How do you prove basic facts on the Schrodinger equation using the spectral theorem? More precisely, here is what I have in mind.

The version of the Spectral Theorem I am familiar with is the following.

**Spectral Theorem for normal operators:** Let $H$ be a Hilbert space and $T\in \mathcal{B}(H)$ be a normal operator. Then there is a unique resolution $P$ of the identity of $H$ over $\sigma(T)$ (the spectrum of $T$) such that
$$
T=\int_{\sigma(T)} \lambda\, dP.
$$

I would like a theorem like the following.

**Theorem:** Let $\psi$ satisfy the 1-D Schrodinger equation
$$\left(-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} +u(x)\right)\psi=i\hbar \frac{\partial}{\partial t}\psi$$
where $u$ is some "nicely behaved" function, and let $f=\psi(x,0)$.
Then
$$(1)\qquad f=\sum_{\lambda} c_{\lambda}\phi_{\lambda}+\int_{\lambda} c_{\lambda}\phi_{\lambda}$$
where the first sum is over the discrete spectrum, the integral is over the continuous spectrum, $\phi_{\lambda}$ is the eigenfunction for $\lambda$, and $c_{\lambda}=\int f\overline{\phi_{\lambda}}\,dx$. Therefore,
$$\psi=\sum_{\lambda} c_{\lambda}\phi_{\lambda}e^{-\frac{i\lambda}{\hbar}t}+\int_{\lambda} c_{\lambda}\phi_{\lambda}e^{-\frac{i\lambda}{\hbar}t}.$$

What is the mathematically correct way to state the above, and how can it be proved?

For instance, if $u_{\infty}=\lim_{x\to \pm \infty} u(x)$, for generic $u$, the discrete spectrum would consist of some finite set in $(\min u, u_{\infty})$ and the continuous spectrum would be $(u_{\infty},\infty)$.

Two issues are the following.

- The Spectral Theorem as stated above only works for bounded operators, and the operator is not bounded. (We probably also need growth conditions on $f$.) I know there is a Spectral Theorem for unbounded operators but it seems to require more technicalities; if that is the way to go, how are those technicalities satisfied in this case?
- The theorem doesn't describe what the measure $dP$ is explicitly, so how do I know that the resolution of the identity gives (1)?

Functional Analysis. $\endgroup$ – Fan Zheng Oct 19 '15 at 2:46