Is there a generic term for functions that are proportional to their own Laplacians, and/or for basis sets composed of such functions?
Simple examples would include the well-known Fourier decomposition e.g.: $$\sin(mx+\alpha) \text{ (for one dimension)}$$ $$\sin(mx+\alpha)\sin(ny+\beta) \text{ (for two dimensions)}$$
but I'm also interested in other such functions defined on oddly-shaped domains: anything that satisfies $f(x,y, ...) = k\left(\frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} + ...\right), k \in \mathbb{R} $.
In mechanical engineering these functions correspond to vibrational modes of structures, and where the structure is discretised they correspond to eigenvectors of its stiffness matrix. They're also relevant in other fields e.g. heat flow and gas diffusion, where representing a system state in terms of such basis functions can greatly simplify analysis. I'd be surprised if there wasn't a general term for them, but I'm having difficulty finding one.
(If it makes any difference, I'm particularly interested in those where the proportionality constant is non-positive.)