such that $(Se_{1}^{+}, e_{1}^{+})=1$.

• (ii) Let

  $$ \beta_{2}^{+}:= \sup_{ \{ x\in H;\ (Sx,x)=1,(Sx,e_{1}^{+})=0 \} }(SGx,x). $$

  If $\beta_{2}^{+}>\alpha^{+}(G)$ then $\beta_{2}^{+}$ is an isolated eigenvalue of $G$ associated to an eigenvector $e_{2}^{+}$ such that $(Se_{2}^{+},e_{2}^{+})=1$ and $(Se_{2}^{+},e_{1}^{+})=0$. More generally, if we have built eigenvalues

  $$ \beta_{1}^{+},\beta_{2}^{+},\ldots, \beta_{n-1}^{+}>\alpha^{+}(G) $$

  whose corresponding eigenvectors $e_{1}^{+},e_{2}^{+},\ldots,e_{n-1}^{+}$ satisfy

  $$ \bigl(Se_{i}^{+},e_{j}^{+}\bigr)= \delta_{ij},\ i,j=1,2,\ldots,n-1 $$

  and if

  $$ \beta_{n}^{+}:= \sup_{ \{ x\in H;\ (Sx,x)=1,(Sx,e_{j}^{+})=0\ \forall j=1,2,\ldots,n-1 \} }(SGx,x)> \alpha^{+}(G) $$

  then $\beta_{n}^{+}$ is also an isolated eigenvalue of $G$ associated to an eigenvector $e_{n}^{+}$ such that $(Se_{n}^{+},e_{n}^{+})=1$ and $(Se_{n}^{+},e_{j}^{+})=0,\ \forall j=1,2,\ldots,n-1$.

We recall more practical $\sup \inf$ characterizations of the positive eigenvalues of $G$ located outside its essential disc. Let $\mathcal{V}_{n}$ denotes the class of $n$-dimensional subspaces $E_{n}$ of $\overline{\operatorname{Im}(S)}$. We define the sequence of reals

$$ \rho_{n}^{+}=\sup_{E_{n}\in \mathcal{V}_{n}}\inf _{x\in E_{n},(Sx,x)=1}(SGx,x)\quad (n\in \mathbb{N} ). $$

Theorem 40

([13, Theorem 14])

Let the assumptions of Theorem 39 be satisfied. Let $\beta_{1}^{+}\geqslant \beta _{2}^{+}\geqslant \cdots\geqslant \beta_{n}^{+}\geqslant ..>\alpha^{+}(G)$ be the eigenvalues of $G$ (repeated according to their geometrical multiplicities) located on the right of $\alpha^{+}(G)$. Then

$$ \beta_{n}^{+}=\rho_{n}^{+}. $$

Theorem 41

([13, Theorem 34])

Let (37) be satisfied. If $G$ has exactly $m$ eigenvalues $>\alpha^{+}(G)$ (counted according to their multiplicities) then $\rho_{n}^{+}=\alpha^{+}(G)$ for all $n>m$.

Finally, we recall that the Schechter essential spectrum of a general closed operator $B$ on $H$ is defined by

$$ \sigma_{ess}(B):=\bigcap_{K\in \mathcal{K}(H)}\sigma (B+K) $$

(41)

where $\mathcal{K}(H)$ denotes the space of linear compact operators on $H$ (or equivalently by the complement of the Fredholm domain of $B$ with index zero, see [2, Theorem 1.4, p. 415]). In particular, if $\lambda \in \mathbb{C}$ admits a singular Weyl sequence $( x_{n} ) _{n} \subset H$, i.e.

$$ \vert x_{n} \vert =1,\quad x_{n}\rightharpoonup 0 \quad (\textit{weak} \text{ convergence}) \text{ and}\quad \vert Bx_{n}-\lambda x_{n} \vert \rightarrow 0 $$

(42)

then $\lambda \in \sigma_{ess}(B)$; this follows from the fact that, for any singular Weyl sequence $( x_{n} ) _{n}$, $\vert Kx _{n} \vert \rightarrow 0$ for any $K\in \mathcal{K}(H)$. We call Weyl essential spectrum of $B$ the set of $\lambda^{\prime }s$ admitting a singular Weyl sequence and denote it by $\sigma_{W}(B)$. Thus $\sigma_{W}(B)\subset \sigma_{ess}(B)$ and the inclusion is proper in general.

Theorem 42

([13, Theorem 33])

Let (37) be satisfied. Then the Schechter essential spectrum of $G$ is equal to its Weyl essential spectrum.