HYBRID EXPONENTIAL PRODUCT-FORMULAS FOR UNBOUNDED OPERATORS WITH POSSIBLE APPLICATIONS TO MONTE-CARLO SIMULATIONS

Some new schemes of exponential product formulas are proposed together with a basic theorem. A typical hybrid fourth-order product formula is given by exp[-x(A + B)] = S-a(x/3)Q(x)S-b(x/3)Q(x)S-a(x/3)+O(x(5)) = Q(x)S-a(x/3)S-b(x/3)S-a(x/3)Q(x) + O(x(5)), where S-a(x) = e(-xA/2)e(-xB)e(-xA/2), S-b(x) = e(-xB/2)e(-xA)e(-xB/2), and Q(x) = exp(-1/2(x/6)(3)[B,[A, B]]).