Monday, 25 December 2006

sp.spectral theory - Is there a name for this differential operator and/or its corresponding spectrum?

Let $mathcal{M}$ be a real, compact, orientable manifold and let $X$ be a vector field on $mathcal{M}$. Consider the functional



$$E(f) = int_{mathcal{M}} X_p(f)^2 dV$$



where $X_p(f)$ is the directional (Lie) derivative of $f$ along $X$ at the point $p$ and $dV$ is a volume form on $mathcal{M}$ -- this functional essentially measures the total amount of change in $f$ along $X$ over all of $mathcal{M}$ in the $L^2$ sense. Then $delta E(f)$ is a differential operator whose eigenspectrum



$$delta E(f) = lambda f$$



(for $lambda in mathbb{R}$) yields the critical points of $E$ over the set of functions with unit norm. Is there an established name for this operator (or functional) and/or its corresponding eigenspectrum?



The prototype for this operator is Dirichlet's energy



$$int_{mathcal{M}} ||nabla f||^2 dV$$



which has as its (unit-norm) critical points the Laplacian eigenspectrum



$$nabla^2 f = lambda f,$$



the main difference being that Dirichlet's energy measures the total gradient, i.e., the change in all directions, rather than just the change along a particular direction at each point.

No comments:

Post a Comment