MCS-MP Seminar: Marvin Schulz

The Lieb–Thirring inequality bounds moments of the negative eigenvalues of Schrödinger operators in terms of their potentials. Determining its optimal constants is a longstanding problem in spectral theory, with the Lieb–Thirring conjecture [LT76] dating back half a century. In this talk, I will present joint work with Larry Read establishing the sharp constant for the sum of the absolute values of the negative eigenvalues in one dimension, thereby confirming the conjecture in this case.

The emphasis will be on the main ideas behind the proof. In particular, I will explain how a local Poincaré inequality, obtained through a ground-state transform, allows us to extend the Benguria–Loss transform [BL04] from a single function to families of functions. I will also discuss extensions to higher dimensions, where our approach yields the best currently available constants.

An initial argument emerged through interactions with ChatGPT, lacking the underlying mathematical mechanism. It first appeared in Larry Read’s conversations and independently arose in my investigations of the oval conjecture. We subsequently joined forces to verify the argument, clarify its mechanism, and develop the proof presented here.

[LT76] E. H. Lieb and W. E. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, in Studies in Mathematical Physics, edited by E. H. Lieb, B. Simon and A. S. Wightman, Princeton University Press, 1976, pp. 269–303.

[BL04] R. D. Benguria and M. Loss, Connection between the Lieb–Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane, in Partial Differential Equations and Inverse Problems, Contemporary Mathematics 362, American Mathematical Society, 2004, pp. 53–61.