6 - Sharp Lieb–Thirring Inequalities in Higher Dimensions

Summary

We prove Lieb–Thirring inequalities with optimal, semiclassical constant in higher dimensions by following the Laptev–Weidl approach of "lifting in dimension." We introduce Schrödinger operators with matrix-valued potentials and show how Lieb–Thirring inequalities with semiclassical constants for such operators in one dimension imply the Lieb–Thirring inequality with semiclassical constant in higher dimensions. Subsequently, we prove a sharp Lieb–Thirring inequality in one dimension with exponent 3/2 for Schrödinger operators with matrix-valued potentials. We give a complete proof using the commutation method by Benguria and Loss. We also sketch the original proof by Laptev and Weidl based on trace formula for Schrödinger operators with matrix-valued potentials.