I've been looking at one of the simplest systems I can think of: a one-dimensional infinite square well on $[0,1]$ with Hamiltonian given by the following: $$\hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \alpha \, \delta\left(x-a\right)$$ with $\alpha>0, \; 0<a<1$. The typical way to solve this would be through eigenfunction expansion. For this question, it is possible to find the eigenfunctions explicitly; they solve the time-independent Schrödinger equation in a distributional sense. The conditions I've applied to look for the eigenfunctions are: $\psi(x)$ continuous on $[0,1]$, $\psi(x)=0$ at $x=0$ and $x=1$, and $$\psi'(a^+)-\psi'(a^-)=\frac{2m\alpha}{\hbar^2}\psi(a)$$ (found by integrating the time-independent Schrödinger equation from $a-\varepsilon$ to $a+\varepsilon$ and sending $\varepsilon\rightarrow 0$, assuming $\psi\in L^2[0,1]$).

I have plenty of questions about this system, but my most pressing is the following: for most systems (such as the infinite square well without delta potential), Sturm-Liouville theory can be applied to show the closure in $L^2[0,1]$ of the span of the eigenfunctions of $\hat{H}$ is all of $L^2[0,1]$, and so I can write any initial condition as an eigenfunction expansion that time-evolves trivially and have solved the full time-dependent PDE. Is the closure of the span of my eigenfunctions all of $L^2[0,1]$? I can write down the explicit form of my eigenfunctions if necessary, but I don't think that will help solve the question.

As an extension to the above, what if I consider $H_0^1[0,1]$ with $H^1$ norm?