# Rigorous delta potential – a formulation using distributions?

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 $$\delta$$?

• Related question Commented Feb 1, 2021 at 15:05
• 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 $V\in L^2_{\mathrm{loc}}(\mathbb{R}^d)$. Commented Feb 1, 2021 at 17:56

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.