Consider the One dimensional Schrodinger Operator

$$ -\frac{d^2}{dx^2} + V(x) $$

Where the Potential Function $V$ is of the form $V(x) = x^4 - a^2x^2$ , $a \in \mathbb{R} $. Now of course,the minimum eigenvalue $E_0 \geq -a^4/4$ , since $-a^4/4$ is the minimum of $V$. My Question is : Can we Improve this bound?
Thanks in advance for any comment,suggestion or reference in that direction.

Consider the One dimensional Schrodinger Operator

$$ -\frac{d^2}{dx^2} + V(x) $$

Where the Potential Function $V$ is of the form $V(x) = x^4 - a^2x^2$ , $a \in \mathbb{R} $. Now of course,the minimum eigenvalue $E_0 \geq -a^4/4$ , since $-a^4/4$ is the minimum of $V$. My Question is : Can we Improve this bound for $a$ sufficiently large?
Thanks in advance for any comment,suggestion or reference in that direction.