By a multiplication operator here we mean an operator
$Af(t)=m(t)f(t), \qquad f \in D(A)=\{x \in L^2(\mathbb{R} \mid m(t)f(t) \in L^2(\mathbb{R})\}$
where $m$ is a Borel measurable function on the line. Motivated by Exercise 3.13, pag.104 of Teschl's book (link) I'm trying to construct such an operator having all rationals as eigenvalues.
This amounts to find a Borel measurable function $m$ such that $m^{-1}(\lambda)$ has positive measure for all rational $\lambda$. How to do that?
I find constructing an abstract linear operator on some separable Hilbert space $H$ with $\mathbb{Q}$ as point spectrum not too difficult: take an orthonormal basis $\{e_n\}$ and an enumeration $\{r_n\}$ of $\mathbb{Q}$ then define
$Af=\sum_{n=0}^\infty r_n(f, e_n)e_n, \qquad f \in D(A)=\left\{g \in H \mid \sum_{n=0}^\infty r_n^2\left\lvert(g, e_n)\right\rvert^2 < \infty\right\}.$
But how to go from this to a function $m$ as prescribed above?
Thank you.