Solution of the Schrödinger equation with the use of the translation operator

Formulas that Represent Cauchy Problem Solution for Momentum and Position Schrödinger Equation

“…Since 2000, O. G. Smolyanov and members of his group succeeded in representing solutions of the Cauchy problem for many evolution equations in form of Feynman formulas (see [39,40,41,42,43,47,48,49,51,52,48,54,61,57,58,44] and refereces therein). The key idea in these representations lies in finding the Chernoff function G for operator L and then applying Chernoff's theorem to obtain the equality e tL u 0 = lim n→∞ G(t/n) n u 0 which apperas to be a Feynman formula, because in all known examples (until [50] was published in 2016, see also [44,63]) G(t) from the equation above was an integral operator, so G(t/n) n was an n-tuple integral operator, giving us a limit of multiple integral where miltiplicity tends to infinity.…”

“…We come to formulas that do not include integrals at all, but then interpret expressions obtained as quasi-Feynman formulas with Dirac δ-functions under the integral sign. This approach was used first in [50] for a simple case of one-dimensional Schrödinger equation with the second derivative only and bounded potential in the Hamiltonian. In the present paper we develop methods of [50] in two directions.…”

“…This approach was used first in [50] for a simple case of one-dimensional Schrödinger equation with the second derivative only and bounded potential in the Hamiltonian. In the present paper we develop methods of [50] in two directions. Firsly, we cover the case of the Hamiltonian with derivatives of higher order in one-dimensional case.…”

“…Secondly, we consider a multidimensional space in the case when Hamiltonian has only two terms: the Laplacian and potential. In both cases the potential may be unbounded which covers the Hamiltonian of quantum (an)harmonic oscillator, this was not done in [50]. See also [62] for short introduction to quasi-Feynman formulas and the calculus of Chernoff functions.…”

“…which apperas to be a Feynman formula, because in all known examples (until [50] was published in 2016, see also [44,63]) G(t) from the equation above was an integral operator, so G(t/n) n was an n-tuple integral operator, giving us a limit of multiple integral where miltiplicity tends to infinity.…”