• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Physical interpretation of different selfadjoint extensions

+ 30 like - 0 dislike

Given a symmetric (densely defined) operator in a Hilbert space, there might be quite a lot of selfadjoint extensions to it. This might be the case for a Schrödinger operator with a "bad" potential. There is a "smallest" one (Friedrichs) and a largest one (Krein), and all others are in some sense in between. Considering the corresponding Schrödinger equations, to each of these extensions there is a (completely different) unitary group solving it. My question is: what is the physical meaning of these extensions? How do you distinguish between the different unitary groups? Is there one which is physically "relevant"? Why is the Friedrichs extension chosen so often?

This post has been migrated from (A51.SE)
asked Sep 15, 2011 in Theoretical Physics by András Bátkai (275 points) [ no revision ]
retagged Mar 24, 2014 by dimension10
I am asking this question as a mathematician trying to understand the meaning and motivation of the objects I am working with.

This post has been migrated from (A51.SE)

An interesting early example in the physics literature involves a relativistic electron interacting with a Dirac magnetic monopole. A self-adjoint extension is required, in the form of a boundary condition at the location of the monopole. The extension  parameter $\theta$ results in the breaking of some symmetries and as a consequence the magnetic monopole gets an induced electric charge, becoming a dyon!  
The references and details are in this paper.

2 Answers

+ 23 like - 0 dislike

The differential operator itself (defined on some domain) encodes local information about the dynamics of the quantum system . Its self-adjoint extensions depend precisely on choices of boundary conditions of the states that the operator acts on, hence on global information about the kinematics of the physical system.

This is even true fully abstractly, mathematically: in a precise sense the self-adjoint extensions of symmetric operators (under mild conditions) are classified by choices of boundary data.

More information on this is collected here


See the references on applications in physics there for examples of choices of boundary conditions in physics and how they lead to self-adjoint extensions of symmetric Hamiltonians. And see the article by Wei-Jiang there for the fully general notion of boundary conditions.

This post has been migrated from (A51.SE)
answered Sep 15, 2011 by Urs Schreiber (6,095 points) [ no revision ]
+ 14 like - 0 dislike

A typical interpretation of the self-adjoint extensions for the free hamiltonian in a line segment is that you get a four parametric family of possible boundary conditions, to preserve unitarity. Some of them just "bounce" the wave, some others "teletransport" it from one wall to the other. So it is also traditional to imagine this segment as a circle where you have removed a point, and then you are in the mood of studying "point interactions" or generalisations of dirac-delta potentials. The topic resurfaces from time to time, but surely some old references can be digged starting from M. Carreau. Four-parameter point-interaction in 1d quantum systems. Journal of Physics A, 26:427, 1993. In some works, I quote also Seba and Polonyi.

Sometimes the extensions are linked to the question of the domain of definition for the operator and then to the existence of anomalies. Here Phys.Rev.D34: 674-677, 1986, "Anomalies in conservation laws in the Hamiltonian formalism", revisited by the same autor, J G Esteve, later in Phys.Rev.D66:125013,2002 ( http://arxiv.org/abs/hep-th/0207164 ). These topics have been live for years in the university of Zaragoza; some related material, perhaps more about boundary conditions than about extensions, is http://arxiv.org/abs/0704.1084, http://arxiv.org/abs/quant-ph/0609023, http://arxiv.org/abs/0712.4353

This post has been migrated from (A51.SE)
answered Sep 16, 2011 by anonymous [ no revision ]
I hadn't been aware of the reference by Esteve. I have added it to the references of the nLab entry http://ncatlab.org/nlab/show/quantum+anomaly (many more references are currently still missing there, of course).

This post has been migrated from (A51.SE)
@Urs Schreiber Thanks for the add. The topic was common folklore in Zaragoza in the nineties and it was not infrequent in PhD theses, but I think that its main role was motivational, either aiming towards other topics, or used as a guide when exploring some other concept. For instance, it was very valuable to me in order to navigate Albeverio et al, who had got into a confusing notation/naming for some self adjoint extensions classifying these "1D point interactions".

This post has been migrated from (A51.SE)
Thank, I like both answers very much. The references are great. Unfortunately, I have to choose an answer to accept...

This post has been migrated from (A51.SE)

Your answer

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):
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:
Then 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).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights