I'm currently working on a project that relates to the work of Sandstede, School and Wulff (1997) concerning centre manifold reductions.
Background: In this work we consider an Reaction Diffusion Equation of the form
$u_t = u_{xx} + F(u), x \in \mathbb{R}$
which is translation invariant. That is, if $\mathcal{T}_a u(x) = u(x+a)$ then we have
$\frac{\partial}{\partial t} (\mathcal{T}_a u) = \frac{\partial^2}{\partial x^2} (\mathcal{T}_a(u)) + F(\mathcal{T}_a u) = \mathcal{T}_a \frac{\partial ^2 u}{\partial x^2} + \mathcal{T}_a F(u) = \mathcal{T}_a \frac{\partial u}{\partial t}$.
Thus, the dynamical system is equivariant with respect to the group action of SE(1). This is particularly useful in the analysis of travelling waves as the time evolution of the wave can be described as a translate of the initial wave profile. That is, $u(t,x) = T_{a(t)} u^*(x)$ where $u(0,x) = u^*(x)$. Linearizing around the what the authors call a "relative equilibrium" we have the linear operator
$L(w) = \mathcal{A}w + F_u(u^*)w$
where $\mathcal{A} = \partial^2 / \partial x^2$.
Statement: The group action always enforces spectrum on the imaginary axis.
Question: Why is this statement true? Can anyone direct me to a formal proof of this statement or a more general statement regarding the spectrum in the presence of a group action?