On the spectrum of the transport operator with abstract boundary conditions in slab geometry
This paper is concerned with the spectral properties of the operator \[ A_H\psi(x,\xi)= -\xi{\partial \psi\over \partial x}-\sigma(\xi) \psi (x,\xi)+ \int^1_{-1} \kappa(\xi,\xi') \psi(x,\xi') d\xi'=T_H\psi+ K\psi \] with the boundary conditions \[ {\psi^i_1 \choose\psi^i_2} =H{\psi^o_1 \choose \psi^o_2} =\left(\begin{matrix} H_{11} & H_{12}\\ H_{21} & H_{22} \end{matrix} \right) {\psi^o_1 \choose\psi^o_2}, \] where \(\psi^i_1, \psi^i_2,\psi^o_1,\psi^o_2\) are the values of \(\psi\) on the sets \(D^i_1=\{-a\} \times[0,1]\), \(D^i_2=\{a\} \times [-1,0]\), \(D^o_1=\{-a\} \times[-1,0]\), \(D^o_2=\{a\} \times[0,1]\), respectively. \(H\) is an operator on \(X^o_p=L_p(D^o_1; |\xi|d\xi)\oplus L_p(D^o_2; |\xi |d\xi)\) to \(X^i_p=L_p(D^i_1;|\xi|d\xi)\oplus L_p(D^i_2; |\xi |d\xi)\), \(p\in[1,\infty)\). \(K\) is a bounded operator on \(X_p=L_p (D;dxd \xi)\), where \(D=[-a,a] \times[-1,1]\). Let \(\lambda^*-\lim\inf_{|\xi |\to 0}\sigma(\xi)\). The first result is that if the integral operator with the kernel \(\kappa(\xi,\xi')/ |\xi|\) is bounded on \(X_p\) and \(H\) is bounded from \(X^o_p\) to \(L_p[(-1,0);d\xi]\times L_p[(0,1); d\xi]\) with \(\|H\|<1\), then \(\sigma(A_H) \cap\{\lambda: \text{Re} \lambda >-\lambda^*\}= \emptyset\) for \(a\) small enough. The next result is concerned with the eigenvalues of \(A_H\). Let \[ \begin{aligned} &\widehat K,B:L_p\bigl([-1,1];d \xi\bigr) \to L_p\bigl([-1,1]; d\xi\bigr)\\ &\widehat K\varphi(\xi)= \int^1_{-1}\kappa (\xi,\xi')\psi (\xi')d\xi',\\ & B\varphi(\xi)=-\sigma (\xi) \psi(-\xi)+ \widehat K\varphi(\xi). \end{aligned} \] The intersection of the point spectrum of \(A_H\) (resp. \(B)\) and \(\{ \lambda: \text{Re} \lambda> -\lambda^*\}\) is denoted by \(P(A_H)\) (resp. \(P(B)) \). Suppose that \(\widehat K\) is compact, \(H_{11}= H_{22}=0\) and \(H_{i,j}\leq\text{Id}\), \(i\neq j\). Then, \(P(B)=\emptyset\) implies \(P(A_H)= \emptyset\) for any \(a> 0\), and the leading eigenvalue of \(A_H\leq\) the leading eigenvalue of \(B\leq-\lambda^*+r_\sigma(K)\), where \(r_\sigma(K)\) denotes the spectral radius of \(K\). A number of results on the irreducibility of the semigroup \(e^{tA_H}\) and the strict monotonicity of the leading eigenvalue are also established.
scientific article; zbMATH DE number 1563445
scientific article