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?
Answer
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
http://ncatlab.org/nlab/show/self-adjoint+extension
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.
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