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The analytic subtleties for Schroedinger operators on the usual noncompact setting of $\mathbf{R}^3$ arises from the noncompactness of the space.  I have argued with quantitative evidence that the actual universe is a hypersurface of a scaled 4-sphere.  On a compact manifold, one has the orthonormal basis of eigenfunctions of the Laplacian which can reduce solving for eigenfunctions of Schroedinger operators to linear algebra as follows.  The key point is that any potential function $V(x)$ has an expansion by eigenfunctions.  Suppose $V = \sum_k \phi_k$ where $\Delta \phi_k = \lambda_k \phi_k$.  Suppose $\Delta \psi + V\psi = \mu \psi$.  Expand both $\psi$ and $V$ into eigenfunctions, then solve for $\psi$ using linear algebra.