We discuss the problem of finding the optimal constant in Lieb–Thirring and Cwikel–Lieb–Rozenblum inequalities, thereby introducing, in particular, the semiclassical constant and the one-particle constants, which appear in the Lieb–Thirring conjecture. We discuss Keller's problem of minimizing the lowest eigenvalue of a Schrödinger operator among all potentials with a given L^p norm. We present the Aizenman–Lieb monotonicity argument, as well as semiexplicit computations for eigenvalues of the harmonic oscillator (including the counterexample of Helffer and Robert) and the Pöschl–Teller potential. In the one-dimensional case, we present the optimal bounds due to Hundertmark–Lieb–Thomas and Gardner–Greene–Kruskal–Miura. We provide two proofs of the latter bound, namely, the original one based on trace formulas and a more recent one by Benguria and Loss based on the commutation method.

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.