Спектральные свойства операторов односкоростного переноса нейтронов

In this master’s thesis, we examine the spectral properties of a particular class of so-called one-speed transport operators. In theoretical and mathematical physics, transport theory stud- ies equations governing the motion of particles, explicitly taking into account the possibility of interaction of the particles under consideration and the medium. In contrast to classical me- chanics and fluid dynamics, transport theory operates on a mesoscopic scale. Upon collision of the particles with the medium, three outcomes are possible: absorption, scattering and fission. If we denote by c the mean number of secondary particles per collision, we obtain the following characterization of the medium:
• The medium is called purely absorbing, if c = 0 • absorbing if c ∈ (0, 1)
• conservative if c = 1
• multiplying if c > 1
Now, rather than describing each particle separately, we deal with distribution functions. The
setup is as follows: Consider a region Ω ⊆ R3 (bounded or unbounded). Suppose this region is
filled out with a medium and that an ensemble of particles, separate from the medium, is given.
Let V ⊆ R3 denote the space of admissible velocities of each particle. A distribution function
is a map f: [0,∞)×Ω×V ⊆ R7 → [0,∞) describing the ensemble in the following way: for
each point in time t ∈ [0, ∞) the expression f (t, x, v) over is the particle density at (x, v), i.e.
f(t,x,v)dvdx = # particles in the region B with velocity in W at time t. BW
How f changes over time is then governed by the Boltzmann equation1:
.
The first term on the right-hand side describes the motion of the ensemble when there is no interaction between the particles or with the medium involved and when no external force is present, since the equation
∂f =−v·∇xf, ∂t
is solved by f (t, x, v) = f (0, x − vt, v) which amounts to the particles moving along straight
lines, which is precisely what is dictated by Newton’s law. This is called the free transport
equation. Now, the expression ∂f describes the contribution to ∂f that interaction between ∂t coll ∂t
the particles and with the medium has. Generally, one can represent ∂f as the difference ∂t coll
between the number of secondary particles gained per collision and the number of particles that are lost in a collision, i.e.
∂f
∂f ∂f
∂t =−v·∇xf+ ∂t
coll
= gains(scattering, fission) − losses(scattering, absorption).
If ∂f is linear in f, we speak of linear transport theory. For this to be a reasonable
∂t
coll
∂t coll
approximation to physical reality, the interaction between particles should be negligible, i.e.
only particle–medium interactions should be of significance. This is the case, when studying neutron transport: While neutrons might well hit nuclei of the medium in which they move, they
1also called transport equation

SPECTRAL PROPERTIES OF SPEED-ONE NEUTRON TRANSPORT OPERATORS 5
are too small and generally too far apart from one another for the probability of them hitting each other to be significant. Usually, ∂f is of the form −σ(x,v)f+ k(x,v,v′)f(t,x,v′)dv′,
∂t coll V
where σ(x,v) is the collision frequency and k(x,v,v′) the scattering kernel. We shall always
assume that the collision frequency depends on the position only, i.e. σ(x, v) = σ(x) and that the collision kernel is separable in the sense that, i.e. k(x, v, v′) = c(x)K(v, v′). We are therefore interested in equations of the form
V
We shall introduce reasonable restrictions on σ, c and k later. As is common in the theory of evolution equations, we would thus like to examine the spectral properties of the operator T given by

V
∂f ′′′
∂t (t, x, v) = −v · ∇xf(t, x, v) − σ(x) + c(x)
K(v, v )f(t, x, v )dv .
Tf(x,v) = −v · ∇xf(x,v) − σ(x) + c(x)
on a suitable function space.
K(v,v′)f(x,v′)dv′
The particular setting we deal with in this thesis is the following: Consider the entire space R3
and that σ(x, y, z) = σ(x). This means we are given a slab of multiplying media, that extends
infinitely in the y- and z-directions. Separating variables, we end up with

V
where μ is the cosine of the angle between the velocity v and the x-axis. See [KLH82] for further physical motivation.