There is no exact expression for the ground state energy $E_0$ for any nonzero $g$, but there are upper and lower bounds: for $g=1/2$ the upper bound for $2E_0$ is 1.3923516415302918570  and the lower bound for $2E_0$ is 1.3923516415302918502 , see <A HREF="https://doi.org/10.1063/1.529452">Upper and lower bounds of the ground state energy of anharmonic oscillators using renormalized inner projection.</A> There is no indication that $E_0$ can be expressed as the root of a polynomial, for all we know it's a transcendental number.