My question is about linear stability analysis of dynamical systems obtained by discretizing linear(ized) partial differential equations. Consider, $\dot{x}=Ax$, where $x$ is the infinite dimensional vector of coefficients, and $A$ a linear operator in an countably finite basis of a nice function space (using, say Fourier series on a circle, or Hermite on $\mathbb{R}$). Assume that the PDE system is of the type that the full spectrum of $A$ determines the stability.
Usually, physics/engineering papers impose an arbitrary cut-off on number of modes considered, and give analytical/numerical results on the spectrum of this truncation of $A$ to prove/disprove stability.
My question is: How does one make such arguments rigorous ? Are there some standard tricks that can enable one to take into account the neglected modes in a rigorous manner ?
Specifically, I am looking for examples/papers where stability was deduced based on considering the operator acting on a subset of modes, and estimating the effect of "tails" (or neglected modes).
