# Spectral theory and Taylor expansion in quantum mechanics for unbounded operators

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

$$$$\exp(iAt) = \sum_n \frac{(iAt)^n}{n!}.$$$$

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:

$$$$H = \frac{\hat{p}^2}{2m} + V(x) =: H_0 + V,$$$$

where $$\hat{p} = - i \hbar \partial_x$$ and some (unbounded) potential $$V(x)$$. How could we justify the expansion of an $$U$$-matrix:

$$$$U(t) := \exp(i H_0 t /\hbar )\exp(-i H t / \hbar) \to \sum_n \frac{(iVt/\hbar)^n}{n!},$$$$

in general?

Indeed, neither the exponential series nor the limit definition of the exponential $$e^{tA}=\lim_{n\to\infty}(1+\frac{t}nA)^n$$ makes any sense for unbounded operators. However, if $$A$$ is the generator of a strongly continuous one-parameter semigroup (e.g., a self-adjoint Hilbert space operator) the "exponential" can be evaluated via the Post-Widder Inversion Formula: $$e^{tA}=\underset{n\to\infty}{\operatorname{s-lim}}\Big({\bf 1}-\frac{t}nA\Big)^{-n}=\underset{n\to\infty}{\operatorname{s-lim}}\Big(\Big({\bf 1}-\frac{t}nA\Big)^{-1}\Big)^n$$ Here $$({\bf 1}-\frac{t}nA)^{-1}$$ is defined via the resolvent formalism, and $$\operatorname{s-lim}$$ is the limit in the strong operator topology, cf., e.g., Corollary 5.5 in Chapter III of Engel & Nagel's, One-Parameter Semigroups for Linear Evolution Equations (2000).
Theorem. Given a sequence $$\{(A_n(t))_{t\geq 0}:n\in\mathbb N\}$$ as well as $$(A(t))_{t\geq 0}$$ of generators of quasi-bounded semigroups ($$\|(A+z)^{-k}\|\leq M(z-\beta)^{-k}$$), if $$\operatorname{s-lim}_{n\to\infty}(A_n-z{\bf 1})^{-1}=(A-z{\bf 1})^{-1}$$ for some $$z$$ with $$\operatorname{Re}z>\beta$$, then $$\underset{n\to\infty}{\operatorname{s-lim}}\;e^{-tA_n}=e^{-tA}$$
This is Theorem 2.16 in Chapter 9 of Kato's Perturbation Theory for Linear Operators (1980). As an application (Example 2.22 in Ch.9) Kato shows that given $$H$$ self-adjoint and $$V$$ symmetric and $$H$$-bounded, for all $$\varepsilon$$ from a sufficiently small neighbourhood of zero one obtains a Taylor(-esque) expansion $$e^{it(H+\varepsilon V)} u = e^{itH} u + \varepsilon u^{(1)} (t) + \varepsilon^2 u^{(2)}(t) + o(\varepsilon^2)$$ where $$u^{(1)}(t)=-\int_0^te^{i(t-s)(H+\varepsilon V)}iVe^{is(H+\varepsilon V)}u\,ds$$ (and $$u^{(2)}$$ similarly).