It is common in QM, even in mathematical physics, to consider Hamiltonians with a Dirac delta in the potential: $$ \hat H = -\frac{\mathrm{d}^2}{\mathrm{d}x^2} + f(x) + \delta(x-a) \: . $$ The most common interpretation of this is the self-adjoint operator $H$, such that: $$ H: \mathrm{D}(H) \subset L^2(\mathbb{R}) \to L^2(\mathbb{R}) \\[5pt] \mathrm{D}(H) = \big\{\; f \in W^{1,2}(\mathbb{R}) \cap W^{2,2}(\mathbb{R} \setminus \{0\}) \;\; \big| \;\; f'(a+) - f'(a-) = f(a) \;\big\} \\[5pt] H\psi(x) = -\psi''(x) + f(x)\psi(x) \quad \text{on } \mathbb{R} \setminus \{0\} $$ This is a very practical way to interpret the problem, however I feel like this approach sweeps the distributional aspect „under the rug“. I would love to understand the problem in terms of distributions.

My gut feeling is that something like this should be possible:

Let $\mathcal{H} = L^2(\mathbb{R})$ and $\Phi = C^\infty_0(\mathbb{R})$ be the space of smooth functions with compact support, then $\Phi \subset \mathcal{H} \subset \Phi'$ is a Gelfand triple. We define a map $h: \Phi \to \Phi'$, such that $$ h(\varphi) = T_{\varphi'' + f\varphi} + \varphi(a) \, \delta_a \: , $$ where $T_g(\psi) := \int \! g \, \psi$ is the regular distribution of function $g$. Now we define a sesquilinear form $\omega: \Phi \times \Phi \to \mathbb{C}$, such that $$ \omega(\varphi, \psi) = \big< h(\overline\varphi), \; \psi \big>_\Phi = \big( h(\overline\varphi) \big)(\psi) \: . $$ Now, if we extend $\omega$ to a maximum domain (in some sense), it uniquely determines the self-adjoint operator $H$: $$ \Omega|_{\Phi\times\Phi} = \omega \\[10pt] \Omega(\varphi, \psi) = \big( H\varphi, \; \psi \big)_{\mathcal{H}} = \big( \varphi, \; H\psi \big)_{\mathcal{H}} $$

Is there any author that proves this rigorously, or approaches the problem in a similar way? Does this approach extend to potentials with distributions other than $\delta$?

Recommendations for books which deal with such topics are welcome – none of the books on the mathematical-physics of QM that I have read seem to mention delta potentials, yet in practise they appear everywhere.