Abstract:

The solution to the problem of stability of matter is one of the fine successes of mathematical physics. The statement of the problem reads: show that the energy of a system of N electrons and K nuclei interacting through Coulomb forces is bounded from below by a (negative) constant times N+K. The constant is only allowed to depend on the charge of the nuclei. Simply, the problem asks, “Why don’t the point-like electrons fall into the (nearly) point-like nuclei?”, or in other words, “why is the energy of an atom with a point-like nucleus not −∞?” While working on this problem, Elliott H. Lieb and Walter Thirring developed inequalities that provide an upper bound on the sums of powers of the negative eigenvalues of a Schrödinger operator in terms of integrals of the potential. These inequalities are famously called the Lieb-Thirring inequalities, and are well celebrated in mathematical physics. A special case of these Lieb-Thirring inequalities is equivalent to a Sobolev inequality for orthonormal functions which gives a lower bound on the kinetic energy of N quantum mechanical particles. This inequality plays an important role in the proof of stability of matter. We aim to explain the proof of equivalence between the special case of the Lieb-Thirring inequalities and the Sobolev inequality for orthonormal functions. In future, we shall expound the proof of the Lieb-Thirring inequalities in detail, which will then also prove the Sobolev inequality for orthonormal functions.

Evaluation Committee   

• Dr. Muhammad Usman (Supervisor) 
• Dr. Ali Ashher Zaidi (Evaluator)