# Dynamics seminar by Sarina Sutter (VU Amsterdam) on 15 May

On Wednesday the 15th of May at 4 pm, Sarina Sutter (VU Amsterdam) will give an Amsterdam Dynamics seminar.

The talk will take place in the Maryam seminar room (9A-46).

**Title:** %%\upsilon%%-Representability on Periodic Domains: a Sobolev Space Approach

**Abstract: **An important tool in theoretical chemistry is density functional theory (DFT). The idea is that the wave function of the system, depending on all spatial coordinates of the particles, is replaced by the density function of the particles. This is possible, since the density contains sufficient information to reconstruct the external potential (system) it came from [1]. An important question is, how to characterize the class of densities for which there is a certain potential such that the ground state wave function of the system yields that density. This problem is known as the %%\upsilon%%-representability problem.

I will present our recent work about finding functional-analytic properties of the potentials that allow upsilon-representability of a reduced but still large set of densities. It is known that densities need to be in %%L^1 \cap L^3%% (or in %%L^3%% for the periodic case) [2]. However, for particles in one dimension, they are contained in a smaller subspace of %%L^3%%, namely the Sobolev space %%H^1%%. This space already arose naturally from the finite kinetic energy condition which guarantees that %%\sqrt{\rho} \in H^1%% (this holds also in 2 and 3 dimensions). The new density space %%H^1%% leads to a different, more general potential space than the usual used %%L^{3/2}%% space if a dual setting for densities and potentials is constructed. Within this dual setting it is possible to show v-representability for a certain class of densities. The idea is to use standard techniques from convex analysis to find a suitable potential. In the usual %%L^3%% setting these techniques lead to some fundamental problems which motivate our choice of the new setting.

**REFERENCES**

[1] P. Hohenberg and W. Kohn, *Inhomogeneous electron gas, *Phys. Rev. **136**, B864 (1964).

[2] E. H. Lieb, *Density functionals for Coulomb-systems, *Int. J. Quantum Chem. **24**, 243 (1983).