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.

In this chapter, we derive the currently best known bounds on the constants in the Lieb–Thirring inequality following Hundertman–Laptev–Weidl and Frank–Hundertmark–Jex–Nam. These arguments proceed by proving bounds for one-dimensional Schrödinger operators with matrix-valued potentials and then using the method of "lifting in dimension." In the final section, we summarize the results in the book and provide an overview of what is known about the sharp constants in the Lieb–Thirring and Cwikel–Lieb–Rozenblum inequalities.