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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 Upper and lower bounds of the ground state energy of anharmonic oscillators using renormalized inner projection. There is no indication that $E_0$ can be expressed as the root of a polynomial.