I would like to find solutions of the following differential equation:

$ \sum_{1}^{\infty} a_n f(nx) + f''(x)+ x^2 f(x)=\lambda f(x)$

For example in space of function from $\mathbb R^*$ to $\mathbb C$

If we modify the sum in the differential equation by posing $g(t)=f(e^{t})$, and make a Fourier transform like advise here (for an equation with infinite sum but without derivation) it seems it does not work.

Which method would you suggest ?