Solution of the Schrödinger equation with the use of the translation operator (Q509157)

The author studies the Cauchy problem for the Schrödinger equation in the complex space \(L_2(\mathbb{R}^1)\). A new technique for solving this problem is given using the standard technique of \(C_0\) semigroups, a theorem due to \textit{P. R. Chernoff} [J. Funct. Anal. 2, 238--242 (1968; Zbl 0157.21501)], and the notion of Chernoff tangency. A new type of formula is obtained for the solution of the Cauchy problem, which does not contain integrals. The solution formula is neither a Feynman nor quasi-Feynman formula, but rather resembles a multiple weighted infinite sum of translations of the initial condition with respect to the space variable.