Next: References Up: A remark on quantum Previous: Numerical experiments

# Relations with other numerical methods

The usual Schrödinger evolution needs a numerical integration like , where n is so large that is smaller than the desired accuracy . Any such truncation produces high-frequency noise after a relatively small number of time steps. An other method used in quantum dynamics is to diagonalize a finite dimensional approximation of L and to evolve its eigenfunctions (see for example [14]). It is quantitatively not clear, how well a finite dimensional Galerkin cut-off respects the actual dynamics. Moreover, in dimension d, the number of grid points N has to be so small that an eigensystem of a matrix can be found.

While numerical approximations of are not unitary, the evolution (2.1) is conjugated to a unitary evolution. Other discrete unitary time evolutions have been considered in [4]. The problem to preserve unitarity is similar to numerical integration problems for ODE's, where for example symplecticity should be preserved during the discretisation of a Hamiltonian system.

If has compact support, then has this property too. This leads to a finite propagation speed as in a relativistic set-up. This fact has computational advantages. For example, we know exactly, after which time, boundary effects begin to influence the value of a wave at some point.

The discrete evolution preserves the (not necessarily closed) algebraic field in which L is defined. For example, if L is an operator defined over the rationals and if the coordinates of are rational, then is rational and can be determined exactly.

The evolution (2.1) can be defined on all bounded sequences and not only on . For example, the evolution leaves almost periodic configurations invariant, elements , for which the closure of all translated sequences is compact in the uniform topology ). This is useful, because solutions of (2.1) define generalized eigenfunctions of an operator K defined on space-time.

The unitary operator V=- i U solves which is a discretisation of the Schrödinger equation . Since , the evolutions U and V are essentially the same. The discrete evolution is a second order approximation to in the sense that and agree up to second order. This second order approximation is more efficient than the second order but computationally more expensive Cayley method

Next: References Up: A remark on quantum Previous: Numerical experiments

Oliver Knill, Tue Aug 18, 1998