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I'm a physics student trying to do the maths of the Hilbert space in quantum mechanics with a bit more rigour than I'm accustomed to. I am trying to find ways to extend the spectral theorem for unbounded self-adjoint operators on an infinite dimensional Hilbert space to other operators. The form of the spectral theorem I'm working with is from Hall, which says that a self-adjoint operator is unitarily equivalent to a multiplication operator on some other Hilbert space. (Page 207, Theorem 10.10.) Specifically, I have an operator $\hat{H}$ on the infinite dimensional Hilbert space, which is not itself self-adjoint but is related to self-adjoint operators by at least two simple relations. First, both

$$ \hat{H}\hat{\gamma}^0 \; \text{and} \; \hat{\gamma}^0\hat{H} $$

are self-adjoint, where $\hat{\gamma}^0$ can be written in block-diagonal form as

$$ \hat{\gamma}^0 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. $$

It therefore has all sorts of useful properties, such as being bounded, Hermitian, invertible, unitary, and $(\hat{\gamma^0})^2=1$. I also have the knowledge that

$$ (\hat{H})^2 $$

is self-adjoint. Is there any way of proving that $\hat{H}$ must be isomorphic (though presumably not in general by a unitary transformation) to a multiplication operator on a Hilbert space, from the fact that $\hat{H}^2$ and/or $\hat{H}\hat{\gamma}^0$ are? It feels like there might or might not be something obvious I'm missing. Would any further properties of $\hat{H}$ be needed?

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1 Answer 1

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I'd suggest checking Frederic Schuller's Youtube Channel. He covers those kinds of topics from a physics perspective.

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