On the sum of eigenvalues of the discrete Laplace operator

Daniel Ueltschi University of California, Davis

The ground state energy of N non-interacting, spinless electrons in a finite domain D is given by the sum of the N lowest eigenvalues of the Laplace operator with Dirichlet boundary conditions. Li and Yau proved a lower bound in 1983, namely that this sum is larger than a ‘bulk term’ that is proportional to |D|.

Motivated by a study of the Falicov-Kimball model, we considered the analogous problem on a lattice. One can show that the ground state energy of N electrons in an arbitrary domain D ⊂ ℤ^{d} goes like a bulk term proportional to |D|, plus a positive term proportional to the boundary of D.

This work was done in collaboration with Jim Freericks and Elliott Lieb and is archived in math-ph/0107003.