Inmathematics andphysics,Lieb–Thirring inequalities provide an upper bound on the sums of powers of the negativeeigenvalues of aSchrödinger operator in terms of integrals of the potential. They are named afterE. H. Lieb andW. E. Thirring.

The inequalities are useful in studies ofquantum mechanics anddifferential equations and imply, as a corollary, a lower bound on thekinetic energy of${\displaystyle N}$ quantum mechanical particles that plays an important role in the proof ofstability of matter.^[1]

For the Schrödinger operator${\displaystyle -\mathrm{\Delta }+V(x)=-{\mathrm{\nabla }}^2+V(x)}$ on${\displaystyle {\mathbb{R}}^n}$ with real-valued potential${\displaystyle V(x):{\mathbb{R}}^n\to \mathbb{R},}$ the numbers${\displaystyle {\lambda }_1\le {\lambda }_2\le \cdots \le 0}$ denote the (not necessarily finite) sequence of negative eigenvalues. Then, for${\displaystyle \gamma }$ and${\displaystyle n}$ satisfying one of the conditions

there exists a constant${\displaystyle L_{\gamma ,n}}$, which only depends on${\displaystyle \gamma }$ and${\displaystyle n}$, such that

${\displaystyle \sum _{j\ge 1}|{\lambda }_j{|}^{\gamma }\le L_{\gamma ,n}{\int }_{{\mathbb{R}}^n}V(x{)}_{-}^{\gamma +\frac{n}{2}}{\mathrm{d}}^{n}x}$

1

where${\displaystyle V(x{)}_{-}:=max(-V(x),0)}$ is the negative part of the potential${\displaystyle V}$. The cases${\displaystyle \gamma >1/2,n=1}$ as well as${\displaystyle \gamma >0,n\ge 2}$ were proven by E. H. Lieb and W. E. Thirring in 1976^[1] and used in their proof of stability of matter. In the case${\displaystyle \gamma =0,n\ge 3}$ 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. Rozenbljum.^[4] The resulting${\displaystyle \gamma =0}$ inequality is thus also called the Cwikel–Lieb–Rosenbljum bound. The remaining critical case${\displaystyle \gamma =1/2,n=1}$ was proven to hold by T. Weidl^[5]The conditions on${\displaystyle \gamma }$ and${\displaystyle n}$ are necessary and cannot be relaxed.

The Lieb–Thirring inequalities can be compared to the semi-classical limit. The classicalphase space consists of pairs${\displaystyle (p,x)\in {\mathbb{R}}^{2n}.}$ Identifying themomentum operator${\displaystyle -\mathrm{i}\mathrm{\nabla }}$ with${\displaystyle p}$ and assuming that everyquantum state is contained in a volume${\displaystyle (2\pi {)}^n}$ in the${\displaystyle 2n}$-dimensional phase space, the semi-classical approximation

${\displaystyle \sum _{j\ge 1}|{\lambda }_j{|}^{\gamma }\approx \frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}(p^2+V(x){)}_{-}^{\gamma }{\mathrm{d}}^{n}p{\mathrm{d}}^{n}x=L_{\gamma ,n}^{\mathrm{c}\mathrm{l}}{\int }_{{\mathbb{R}}^n}V(x{)}_{-}^{\gamma +\frac{n}{2}}{\mathrm{d}}^{n}x}$

is derived with the constant

${\displaystyle L_{\gamma ,n}^{\mathrm{c}\mathrm{l}}=(4\pi {)}^{-\frac{n}{2}}\frac{\mathrm{\Gamma }(\gamma +1)}{\mathrm{\Gamma }(\gamma +1+\frac{n}{2})}.}$

While the semi-classical approximation does not need any assumptions on${\displaystyle \gamma >0}$, the Lieb–Thirring inequalities only hold for suitable${\displaystyle \gamma }$.

Numerous results have been published about the best possible constant${\displaystyle L_{\gamma ,n}}$ in (1) but this problem is still partly open. The semiclassical approximation becomes exact in the limit of large coupling, that is for potentials${\displaystyle \beta V}$ theWeyl asymptotics

${\displaystyle \underset{\beta \to \mathrm{\infty }}{lim}\frac{1}{{\beta }^{\gamma +\frac{n}{2}}}\mathrm{t}\mathrm{r}(-\mathrm{\Delta }+\beta V{)}_{-}^{\gamma }=L_{\gamma ,n}^{\mathrm{c}\mathrm{l}}{\int }_{{\mathbb{R}}^n}V(x{)}_{-}^{\gamma +\frac{n}{2}}{\mathrm{d}}^{n}x}$

hold. This implies that${\displaystyle L_{\gamma ,n}^{\mathrm{c}\mathrm{l}}\le L_{\gamma ,n}}$. Lieb and Thirring^[1] were able to show that${\displaystyle L_{\gamma ,n}=L_{\gamma ,n}^{\mathrm{c}\mathrm{l}}}$ for${\displaystyle \gamma \ge 3/2,n=1}$.M. Aizenman and E. H. Lieb^[6] proved that for fixed dimension${\displaystyle n}$ the ratio${\displaystyle L_{\gamma ,n}/L_{\gamma ,n}^{\mathrm{c}\mathrm{l}}}$ is amonotonic, non-increasing function of${\displaystyle \gamma }$. Subsequently${\displaystyle L_{\gamma ,n}=L_{\gamma ,n}^{\mathrm{c}\mathrm{l}}}$ was also shown to hold for all${\displaystyle n}$ when${\displaystyle \gamma \ge 3/2}$ byA. Laptev and T. Weidl.^[7] For${\displaystyle \gamma =1/2,n=1}$ D. Hundertmark, E. H. Lieb and L. E. Thomas^[8] proved that the best constant is given by${\displaystyle L_{1/2,1}=2L_{1/2,1}^{\mathrm{c}\mathrm{l}}=1/2}$.

On the other hand, it is known that for^[1] and for.^[9] In the former case Lieb and Thirring conjectured that the sharp constant is given by

${\displaystyle L_{\gamma ,1}=2L_{\gamma ,1}^{\mathrm{c}\mathrm{l}}{\left(\frac{\gamma -\frac{1}{2}}{\gamma +\frac{1}{2}}\right)}^{\gamma -\frac{1}{2}}.}$

The best known value for the physical relevant constant${\displaystyle L_{1,3}}$ is${\displaystyle 1.456L_{1,3}^{\mathrm{c}\mathrm{l}}}$^[10] and the smallest known constant in the Cwikel–Lieb–Rosenbljum inequality is${\displaystyle 6.869L_{0,3}^{\mathrm{c}\mathrm{l}}}$.^[3] A complete survey of the presently best known values for${\displaystyle L_{\gamma ,n}}$ can be found in the literature.^[11]

The Lieb–Thirring inequality for${\displaystyle \gamma =1}$ is equivalent to a lower bound on the kinetic energy of a given normalised${\displaystyle N}$-particlewave function${\displaystyle \psi \in L^2({\mathbb{R}}^{Nn})}$ in terms of the one-body density. For an anti-symmetric wave function such that

${\displaystyle \psi (x_1,\dots ,x_i,\dots ,x_j,\dots ,x_N)=-\psi (x_1,\dots ,x_j,\dots ,x_i,\dots ,x_N)}$

for all${\displaystyle 1\le i,j\le N}$, the one-body density is defined as

${\displaystyle {\rho }_{\psi }(x)=N{\int }_{{\mathbb{R}}^{(N-1)n}}|\psi (x,x_2\dots ,x_N){|}^2{\mathrm{d}}^n{x}_2\cdots {\mathrm{d}}^n{x}_N,x\in {\mathbb{R}}^n.}$

The Lieb–Thirring inequality (1) for${\displaystyle \gamma =1}$ is equivalent to the statement that

${\displaystyle \sum _{i=1}^N{\int }_{{\mathbb{R}}^n}|{\mathrm{\nabla }}_i\psi {|}^2{\mathrm{d}}^n{x}_i\ge K_n{\int }_{{\mathbb{R}}^n}{\rho }_{\psi }(x{)}^{1+\frac{2}{n}}{\mathrm{d}}^{n}x}$

2

where the sharp constant${\displaystyle K_n}$ is defined via

${\displaystyle {\left(\left(1+\frac{2}{n}\right)K_n\right)}^{1+\frac{n}{2}}{\left(\left(1+\frac{n}{2}\right)L_{1,n}\right)}^{1+\frac{2}{n}}=1.}$

The inequality can be extended to particles withspin states by replacing the one-body density by the spin-summed one-body density. The constant${\displaystyle K_n}$ then has to be replaced by${\displaystyle K_n/q^{2/n}}$ where${\displaystyle q}$ is the number of quantum spin states available to each particle (${\displaystyle q=2}$ for electrons). If the wave function is symmetric, instead of anti-symmetric, such that

${\displaystyle \psi (x_1,\dots ,x_i,\dots ,x_j,\dots ,x_n)=\psi (x_1,\dots ,x_j,\dots ,x_i,\dots ,x_n)}$

for all${\displaystyle 1\le i,j\le N}$, the constant${\displaystyle K_n}$ has to be replaced by${\displaystyle K_n/N^{2/n}}$. Inequality (2) describes the minimum kinetic energy necessary to achieve a given density${\displaystyle {\rho }_{\psi }}$ with${\displaystyle N}$ particles in${\displaystyle n}$ dimensions. If${\displaystyle L_{1,3}=L_{1,3}^{\mathrm{c}\mathrm{l}}}$ was proven to hold, the right-hand side of (2) for${\displaystyle n=3}$ would be precisely the kinetic energy term inThomas–Fermi theory.

The inequality can be compared to theSobolev inequality. M. Rumin^[12] derived the kinetic energy inequality (2) (with a smaller constant) directly without the use of the Lieb–Thirring inequality.

(for more information, read theStability of matter page)

The kinetic energy inequality plays an important role in the proof ofstability of matter as presented by Lieb and Thirring.^[1] TheHamiltonian under consideration describes a system of${\displaystyle N}$ particles with${\displaystyle q}$ spin states and${\displaystyle M}$ fixednuclei at locations${\displaystyle R_j\in {\mathbb{R}}^3}$ withcharges${\displaystyle Z_j>0}$. The particles and nuclei interact with each other through the electrostaticCoulomb force and an arbitrarymagnetic field can be introduced. If the particles under consideration arefermions (i.e. the wave function${\displaystyle \psi }$ is antisymmetric), then the kinetic energy inequality (2) holds with the constant${\displaystyle K_n/q^{2/n}}$ (not${\displaystyle K_n/N^{2/n}}$). This is a crucial ingredient in the proof of stability of matter for a system of fermions. It ensures that theground state energy${\displaystyle E_{N,M}(Z_1,\dots ,Z_M)}$ of the system can be bounded from below by a constant depending only on the maximum of the nuclei charges,${\displaystyle Z_{max}}$, times the number of particles,

${\displaystyle E_{N,M}(Z_1,\dots ,Z_M)\ge -C(Z_{max})(M+N).}$

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 bebosons (i.e. the wave function${\displaystyle \psi }$ is symmetric), then the kinetic energy inequality (2) holds only with the constant${\displaystyle K_n/N^{2/n}}$ and for the ground state energy only a bound of the form${\displaystyle -C{N}^{5/3}}$ holds. Since the power${\displaystyle 5/3}$ can be shown to be optimal, a system of bosons is stable of the first kind but unstable of the second kind.

If the Laplacian${\displaystyle -\mathrm{\Delta }=-{\mathrm{\nabla }}^2}$ is replaced by${\displaystyle (\mathrm{i}\mathrm{\nabla }+A(x){)}^2}$, where${\displaystyle A(x)}$ is a magnetic fieldvector potential in${\displaystyle {\mathbb{R}}^n,}$ the Lieb–Thirring inequality (1) remains true. The proof of this statement uses thediamagnetic inequality. Although all presently known constants${\displaystyle L_{\gamma ,n}}$ remain unchanged, it is not known whether this is true in general for the best possible constant.

The Laplacian can also be replaced by other powers of${\displaystyle -\mathrm{\Delta }}$. In particular for the operator${\displaystyle \sqrt{-\mathrm{\Delta }}}$, a Lieb–Thirring inequality similar to (1) holds with a different constant${\displaystyle L_{\gamma ,n}}$ and with the power on the right-hand side replaced by${\displaystyle \gamma +n}$. Analogously a kinetic inequality similar to (2) holds, with${\displaystyle 1+2/n}$ replaced by${\displaystyle 1+1/n}$, which can be used to prove stability of matter for the relativistic Schrödinger operator under additional assumptions on the charges${\displaystyle Z_k}$.^[13]

In essence, the Lieb–Thirring inequality (1) gives an upper bound on the distances of the eigenvalues${\displaystyle {\lambda }_j}$ to theessential spectrum${\displaystyle [0,\mathrm{\infty })}$ in terms of the perturbation${\displaystyle V}$. Similar inequalities can be proved forJacobi operators.^[14]

• Birman–Schwinger principle

• Lieb, E.H.; Seiringer, R. (2010).The stability of matter in quantum mechanics (1st ed.). Cambridge: Cambridge University Press.ISBN 9780521191180.
• Hundertmark, D. (2007). "Some bound state problems in quantum mechanics". In Fritz Gesztesy; Percy Deift; Cherie Galvez; Peter Perry; Wilhelm Schlag (eds.).Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday. Proceedings of Symposia in Pure Mathematics. Vol. 76. Providence, RI: American Mathematical Society. pp. 463–496.Bibcode:2007stmp.conf..463H.ISBN 978-0-8218-3783-2.

• Inequalities (mathematics)