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It is common in QM, even in mathematical physics, to consider Hamiltonians with a Dirac delta in the potential: $$ \hat H = -\frac{\mathrm{d}^2}{\mathrm{d}x^2} + V(x) + \delta(x-a) \: . $$ The most common interpretation of this is the self-adjoint operator $H$, such that: $$ H: \mathrm{D}(H) \subset L^2(\mathbb{R}) \to L^2(\mathbb{R}) \\[5pt] \mathrm{D}(H) = \big\{\; \psi \in W^{1,2}(\mathbb{R}) \cap W^{2,2}(\mathbb{R} \setminus \{a\}) \;\; \big| \;\; \psi'(a+) - \psi'(a-) = \psi(a) \;\big\} \\[5pt] H\psi(x) = -\psi''(x) + V(x)\psi(x) \quad \text{on } \mathbb{R} \setminus \{a\} $$ 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 $$ h(\varphi) = T_{-\varphi'' + V\varphi} + \varphi(a) \, \delta_a \: , $$ where $T_f(\psi) := \int \! f \, \psi$ is the regular distribution of function $f$. 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 $\delta$?

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    $\begingroup$ Related question $\endgroup$
    – J. Murray
    Feb 1, 2021 at 15:05
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    $\begingroup$ A small comment concerning potentials involving distributions other than $\delta$: typically $V$ in $H=-\Delta +V$ is taken to be a distribution, rather than a (smooth) function. Essential self-adjointness of $H$ on $C_0^\infty$ for example holds in $d=3$ for all $V\in L^2(\mathbb{R}^3)+L^\infty(\mathbb{R}^3)$, or for all $V\geq 0$ pointwise such that $V\in L^2_{\mathrm{loc}}(\mathbb{R}^d)$. $\endgroup$
    – yuggib
    Feb 1, 2021 at 17:56

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First of all, let me mention that a good reference for a rigorous treatment of point interactions is this book by Albeverio, Gesztesy, Høegh-Krohn, and Holden.

The construction the OP outlines of a densely defined quadratic form $\omega$ associated to the Schrödinger operator with $\delta$-potential is fairly standard, the crucial point is then to prove there are self-adjoint extensions.

It turns out that with point interactions there are usually infinitely many possible self-adjoint extensions corresponding, roughly speaking, to different boundary conditions at the contact point. There are also interesting and very explicit characterizations of the action of the self-adjoint extensions on wavefunctions, in terms of a regular and a singular part (charge).

It would be too long and too technical to outline the details in an answer here (there is a crucial dependence on the space dimension considered, on whether the point interaction is happening in a fixed or moving point, on whether there are several or just one particle involved, ...). The aforementioned book and references thereof contained should provide plenty of details to anyone interested, though.

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