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The Löwdin method to obtain lower bounds to energy eigenvalues of the Schrödinger equation has been reviewed in Lower Bounds to Energy Eigenvalues (1976). It has been applied to the quartic potential in Upper and Lower Bounds to the Eigenvalues of Double-Minimum Potentials (1965). See table I, where $v_2$ and $v_4$ are the $x^2$ and $x^4$ coefficients of the potential.