On the Spectrum of a Class of Integro-Differential Operators in L^p Space

Consider the following integro-differential operator arising from neutron transport in a finite closed convex medium V surrounded by vacuum: A·=-vΩ·grad_r·-v∑(r, v)·+∫_D∫_E k(r, v, Ω, v’, Ω’)·dv’ dΩ’, D(A)={ψ∈L^P(G)|Aψ∈L^P(G); ψ(r, v, Ω)=0 for r∈V and Ω entering V}, where (r, v, Ω)∈ G=V×E×D, E=(0, v_M], 0【V_M【+∞, D is the surface of the unit sphere in R^3 and 1【p【+∞.