We consider $x\in B(H)$ a self-adjoint operator, then we know that it can determine a unique spectral measure $E$. If we know that $E(\Delta)=0$ or $1$ for all measurable $\Delta\subset \sigma(x)$, then it is not hard to show that $x=\lambda I$ for some $\lambda\in \mathbb{R}$.
My question is, if $x$ is not a bounded operator now, it is only a self-adjoint (unbounded) operator, then we know that we can still have a unique spectral measure E. If we know that $E(\Delta)=0$ or $1$ for all measurable $\Delta\subset \sigma(x)$, can we show that $x=\lambda I$ for some $\lambda\in \mathbb{R}$.
Any help will be truly grateful!