We provide a short introduction to the area of Lieb–Thirring inequalities and their applications. We also explain the structure of the book and summarize some of our notation and conventions.

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.

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.

We discuss various independent aspects of sharp Lieb–Thirring inequalities. First, we present an argument of Stubbe which shows that Riesz means of order two and higher approach their semiclassical limit monotonically, thus leading to an alternative proof of sharp Lieb–Thirring inequalities. Next, we discuss the number of negative eigenvalues of Schrödinger operators with radial potentials, following Glaser, Grosse, and Martin. This leads, on the one hand, to a sharp CLR inequality for radial potentials in dimension 4 and, on the other hand, to a counterexample to the Lieb–Thirring conjecture with exponent zero in sufficiently high dimensions. Next, we discuss briefly an approach that disproves the Lieb–Thirring conjecture in a certain range of positive exponents. Finally, we discuss the Lieb–Thirring inequality with exponent one in its dual formulation, also known as kinetic energy inequality, in which it enters in many applications.

We discuss the definition of the Schrödinger operator on Euclidean space as a self-adjoint operator and discuss basic properties of its spectrum, including criteria for discreteness and finiteness of its negative spectrum. We analyze in detail the classical examples of the harmonic oscillator, the Coulomb Hamiltonian and the Pöschl–Teller potential, which can be solved using a commutation method. Returning to general potentials, we use Dirichlet–Neumann bracketing to prove Weyl asymptotics for the number and Riesz means of negative eigenvalues in the strong coupling constant limit. These asymptotic results are complemented by the nonasymptotic results of Lieb–Thirring, Cwikel–Lieb–Rozenblum, and Weidl. We present a unified method of proof of these bounds, based on Sobolev inequalities and the Besicovitch covering lemma. As an application of these bounds, we extend Weyl asymptotics to a large class of potentials.

We provide a brief, but self-contained, introduction to the theory of self-adjoint operators. In a first section we give the relevant definitions, including that of the spectrum of a self-adjoint operator, and we discuss the proof of the spectral theorem. In a second section, we discuss the connection between lower semibounded, self-adjoint operators and lower semibounded, closed quadratic forms, and we derive the variational characterization of eigenvalues in the form of Glazman’s lemma and of the Courant–Fischer–Weyl min-max principle. Furthermore, we discuss continuity properties of Riesz means and present in abstract form the Birman–Schwinger principle.

We discuss the definition of the Laplace operator on an open subset in Euclidean space as a self-adjoint operator with Dirichlet or Neumann boundary conditions and we derive its basic spectral properties. Among others, we include the spectral inequalities of Faber–Krahn, Hersch, and Friedlander. Then, using the technique of Dirichlet–Neumann bracketing, we derive Weyl's law for the asymptotic distribution of eigenvalues. We supplement this with a discussion of non-asymptotic bounds, including Pólya's conjecture and its proof for tiling domains and domains of product form. We present the sharp eigenvalue bounds of Berezin and Li–Yau. Finally, using separation of variables in spherical coordinates, we discuss the Laplacian on a ball.

We provide a brief, but self-contained, introduction to the theory of Sobolev spaces. We prove some facts from the calculus with weak derivatives, including product and chain rules. We discuss various kinds of Sobolev inequalities, including those by Gagliardo–Nirenberg, Poincaré, Friedrichs, and Hardy, both on the whole space and on domains, and include some information on their sharp constants. Furthermore, we discuss Rellich’s compactness theorem, the Sobolev extension property of a domain, as well as homogeneous Sobolev spaces.

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.