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Self-adjoint operators on a Hilbert space are used to represent observable quantities associated to particles in quantum mechanics.  As most observable operators are unbounded, it’s generally difficult to find a domain on which the operator is self-adjoint while it’s easier to find domains on which the operator satisfies a weaker condition called symmetry.  I’ll present an extension theory to start from an unbounded symmetric operator and extend its domain to find a true self-adjoint operator.

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