SoSe 14: Regularity and Approximability of Electronic Wave Functions
Rupert Klein
Comments
Solutions to the quantum-mechanical Schroedinger equation for N electrons
in the Born-Oppenheimer approximation are complex, time-dependent functions
on a 3N-dimensional space. For large N even the storage of a good approximation
to such functions on a computer is hardly possible, much less their determination
as solutions of the Schroedinger problem -- unless they have particular
N-dependent smoothness properties that come to our rescue.
Harry Yserentant (TU-Berlin) rigorously investigated the regularity of
eigenfunctions of the N-electron problem and found that such a rescue might, in
fact, be in reach. He documented the results of his studies in detail in [1,2,3].
In this course, we will retrace the steps of this analysis by considering, amongst others, the following questions:
- How can we characterize the "smoothness" of a function systematically?
- How does the computer storage needed to approximate a function with a given accuracy depend on its smoothness?
- What is the smoothness of a 3N-dimensional function if it can be represented
by a superposition of Slater-determinants (essentially antisymmetric products)
of 3-dimensional once-differentiable functions?
- What is the role of the Pauli-exclusion principle for electrons in this context?
and finally
- What does all this have to do with solutions to the electronic many-electron
Schroedinger problem?
Harry Yserentant builds the theory in [1] from bottom up, so that the course
should be accessible to students of mathematics as well as to theoretically inclined students of the natural sciences alike.
Suggested reading
[1] H. Yserentant, "Regularity and Approximability of Electronic Wave Functions", Lecture Notes in Mathematics, vol. 2000, 2010
[2] H. Yserentant, "The mixed regularity of electronic wave functions multiplied by explicit correlation factors", ESAIM: Mathematical Modelling and Numerical Analysis, vol. 45, 803-824, 2011
[3] H.-Chr. Kreuser, H. Yserentant, "The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces", Numer. Math., vol. 121, 781–802, 2012
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