2017/10/24 by Ivan D. Remizov, Remizov, Ivan D.
Mathematics · #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1710.08714
In the paper we derive two formulas representing solutions of Cauchy problem\nfor two Schr "odinger equations: one-dimensional momentum space equation with\npolynomial potential, and multidimensional position space equation with locally\nsquare integrable potential. The first equation is a constant coefficients\nparticular case of an evolution equation with derivatives of arbitrary high\norder and variable coefficients that do not change over time, this general\nequation is solved in the paper. We construct a family of translation operators\nin the space of square integrable functions and then use methods of functional\nanalysis based on Chernoff product formula to prove that this family\napproximates the solution-giving semigroup. This leads us to some formulas that\nexpress the solution for Cauchy problem in terms of initial condition and\ncoefficients of the equations studied.\n