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 the coefficient $a$ is given by $a^2=-v_2 v_4^{-2/3}$ and then the lower bound is $\frac{1}{2}E v_4^{-1/3}$ with $E$ the "level 0 energy" from table I. For example, for $a=2.6375$ I find $E_0\geq-8.53234$, well above $-a^4/4=-12.0981$.