Exponentially weighted L2 space ($L_{\gamma }^2=\{u:e^{\gamma x}u\in L^2\}$)

The exponentially weighted Hilbert space consists of measurable functions with ${\int }_{\mathbb{R}}|u(x){|}^2{e}^{2\gamma x}dx<\infty$. Multiplication by $e^{\gamma x}$ is a unitary map onto L2 space. Requiring unweighted square integrability as well produces an intersection that is incomplete under the weighted norm alone. For example, $1_{[-N,0]}$ is a weighted Cauchy sequence for $\gamma >0$, but its weighted limit $1_{(-\infty ,0]}$ is not in unweighted L2 space.

The linearization at a travelling wave $w_c$ is $A={\partial }_z^2+c{\partial }_z+k-2w_c$. Conjugation by $e^{\gamma z}$ changes this to ${\partial }_z^2+(c-2\gamma ){\partial }_z+{\gamma }^2-c\gamma +k-2w_c$. The choice $\gamma =c/2$ removes the first derivative and makes the operator self-adjoint. Its two limiting potentials are $-k-c^2/4$ and $k-c^2/4$, so essential-spectrum invariance under relatively compact perturbations gives ${\sigma }_{\mathrm{ess}}=(-\infty ,k-c^2/4]$. This lies strictly below zero for $c>2\sqrt{k}$ but touches zero at the critical speed. Stability of this essential spectrum alone is not a general nonlinear stability theorem.