**Quantum mechanics and approximation theory**

With the work of Schrödinger and Heisenberg in 1926,
quantum mechanics found its final form and the complete
mathematical description of atoms and molecules had become
possible. Dirac commented this with the words ``The
underlying physical laws necessary for the mathematical
theory of a large part of physics and the whole chemistry
are thus completely known, and the difficulty is only that
the exact application of these laws leads to equations much
too complicated to be soluble.'' The reason is the high
dimensionality of the Schrödinger equation. Its solutions
depend on 3*N* variables for a system consisting of *N*
particles, three spatial coordinates for each particle.
Although very sophisticated and highly successful simplified
models accessible to a numerical treatment have been
developed during the last seventy or seventy-five years,
from a mathematical point of view the situation has not much
changed since Dirac's time. It is shown in this talk why a
refined regularity theory for the Schrödinger equation and
modern principles of approximation theory and numerical
analysis could change this situation and why the construction
of true numerical methods for the Schrödinger equation
comes into reach now.

*Harry Yserentant*

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