Calculation of a bound state wavefunction using free state wavefunctions only

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Abstract
A pedagogical example is presented in which the completeness of the free state wavefunctions of a Hamiltonian can be explicitly checked. For certain values of the potential strength the free states prove to be complete. For other values of the potential strength the free states are shown to be incomplete and the extent of this incompleteness is shown to consist of precisely one function. Using the fact that the totality of all energy eigenfunctions must be complete, this single bound statewavefunction is calculated using the free state wavefunctions only.

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