In mathematics and physics, Lieb–Thirring inequalities provide an upper bound on the sums of powers of the negative eigenvalues of a Schrödinger operator in terms of integrals of the potential.
The inequalities are useful in studies of quantum mechanics and differential equations and imply, as a corollary, a lower bound on the kinetic energy of
quantum mechanical particles that plays an important role in the proof of stability of matter.
denote the (not necessarily finite) sequence of negative eigenvalues.
were proven by E. H. Lieb and W. E. Thirring in 1976 [1] and used in their proof of stability of matter.
the left-hand side is simply the number of negative eigenvalues, and proofs were given independently by M. Cwikel,[2] E. H. Lieb [3] and G. V.
The classical phase space consists of pairs
and assuming that every quantum state is contained in a volume
The semiclassical approximation becomes exact in the limit of large coupling, that is for potentials
D. Hundertmark, E. H. Lieb and L. E. Thomas [8] proved that the best constant is given by
[9] In the former case Lieb and Thirring conjectured that the sharp constant is given by
is equivalent to a lower bound on the kinetic energy of a given normalised
is defined via The inequality can be extended to particles with spin states by replacing the one-body density by the spin-summed one-body density.
is the number of quantum spin states available to each particle (
Inequality (2) describes the minimum kinetic energy necessary to achieve a given density
would be precisely the kinetic energy term in Thomas–Fermi theory.
(for more information, read the Stability of matter page) The kinetic energy inequality plays an important role in the proof of stability of matter as presented by Lieb and Thirring.
The particles and nuclei interact with each other through the electrostatic Coulomb force and an arbitrary magnetic field can be introduced.
If the particles under consideration are fermions (i.e. the wave function
is antisymmetric), then the kinetic energy inequality (2) holds with the constant
This is a crucial ingredient in the proof of stability of matter for a system of fermions.
of the system can be bounded from below by a constant depending only on the maximum of the nuclei charges,
, times the number of particles, The system is then stable of the first kind since the ground-state energy is bounded from below and also stable of the second kind, i.e. the energy of decreases linearly with the number of particles and nuclei.
In comparison, if the particles are assumed to be bosons (i.e. the wave function
is symmetric), then the kinetic energy inequality (2) holds only with the constant
and for the ground state energy only a bound of the form
Analogously a kinetic inequality similar to (2) holds, with
, which can be used to prove stability of matter for the relativistic Schrödinger operator under additional assumptions on the charges
[13] In essence, the Lieb–Thirring inequality (1) gives an upper bound on the distances of the eigenvalues
Similar inequalities can be proved for Jacobi operators.