Some spectral relations

We consider the zeta function of a second-order differential operator which has a second-order turning point:
$$Lu=\frac{d^2u}{dx^2}+[\lambda^2q(x)+R(x)]u,$$
where $q(x)=x^2q_1(x)$, $q_1(x)\ne0$ and $u(0)=u(1)=0$. We construct an asymptotic series and calculate regularized traces for the eigenvalues of this operator.