The following is well known. Given a symmetric differential operator, like partial2x, defined on smooth functions of compact support on mathbbR, Ci0nfty(mathbbR), one can count the number of independent L2-normalizable solutions of partialxpmi and use the von Neumann index theorem to classify possible self-adjoint extensions of this operator on L2(mathbbR). This can be generalized to more complicated differential operators, to mathbbRn as well as bounded open subsets thereof.
On the other hand, suppose that I have a manifold M that is covered by a set of open charts Ui with differential operators Di defined in corresponding local coordinates. It is easy to check if the Di are restrictions of a globally defined differential operator D on M: the transition functions on intersections of charts UicapUj must transform Di into Dj and vice versa. Suppose that is the case and that I am interested in self-adjoint extensions of D to L2(M) (supposing that an integration measure is given and that D is symmetric with respect to it). Now, the question:
Is there way of classifying the self-adjoint extensions of D on L2(M) in terms of its definition in local coordinates, the actions of Di on Ci0nfty(Ui).
A simple example would be the cover of S1 by two overlapping charts. I know that a self-adjoint extension of partial2x on [0,1] with periodic boundary conditions gives the naturally defined self-adjoint Laplacian on S1. Then (0,1) is interpreted as a chart on S1 that excludes one point. However, I don't know how to define the self-adjoint Laplacian on S1 if it's given on two overlapping charts.
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