# Stone's theorem and the spectral theorem

I am struggling to formally derive the expression found in the Stone's theorem for one-parameter unitary groups. I am aware that this can be done by using the spectral theorem. I am mostly interested in the discrete 'version' of the spectral theorem.

Here's a short statement of Stone's theorem:

If $${\cal H}$$ is a Hilbert space and $$U(t)$$ is a strongly-continuous, one-parameter unitary group, then $$U(t)=\exp\bigl(-itH\bigr)$$, where $$H$$ is self-adjoint.

Can anyone help me out or point me to some reference where this is explicitly done?

• What do you mean by "discrete version" of the spectral theorem? – MaoWao Nov 1 '19 at 10:41
• By discrete I mean the case in which an operator in question has a purely discrete spectrum – omsorg Nov 1 '19 at 11:05

The rough idea is as follows. Define a (possibly unbounded) operator $$H$$ by $$D(H)=\left\{\xi\in \mathcal{H}\mid\lim_{t\to 0}\frac{\xi-U(t)\xi}{it}\phantom{x}\text{ exists }\right\},\ \ \ H\xi=\lim_{t\to 0}\frac{\xi-U(t)\xi}{it}.$$
It needs some effort, but one can show that $$H$$ is self-adjoint. Then one can check that if $$\xi\in D(H)$$, both $$u_1(t)=U(t)\xi$$ (by the group property of $$U$$) and $$u_2(t)=\exp(-itH)\xi$$ (by the spectral theorem) solve the initial-value problem \begin{align*} \dot u(t)&=-iHu(t),\\ u(0)&=\xi. \end{align*} In particular, $$\|u_1(t)-u_2(t)\|^2=\int_0^t\frac{d}{ds}\|u_1(s)-u_2(s)\|^2\,ds=-2\int_0^t\operatorname{Re}i\langle u_1(s)-u_2(s),H(u_1(s)-u_2(s)\rangle\,ds=0.$$ Thus $$U(t)\xi=\exp(-itH)\xi$$. The equality for arbitrary $$\xi\in\mathcal{H}$$ follows by continuity.