## Geometry of nonnegative potentials in a closed riemannian manifold

March 14, 2014 by zulfahmed

Our universe is a scaled S4. I have been interested in the movement to ‘wave mechanics’ of Schroedinger from classical mechanics. It seems that the knowledge that our actual universe is a scaled S4 might lead to a better understanding of the quantum revolution by detaching from the idea that wave mechanics must be connected with strange behavior in the small; rather, wave behavior is dictated by global geometry. With this in mind, let’s consider well-behaved (smooth) potentials possible on the 4-sphere. For this exercise, let’s focus on non-negative potentials. In fact let’s restrict attention to potentials that can be written as linear combinations of squares of Laplacian eigenfunctions to make life simpler. Consider the Feynman-Kac formula using Brownian motion. For for some Laplacian eigenfunction, it’s clear that these concentrate higher weights to the zeros of , i.e. the nodal lines. Thus intuitively, one can expect that using potentials which are linear combinations, one expects the Schroedinger eigenfunctions to concentrate to combinations of nodal lines. In other words, we can produce three dimensional shapes in a scaled 4-sphere using nodal lines of various eigenfunctions using this potential class and the corresponding Schroedinger eigenfunction will concentrate on the shapes produced. This intuition can be made rigorous but it is an attempt to understand what Schroedinger operators are doing geometrically. Now the possible shapes that can be constructed using nodal lines of eigenfunctions is quite varied but could perhaps be characterized in some reasonable manner. Note that ‘waves’ become natural in this sort of analysis.

n a scaled 4-sphere, one naturally gets a giant construction kit of three-dimensional shapes by taking, say convex, combinations of squares of Laplacian eigenfunctions via their nodal lines. The Schroedinger eigenfunctions for such potentials concentrate weight to the nodal sets. Thus if we take a broad view about Schroedinger eigenfunctions then there are natural geometries for the eigenfunctions with ‘wave’ behavior. Note that the corresponding classical system ALREADY exhibits the wave characteristics for these potentials.

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