Born-Oppenheimer dynamics is shown to provide an accurate approximation of time-independent Schrödinger observables for a molecular system with an electron spectral gap, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schrödinger equation and stabil...

1 Schrödinger Equation and Time Development Operators 1 1.1 Schrödinger’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Time-dependent kets and operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Solutions to Schrödinger’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Time developme...

A propagation method for the time dependent Schrödinger equation was studied leading to a general scheme of solving ode type equations. Standard space discretization of time-dependent pde’s usually results in system of ode’s of the form ut −Gu= s (0.1) where G is a operator (matrix) and u is a time-dependent solution vector. Highly accurate methods, based on polynomial approximation of a modifi...

A numerical method for solving the time-independent Schrödinger equation of a particle moving freely in a three-dimensional axisymmetric region is developed. The boundary of the region is defined by an arbitrary analytic function. The method uses a coordinate transformation and an expansion in eigenfunctions. The effectiveness is checked and confirmed by applying the method to a particular exam...

We prove an error estimate for a Lie–Trotter splitting operator associated with the Schrödinger–Poisson equation in the semiclassical regime, when the WKB approximation is valid. In finite time, and so long as the solution to a compressible Euler–Poisson equation is smooth, the error between the numerical solution and the exact solution is controlled in Sobolev spaces, in a suitable phase/ampli...