Abstract

Chapter 4
Theory
4.1 Steady-State Theory
In this chapter, we are concerned with theoretically analyzing the infinite-dimensional
steady-state solutions of the Wigner-Poisson equations as the applied voltage drop
V > 0 is varied. We will recast the problem as a fixed point problem of the form
f(x, k) =Z(f)(x, k)+ā(x, k), where the steady-state solutions of the Wigner-Poisson
equations are the fixed points of this map. We will analyze this fixed point map in
the function space
with norm
Kmax
X = {u(x, k)|
−Kmax
|u(x, k ′ )| 2 dk ′ ∈ C [0,L]}
Kmax
uX = max |u(x, k
x∈[0,L] −Kmax
′ )| 2 dk ′
1
2
.
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