I am reading spectral theory and quantum mechanics. We know that for an unbounded self-adjoint operator $A$, operator-valued functions such as $\exp(iAt)$, $t\in\mathbb{R}$ can be defined using spectral theorems. It seems to me that there does not exist a Taylor expansion for the exponentiation here i.e. we cannot write
\begin{equation} \exp(iAt) = \sum_n \frac{(iAt)^n}{n!}. \end{equation}
However, in quantum mechanics, people always consider the case of Taylor expanding to obtain a perturbation theory. How is perturbation theory consistent with the spectral theory? Take, for example, the following simple (1+1)D Schrodinger Hamiltonian:
\begin{equation} H = \frac{\hat{p}^2}{2m} + V(x) =: H_0 + V, \end{equation}
where $\hat{p} = - i \hbar \partial_x$ and some (unbounded) potential $V(x)$. How could we justify the expansion of an $U$-matrix:
\begin{equation} U(t) := \exp(i H_0 t /\hbar )\exp(-i H t / \hbar) \to \sum_n \frac{(iVt/\hbar)^n}{n!}, \end{equation}
in general?