Spectral Properties of the Neutron Transport Equation for Spherical Geometry in the Setting of L¹

The time‐dependent transport equation in a sphere with reflecting boundary conditions is discussed in the setting of L ¹. Some aspects of the spectral properties of the strongly continuous semigroup T(t) generated by the corresponding transport operator A are studied, and it is shown that the spectrum of T(t) outside the disk {λ: |λ|≤exp(−λ∗t)}, where λ∗ is the essential infimum of the total collision frequency σ(r, v), or λ∗=ess inf _r lim  _v→0⁺ σ(r, v), consists of isolated eigenvalues of T(t) with finite algebraic multiplicity, and the accumulation points of σ(T(t))∩{λ: |λ|>exp(−λ∗t)} can only appear on the circle {λ: |λ|=exp(−λ∗t)}. Consequently, the asymptotic behavior of the time‐dependent solution is obtained.

Keywords:

• Neutron transport
• Transport equation
• Spherical geometry