A real-valued description of quantum mechanics with Schrödinger's 4th-order matter-wave equation

Using a variational formulation, we show that Schrödinger's 4th-order, real-valued matter-wave equation which involves the spatial derivatives of the potential $V\left(r\right)$, produces the precise eigenvalues of Schrödinger's 2nd-order, complex-valued matter-wave equation together with an equal number of negative, mirror eigenvalues. The variational forms of the matter-wave equations are computed numerically with a Ritz-spline method and we show how this method handles accurately discontinuous potentials with singular derivatives. Accordingly, the paper concludes that there is a real-valued description of non-relativistic quantum mechanics in association with the existence of negative, mirror energy levels. Schrödinger'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 $V\left(r\right)$, at the expense of missing the negative, mirror energy levels.