The Löwdin method to obtain lower bounds to energy eigenvalues of the Schrödinger equation has been reviewed in <A HREF="http://link.springer.com/chapter/10.1007%2F978-1-4757-1659-7_21#page-1">Lower Bounds to Energy Eigenvalues</A> (1976). It has been applied to the quartic potential in <a href="http://sgpwe.izt.uam.mx/pages/cbi/annik/revistas/1965.pdf">Upper and Lower Bounds to the Eigenvalues of Double-Minimum Potentials</a> (1965). See table I, where $v_2$ and $v_4$ are the $x^2$ and $x^4$ coefficients of the potential.