Professor Daniel Ueltschi, Department of Mathematics, University of Arizona

The sum of lower eigenvalues of the discrete Laplacian describes the ground state energy of non-interacting electrons moving in a solid. This quantity appears in some studies of condensed matter systems.

We consider here the discrete Laplacian defined in a finite subset of the cubic lattice. Extending a result of Li and Yau, one can show that the sum of lower eigenvalues is bounded below by a term proportional to the volume of the domain (`bulk term'), plus a positive correction proportional to the boundary. The bulk term involves the ground state energy per site of lattice electrons on the infinite lattice.

The suggestion here is to consider the domain as a subset of a finite torus, and to derive a lower bound involving the ground state energy per site of electrons on the torus. Such a bound would generalize the infinite-lattice bound described above. I have some situations in mind where this extension could be very useful.

No background of physics is required. As everything is finite-dimensional, basic notions of algebra are enough. The Fourier transform and some analytic inequalities like Holder will be necessary. This material can be learned on the way.