exponential of operators - Trotter product formula

Question

Let $H$ be a self-adjoint operator, bounded from below, with spectrum consisting of isolated eigenvalues $E_n (E_0 < E_1 < ...)$ with finite multiplicities $d(n)$, and $Tr[exp(- \beta H)] < \infty$ for all positive $\beta$.

Furthermore, let $\lambda$ be a small positive constant, and $V$ a relatively bounded self-adjoint operator with respect to H, i.e., there exist constants $a, b > 0$ such that $$ ||V \psi || \leq a ||H \psi || + b ||\psi||$$ for all $\psi \in \mathcal{D}(H) \subseteq \mathcal{D}(V)$. Does Lie product formula (or else known as Trotter product formula) $$ e^{-it(H+\lambda V)} = \lim_{n \to \infty}(e^{\frac{-itH}{n}} e^{\frac{-it \lambda V}{n}})^n$$ hold true in this case ($t$ is just the parameter denoting time)?

Answer 1

You only need self-adjointness for $H$, but you keep the assumptions on $V$.

Then, for $\lambda$ such that $\lambda a < 1$, by the Kato-Rellich theorem $H + \lambda V$ is self-adjoint on $D(H)$ which is equal to $D(H) \cap D(\lambda V)$, hence you can apply the Trotter product formula given for example by Theorem VIII.30 in Reed&Simon Volume I with $A = H$ and $B = \lambda V$ (this one is proven inside the book, while its generalisation Theorem VIII.31 to the case where $A + B$ is essentially self-adjoint on the intersection of the domains isn't).

Statement of the aforementioned theorem to keep the post self-contained:

Theorem VIII.30: Let $A$ and $B$ be self-adjoint operators on $\mathscr{H}$ and suppose that $A+B$ is self-adjoint on $D(A) \cap D(B)$. Then: $${\text{s -}\lim}_{n \to \infty} \left(e^{itA/n}e^{itB/n}\right)^n = e^{it(A+B)}$$

($\text{s-}\lim$ aka "strong limit" means that the result deals with pointwise convergence).