Abstract
Yserentant's results on the effective complexity of many-electron eigenfunctions
Rupert Klein
Freie Universität Berlin
Due to the curse of dimensionality, we will not be able to numerically
solve non-stationary many-body Schroedinger problems for large particle
numbers and for general initial data in the foreseeable future. The
situation seems equally severe for the simpler problem of approximating
eigenfunctions of the non-relativistic electronic Schroedinger equation
with clamped nuclei.
It is known, on the other hand, from numerical analysis that very smooth
functions can be approximated efficiently with few degrees of freedom,
given the "right" ansatz functions for their representation.
Harry Yserentant and co-workers recently analyzed the smoothness of
many-electron eigenfunctions for large particle numbers [1,2,3].
By revealing an interesting competition between increasing dimensionality
and increasing smoothness, they came up with results that seem outright
spectacular to a newcomer in the field like myself.
In this presentation I will summarize these results, the principal
line of analysis, and open issues.
[1] H. Yserentant, Lecture Notes in Mathematics, Vol. 2000 (2010)
[2] H. Yserentant, ESAIM: M2AN, vol. 45, 803-824 (2011)
[3] H.-C. Kreusler, H. Yserentant, Numer. Math., vol. 121, 781-802 (2012)
solve non-stationary many-body Schroedinger problems for large particle
numbers and for general initial data in the foreseeable future. The
situation seems equally severe for the simpler problem of approximating
eigenfunctions of the non-relativistic electronic Schroedinger equation
with clamped nuclei.
It is known, on the other hand, from numerical analysis that very smooth
functions can be approximated efficiently with few degrees of freedom,
given the "right" ansatz functions for their representation.
Harry Yserentant and co-workers recently analyzed the smoothness of
many-electron eigenfunctions for large particle numbers [1,2,3].
By revealing an interesting competition between increasing dimensionality
and increasing smoothness, they came up with results that seem outright
spectacular to a newcomer in the field like myself.
In this presentation I will summarize these results, the principal
line of analysis, and open issues.
[1] H. Yserentant, Lecture Notes in Mathematics, Vol. 2000 (2010)
[2] H. Yserentant, ESAIM: M2AN, vol. 45, 803-824 (2011)
[3] H.-C. Kreusler, H. Yserentant, Numer. Math., vol. 121, 781-802 (2012)