The zeta function of the quartic oscillator

We investigate the levels of the quartic oscillator ($\text{?}\text{d}{}^2\text{d}\text{q}{}^2\text{+ q}{}^4$) by means of the associated zeta function. By using several versions of the semiclassical approach, we prove exact results concerning the locations of the real zeros and of the poles of that function, the residues of which are essentially the coefficients of the semiclassical expansion of the Bohr-Sommerfeld quantization rule. We also compute the derivative of the zeta function at the origin, by relating it to the Fredholm determinant of the operator. These results translate as sum rules for the eigenvalues and thus are helpful for their precise computation, and a further analysis allows one to split them into separate rules for even and odd levels; the extension to higher anharmonic oscillators ($\text{?}\text{d}{}^2\text{d}\text{q}{}^2\text{+ q}{}^{\mathrm{2M}}$) is immediate. Finally we propose an asymptotic formula for large negative arguments of the zeta function, based on the nature and location of the complex singularities of the partition function of the operator $\text{(}\text{?}\text{d}{}^2\text{d}\text{q}{}^2\text{+ q}{}^4\text{)}{}^{\mathrm{<mtext>3</mtext><mtext>4</mtext>}}$.