The eigenfunction problem in higher dimensions: Exact results.

A hermitian integral kernel in N-space may be mapped to a corresponding Hamiltonian in 2N-space by the Wigner transformation. Linear simplectic transformation on the phase space of the Hamiltonian yields a new kernel whose spectrum is unchanged and whose eigenfunctions follow from an explicit unitary transformation. If an integral kernel has a Wigner transform whose surfaces of constant value are concentric ellipsoids, then the Wigner transform yields exact results to the eigenfunction problem. Such behavior is asymptotically generic near extrema of the Wigner transform, from which follow simple and robust asymptotic results for the ends of the eigenvalue spectrum and for the corresponding eigenfunctions.