### Abstract

Coarse-grained spin density functional theory (SDFT) is a version of SDFT which works with number/spin densities specified to a limited resolution - averages over cells of a regular spatial partition - and external potentials constant on the cells. This coarse-grained setting facilitates a rigorous investigation of the mathematical foundations which goes well beyond what is currently possible in the conventional formulation. Problems of existence, uniqueness, and regularity of representing potentials in the coarse-grained SDFT setting are here studied using techniques of (Robinsonian) nonstandard analysis. Every density which is nowhere spin-saturated is V-representable, and the set of representing potentials is the functional derivative, in an appropriate generalized sense, of the Lieb internal energy functional. Quasi-continuity and closure properties of the set-valued representing potentials map are also established. The extent of possible non-uniqueness is similar to that found in non-rigorous studies of the conventional theory, namely non-uniqueness can occur for states of collinear magnetization which are eigenstates of Sz.

Original language | English (US) |
---|---|

Article number | 062104 |

Journal | Journal of Mathematical Physics |

Volume | 54 |

Issue number | 6 |

DOIs | |

State | Published - Jun 3 2013 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics