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I am wondering if beyond physical interpretation, the one dimensional contact interactions (self-adjoint extensions of the the free Hamiltonian when defined everywhere except at the origin) have a geometric interpretation along the lines of non commutative geometry, Lipzchitz distance, etc.

Particularly some extensions (such as Albeverio-Holden pseudodelta) seen as if we have just cut a segment of length $l$ from the free solution and then just pasted the half-lines. So in some sense they could be argued to be just the free solution over two half-lines separated a distance $l$.

Still, the full set of extensions is four parametric, so it is not clear to me if the rest of the parameters have a geometric interpretation, or even if this one can be translated to a Lipzchitz distance. Also, in Non-Commutative Geometry (NCG) sometimes the separation is linked to Higgs potential, definitely not to self-adjoint extensions; Should I be suprised at the implied connection between both concepts?

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The geometric interpretation is that the self-adjoint extensions encode sensible boundary conditions that make the Schroedinger equation uniquely solvable while preserving Hermiticity. (One has similar problems for the Schroedinger equation in a bounded domain in $R^n$. Without boundary conditions it is underdetermined, and with inappropriate boundary conditions it is not Hermitian.)

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