Asymptotic Equipartition of Energy by Nodal Points
of an Eigenfunction
Goong Chen,
Stephen A. Fulling,
Jianxin Zhou
The time-reduced form of a partial differential
equation in vibration or quantum mechanics in one space dimension often
satisfies a Sturm-Liouville (S-L) equation. Nodal points of eigenfunctions
of the S-L equation form energy barriers.
When the S-L equation has constant
coefficients, the energy localized in each nodal interval is the same.
But when the S-L equation has variable coefficients, strict
equipartition of energy by nodal points no longer holds. In this
paper, however, we
formulate an asymptotic form of the principle
of equipartition of energy by nodal points, showing that the energies
stored on connected nodal intervals away from the turning points or
singularities of the governing equation differ at most by an error of
order of magnitude inversely proportion to the frequency. Also, using a
numerical example, we demonstrate this asymptotic equipartition
principle when a potential barrier is present.
In higher dimensions, nodal lines or nodal (hyper)surfaces do not,
in general,
equipartition the energy even in an approximate or asymptotic sense. We
nevertheless suggest that some appropriately reformulated counterpart
of the
equipartition-of-energy principle might still exist.