Spectral theory and Taylor expansion in quantum mechanics for unbounded operators

Question

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?

Answer 1

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).

It turns out that the resolvent is the key to characterizing pertubations of (unbounded) generators. From here on out I'll be a bit sloppy with notation and assumptions---in an attempt to get the concept across a bit better.

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).