Consider an enumeration $\{q_1,q_2,\ldots\}$ of $\mathbb{Q}\cap [1,\infty)$ and a orthogonal Schauder basis $\{e_1,e_2,\ldots\}$ of $\ell^2(\mathbb{N})$. Define
$Ae_{2k-1}=e_{2k-1}$ and $Ae_{2k}=q_ke_{2k}$ for all $k\geq 1$.

** Question 1: ** Is it possible to extend $A$ to a linear self-adjoint operator defined in some infinite dimensional subspace of $\ell^2(\mathbb{N})$ ?

If I am not wrong, this possible linear self-adjoint extension of $A$ can not be defined everywhere in $\ell^2(\mathbb{N})$ and I would like to know if the following set
$$
\left\{
v\in \ell^2(\mathbb{N}); v=\lim_{n\to\infty} \sum_{i=1}^n\alpha_ie_i \ \text{and}\ \ \lim_{n\to\infty}\sum_{i=1}^n q_i\alpha_ie_i \in \ell^2(\mathbb{N})
\right\}
$$
is a good candidate to be the domain of $A$ ?

** Question 2:** Is the point spectrum
$\sigma_p(A)\supset \{q_1,\ldots,q_n,\ldots\}$ ?

** Motivavation: **
I would like to know if there is an example of an unbounded self-adjoint operator such that the point spectrum is not composed only by isolated
points in $\mathbb{R}$ and there is at least one eigenvalue with infinite dimensional eigenspace.