Singular Transport Equations With A Reflecting Boundary Conditions In Slab Geometry

This paper researches singular transport equations for anisotropic continuousenergy homogeneous in slab geometry with reflecting boundary condition by themethods of mordern analysis, such as functional analysis ,the theory of operator andthe theory of semigroup, and it has got a series of new results about the spectrum ofthe transport operator relating to the transport equations. The main results are showedbelow:1. It discusses singular tsansport equations for anisotropic continuous energyhomogenous with reflecting boundary conditions in slab geometry .It proves thatthe singular transport operator generates a strongly continuous C₀ semigroupV(t)(t≥0) and the weak compactness properties of the second-order remainedterm of the Dyson-Phillips expansion for the C₀ semigroup V(t)(t≥0) in L₁space;2. It discusses singular transport equations for anisotropic continuous energyhomogenous with reflecting boundary conditions in slab geometry.It proves that thesingular transport operator generates a strongly continuous C₀ semigroupV(t)(t≥0) and the compactness properties of the second-order remained term ofthe Dyson-Phillips expansion for the C₀ semigroup V(t)(t≥0) in L_p(l＜p＜∞)space:3. It investigates the spectrum of the singular transport operator only consist of,at most, finitely many isolated eigenvalues which have a finite algebraic multiplicityin tripΓboth in L₁ space and in L_p(1＜p＜∞) space, farthmore it proves that thetransport operator has enginvalue.