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| Lieb–Thirring inequality | |
|---|---|
| Name | Lieb–Thirring inequality |
| Area | Mathematical physics |
| Introduced | 1976 |
| Authors | Elliott H. Lieb; Walter Thirring |
| Related | Cwikel–Lieb–Rosenbljum inequality; Sobolev inequality; CLR bound |
Lieb–Thirring inequality The Lieb–Thirring inequality is a fundamental spectral estimate in mathematical physics linking eigenvalue sums of Schrödinger operators to integrals of potentials. It provides bounds on moments of negative eigenvalues for Schrödinger operators and has deep connections to stability of matter, functional inequalities, and partial differential equations. Introduced by Elliott H. Lieb and Walter Thirring, the inequality has driven developments across operator theory, spectral theory, and many-body quantum mechanics.
The canonical form of the inequality bounds the sum of negative eigenvalues {λ_j} of a Schrödinger operator −Δ + V(x) on Euclidean space, asserting that for suitable s > 0 and dimension d there exists a constant L_{s,d} such that Σ_j |λ_j|^s ≤ L_{s,d} ∫_{R^d} [V_-(x)]^{s + d/2} dx, where V_- denotes the negative part of V. This statement is typically formulated in the contexts of self-adjoint operators on L^2(R^d), trace ideals in operator theory, and spectral estimates used in the analysis surrounding the stability results of many-body models studied by figures associated with Princeton University, Harvard University, and the Institute for Advanced Study. Related classical results include the Cwikel–Lieb–Rosenbljum (CLR) bound, the Sobolev inequality, and semiclassical asymptotics arising in Weyl's law and the Thomas–Fermi model.
The inequality was proposed in the mid-1970s by Elliott H. Lieb and Walter Thirring during investigations into the stability of matter, a subject linked to earlier work by Paul Dirac, Enrico Fermi, and Lev Landau in the development of quantum statistical models. Subsequent contributions from Michael Cwikel, Barry Simon, and Gerald Rosenbljum clarified constants and conditions, while further refinements invoked ideas from Jacques Hadamard, Hermann Weyl, and Lars Hörmander. Institutions where progress was made include Princeton University, the Massachusetts Institute of Technology, and the Courant Institute, with later enhancements tied to seminars at the Institut des Hautes Études Scientifiques and conferences organized by the American Mathematical Society and the International Congress of Mathematicians.
Proof techniques blend harmonic analysis, functional analysis, and semiclassical methods. Approaches exploit the Birman–Schwinger principle, heat kernel estimates developed in work related to Kac and Minakshisundaram, and trace ideal methods building on Schatten classes and the theory of compact operators developed by John von Neumann. Proofs often use coherent state decompositions, the Calderón–Vaillancourt theorem, and tools from microlocal analysis influenced by Lars Hörmander and Louis Nirenberg. Alternative proofs employ Lieb’s variational techniques connected to Thomas–Fermi theory and the Riesz–Sobolev rearrangement inequalities that trace lineage to Henri Lebesgue and Jacques-Louis Lions.
Determining the optimal constants L_{s,d} and the threshold moments s for which the inequality holds has stimulated extensive research. Semiclassical (Weyl) asymptotics, originally advanced by Hermann Weyl and later refined by Victor Ivrii, suggest conjectural sharp constants equal to the semiclassical constants for s ≥ 1 and certain d, while counterexamples inspired by work of Barry Simon and Elliott Lieb show subtleties at low moments. Connections to isoperimetric-type spectral optimization problems studied by Georg Polya and Mark Kac further guide sharpness questions, and numerical and variational analyses carried out at institutions such as the Courant Institute have produced bounds approaching conjectured optimal values.
The inequality underpins proofs of stability of matter in the many-body setting, a topic pioneered by Freeman Dyson, Andrew Lenard, and later resolved using Lieb–Thirring estimates by Lieb and collaborators. It is applied to nonlinear dispersive equations, scattering theory, and existence results for bound states in models treated by Walter Kohn, Pierre-Gilles de Gennes, and Elliott Lieb. In the analysis of partial differential equations, the inequality supplies a priori spectral bounds for operators arising in Hartree–Fock theory, density functional theory, and kinetic models connected to Lev Landau and Enrico Fermi, facilitating well-posedness and blow-up criteria in works by Terence Tao and Jean Bourgain.
Numerous extensions generalize the Lieb–Thirring bound to magnetic Schrödinger operators, fractional Laplacians, and operators on manifolds, building on techniques linked to Michael Taylor, Thierry Coulhon, and Gilles Carron. Related inequalities include the Cwikel–Lieb–Rosenbljum bound, the CLR estimate, and the Kato inequality developed by Tosio Kato; connections also exist with the Sobolev inequality, the Gagliardo–Nirenberg inequality, and the Hardy inequality investigated by G. H. Hardy and Edward C. Titchmarsh. Active research continues to relate Lieb–Thirring-type estimates to spectral geometry questions studied by Peter Li and Shing-Tung Yau and to semiclassical analysis approaches due to Victor Ivrii and Johannes Sjöstrand.
Category:Mathematical inequalities Category:Spectral theory Category:Quantum mechanics