Asymptotic equipartition of energy by nodal points of an eigenfunction
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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, the formation of nodal lines or nodal (hyper)surfaces is quite unstable. Therefore, equipartition of energy does not hold in an approximate or asymptotic sense. 1997 American Institute of Physics.
