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Wave Functions and Their Densities in Metric Space

### Irene D'Amico

**Wednesday, 18 April 2012, 11:00**

Matfys library

**Abstract**:

Hilbert space combines the properties of two different types
of mathematical spaces: vector space and metric space. While the
vector-space aspects are widely used, the metric-space aspects are
much less exploited. Here[1] we show that a suitable metric stratifies
Fock space into concentric spheres on which maximum and minimum
distances between states can be defined and geometrically interpreted.
Unlike the usual Hilbert-space analysis, our results apply also to the
reduced space of only ground states and to that of particle densities,
which are metric, but not Hilbert, spaces. The Hohenberg-Kohn mapping
between densities and ground states, which is highly complex and
nonlocal in coordinate description, is found, for three different
model systems, to be simple in metric space, where it becomes a
monotonic and nearly linear mapping of vicinities onto vicinities.

[1] Phys. Rev. Lett. **106**, 050401 (2011).