# Metric interpretation of self-adjoint extensions?

+ 6 like - 0 dislike
82 views

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.

Particulatly some extensions (such as Albeverio-Holden pseudodelta) seen as if we have just cut a segment of lenght $l$ 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 NCG sometimes the separation is linked to Higgs potential, definitely not to self-adjoint extensions; I would be surprised if having a connection between both concepts, should I?

This post has been migrated from (A51.SE)
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOv$\varnothing$rflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.