Error estimate in operator norm of exponential product formulas for propagators of parabolic evolution equations (Q1282232)

An error estimate in the operator norm of exponential product approximation for propagators \(U(t,s)\) of the parabolic evolution equations \[ \partial _t U(t,s)=-C(t)U(t,s), \quad 0\leq s\leq t \leq T, \quad U(s,s)=I \tag{1} \] is studied. The operator \(C(t)\) is assumed to be of the form \(C(t)=A+B(t)\), where \(A, B(t)\geq c>0\) are positive selfadjoint operators in a compact interval \([0,T]\) or semibounded uniformly in \(t\) operators. The following operators are defined \[ K_j(\tau)=\exp(-\tau A/2)\exp(-\tau B(t_{j-1}))\exp(-\tau A/2), \] \[ F_j(\tau)=K_j(\tau)K_{j-1}(\tau) \times\dots \times K_2(\tau)K_{1}(\tau), \quad 1\leq j\leq N. \] If there exists an \(\alpha\in [0,1)\) independent of \(t\in[0, T]\), such that \(D(A^\alpha)\subset D(B(t))\) and that \(B(t)A^{-\alpha}: X\to X\) is uniformly bounded and \[ \| A^{-\alpha}B(t)-B(s)A^{-\alpha}\| \leq d| t-s| , \quad d>0, \] it is proved that the estimate \[ \| U(t,0)-F_N(\tau)\| = O(N^{-1}\log N), \quad N\to \infty \] is fulfilled uniformly in \(t, 0\leq t\leq T\). The obtained result is applied to the Schrödinger operator \(\Delta+V(t, x)\) with a certain class of time dependent singular potentials.