Spectral Properties of a Streaming Operator with Diffuse Reflection Boundary Condition

AbstractIn this paper, the spectrum of a streaming operator with diffuse reflection boundary condition arising in transport theory is observed in L1 space. We associate this unbounded operator with bounded integral operators which are determined by the boundary condition. It is shown that the spectrum of the streaming operator is completely determined by these integral operators and consists of countable isolated eigenvalues, each of which has finite algebraic multiplicity. Furthermore, under the assumption that the diffusion on the boundary is of Maxwell type, it is proved that the streaming operator has only one real eigenvalue which is simple and that each of the complex eigenvalues has geometry multiplicity one. A formula for computing these eigenvalues is also derived, by means of which the algebraic multiplicity of the complex eigenvalues is discussed