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How to show two self-adjoint operators (unbounded) on a Hilbert space with the same eigenvalues and eigenfunctions are the same.
That's not true. Consider the operators $T,S \colon L^2[0,1] \to L^2[0,1]$, given by
$$ Tx(t) = tx(t), \quad Sx(t) = t^2x(t). $$
Both are self-adjoint, and both have no eigenvalues. But $S \ne T$, as $S1 \ne T1$.
Required, but never shown