# Spectral theorem for unbounded self-adjoint (hermitian) operators

It is my understanding that in quantum mechanics we use self-adjoint operators (that is an axiom of the theory). This operators can be either bounded or unbounded, being the latter the more general case.

Has the spectral theorem had been proven for the unbounded case? And if so, what references discuss this?

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• Yes it has. The wikipedia entry on the Spectral Theorem (General Self-adjoint Operators) references Section 10.1 of Quantum Theory for Mathematicians by Brian Hall. – J. Murray Mar 23 '18 at 18:33
• If you'd prefer a free video reference, this lecture (and the ones preceding it) by Frederic Schuller does an excellent job. – J. Murray Mar 23 '18 at 18:36

von Neumann uses the suggestive notation$$H=\int \lambda \, \mathrm{d}{E \left( \lambda \right)}$$ for his spectral integral. von Neumann discusses the difference between symmetric operators and self-adjoint operators; he formulates abstract boundary conditions and discusses how to obtain a self-adjoint operator from the adjoint of a symmetric operator by imposing such conditions.