Remarks on the Trotter–Kato Product Formula for Unitary Groups

Abstract

Let A and B be non-negative self-adjoint operators in a separable Hilbert space such that their form sum C is densely defined. It is shown that the Trotter product formula holds for imaginary parameter values in the L 2-norm, that is, one has $$ \lim_{n\to+\infty} \int\limits^T_{-T} \left\|\left(e^{-itA/n}e^{-itB/n} \right)^nh - e^{-itC}h\right\|^2dt = 0 $$ for each element h of the Hilbert space and any T > 0. This result is extended to the class of holomorphic Kato functions, to which the exponential function belongs. Moreover, for a class of admissible functions: $${\phi(\cdot),\psi(\cdot):{\mathbb R}_+ \longrightarrow {\mathbb C}}$$ , where $${{\mathbb R}_+ := [0,\infty)}$$ , satisfying in addition $${{\Re{\rm e}}\,(\phi(y))\ge 0, {\Im{\rm m}}\,(\phi(y) \le 0}$$ and $${{\Im{\rm m}}\,(\psi(y)) \le 0}$$ for $${y \in {\mathbb R}_+}$$ , we prove that $$ \,\mbox{\rm s-}\hspace{-2pt} \lim_{n\to\infty}(\phi(tA/n)\psi(tB/n))^n = e^{-itC} $$ holds true uniformly on $${[0,T]\ni t}$$ for any T > 0.