  Using a variational formulation, we show that Schrodinger's 4th-order,
real-valued matter-wave equation which involves the spatial derivatives of the
potential V(r), produces the precise eigenvalues of Schrodinger's 2nd-order,
complex-valued matter-wave equation together with an equal number of negative,
mirror eigenvalues. Accordingly, the paper concludes that there is a
real-valued description of non-relativistic quantum mechanics in association
with the existence of negative (repelling) energy levels. Schrodinger's
classical 2nd-order, complex-valued matter-wave equation which was constructed
upon factoring the 4th-order, real-valued differential operator and retaining
only one of the two conjugate complex operators is a simpler description of the
matter-wave, since it does not involve the derivatives of the potential V(r),
at the expense of missing the negative (repelling) energy levels.
