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## Schroedinger eigenfunctions as equilibrium distribution of a heat flow

This is quite well known, but worthwhile recalling for gaining some intuition for Schroedinger eigenfunctions, and applies to compact manifolds.  For the sake of simplicity suppose $V(x)$ is a smooth potential in compact closed manifold and suppose we know that $\mu>0$ is an eigenvalue of $H = -\Delta/2 + V$.  Suppose $\psi$ is an eigenfunction $H\psi = \mu \psi$.  So the way to relate this to the heat equation for $H$ is to consider $\psi$ to be a zero for the operator $H^\mu = H - \mu$ and then note that because $\psi$ is a zero of $H^\mu$, it is an equilibrium distribution for the heat semigroup $\exp(-tH^\mu)$, that is, $\exp(-tH^\mu)\psi = \psi$.  This characterization sheds some light on what Schroedinger’s picture is telling us regarding states of a physical system: they are equilibrium distributions for certain heat flows.  It is worthwhile asking if these equilibrium distributions are averaging some classical systems, i.e. if solving the Schroedinger equation is the approximation rather than the usual picture which is the other way around.  Given that we now know that the universe is a scaled 4-sphere, the original motivation of Schroedinger, of producing a precise ‘wave mechanics’ for which Hamiltonian mechanics is an approximation may have not been fulfilled (regardless of the approximate correctness of the energy levels of actual physical systems).  A classical system on a 4-sphere will automatically have ‘wavelike’ behavior simply by analogy to Fourier analysis on a circle.

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