Solution of spectral problem for Schrödinger equation with degenerate polinomial potential of even power

The symmetry of the stationary Schrödinger equation with a degenerate potential $U(x)=x^{2r}$, $r \in Z_+$, describing phase transitions in quantum systems, is reveled. The analytical procedure of finding the eigenvalues of the potentials in question is constructed and realized numerically for $r=2,3,\dots,18$. The low energy levels are found.