Lieb–Thirring inequality

For the Schrödinger operator $-\backslash Delta+V(x)=-\backslash nabla^2+V(x)$ on $\backslash Reals^n$ with real-valued potential $V(x)\; :\; \backslash Reals^n\; \backslash to\; \backslash Reals,$ the numbers $\backslash lambda\_1\backslash le\backslash lambda\_2\backslash le\backslash dots\backslash le0$ denote the (not necessarily finite) sequence of negative eigenvalues. Then, for $\backslash gamma$ and $n$ satisfying one of the conditions

:$\backslash begin\{align\}$

\gamma\ge\frac12&,\,n=1,\\

\gamma>0&,\,n=2,\\

\gamma\ge0&,\,n\ge3,

\end{align}

there exists a constant $L\_\{\backslash gamma,n\}$, which only depends on $\backslash gamma$ and $n$, such that

{{NumBlk|:|$$

\sum_{j\ge1}|\lambda_j|^\gamma\le L_{\gamma,n}\int_{\Reals^n}V(x)_-^{\gamma+\frac n2}\mathrm{d}^n x

|{{EquationRef|1}}}}

where $V(x)\_-:=\backslash max(-V(x),0)$ is the negative part of the potential $V$. The cases $\backslash gamma>1/2,n=1$ as well as $\backslash gamma>0,n\backslash ge2$ were proven by E. H. Lieb and W. E. Thirring in 1976 and used in their proof of stability of matter. In the case $\backslash gamma=0,\; n\backslash ge3$ the left-hand side is simply the number of negative eigenvalues, and proofs were given independently by M. Cwikel,{{cite journal | last=Cwikel | first=Michael | title=Weak Type Estimates for Singular Values and the Number of Bound States of Schrödinger Operators | journal=The Annals of Mathematics | volume=106 | issue=1 | pages=93–100 | year=1977 | doi=10.2307/1971160 | jstor=1971160 }} E. H. Lieb {{cite journal | last=Lieb | first=Elliott | title=Bounds on the eigenvalues of the Laplace and Schroedinger operators | journal=Bulletin of the American Mathematical Society | volume=82 | issue=5 | date=1 August 1976 | doi=10.1090/s0002-9904-1976-14149-3 | pages=751–754|doi-access=free}} and G. V. Rozenbljum.{{cite journal|first=G. V.| last=Rozenbljum|title=Distribution of the discrete spectrum of singular differential operators|journal=Izvestiya Vysshikh Uchebnykh Zavedenii Matematika |year=1976| issue=1|pages=75–86|url=http://mi.mathnet.ru/eng/ivm/y1976/i1/p75|mr=430557|zbl=0342.35045}} The resulting $\backslash gamma=0$ inequality is thus also called the Cwikel–Lieb–Rosenbljum bound. The remaining critical case $\backslash gamma=1/2,\; n=1$ was proven to hold by T. Weidl {{cite journal | last=Weidl | first=Timo | title=On the Lieb-Thirring constants $L\_\{\backslash gamma,1\}$ for γ≧1/2 | journal=Communications in Mathematical Physics | volume=178 | issue=1 | year=1996 | doi=10.1007/bf02104912 | pages=135–146| arxiv=quant-ph/9504013 | s2cid=117980716 }}

The conditions on $\backslash gamma$ and $n$ are necessary and cannot be relaxed.

The Lieb–Thirring inequalities can be compared to the semi-classical limit.

The classical phase space consists of pairs $(p,\; x)\; \backslash in\; \backslash Reals^\{2n\}.$ Identifying the momentum operator $-\backslash mathrm\{i\}\backslash nabla$ with $p$ and assuming that every quantum state is contained in a volume $(2\backslash pi)^n$ in the $2n$-dimensional phase space, the semi-classical approximation

:$$

\sum_{j\ge 1}|\lambda_j|^\gamma\approx \frac{1}{(2\pi)^n}\int_{\Reals^n}\int_{\Reals^n}\big(p^2+V(x)\big)_-^\gamma\mathrm{d}^n p\mathrm{d}^n x

=L^{\mathrm{cl}}_{\gamma,n}\int_{\Reals^n} V(x)_-^{\gamma+\frac n2}\mathrm{d}^n x

is derived with the constant

:$$

L_{\gamma,n}^{\mathrm{cl}}=(4\pi)^{-\frac n2}\frac{\Gamma(\gamma+1)}{\Gamma(\gamma+1+\frac n2)}\,.

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

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

:$$

\lim_{\beta\to\infty}\frac{1}{\beta^{\gamma+\frac n2}}\mathrm{tr} (-\Delta+\beta V)_-^\gamma=L^\mathrm{cl}_{\gamma,n}\int_{\Reals^n} V(x)_-^{\gamma+\frac n2}\mathrm{d}^n x

hold. This implies that $L\_\{\backslash gamma,n\}^\{\backslash mathrm\{cl\}\}\backslash le\; L\_\{\backslash gamma,n\}$. Lieb and Thirring were able to show that $L\_\{\backslash gamma,n\}=L\_\{\backslash gamma,n\}^\{\backslash mathrm\{cl\}\}$ for $\backslash gamma\backslash ge\; 3/2,\; n=1$. M. Aizenman and E. H. Lieb {{cite journal | last1=Aizenman | first1=Michael | last2=Lieb | first2=Elliott H. | title=On semi-classical bounds for eigenvalues of Schrödinger operators | journal=Physics Letters A | volume=66 | issue=6 | year=1978 | doi=10.1016/0375-9601(78)90385-7 | pages=427–429| bibcode=1978PhLA...66..427A }} proved that for fixed dimension $n$ the ratio $L\_\{\backslash gamma,n\}/L\_\{\backslash gamma,n\}^\{\backslash mathrm\{cl\}\}$ is a monotonic, non-increasing function of $\backslash gamma$. Subsequently $L\_\{\backslash gamma,n\}=L\_\{\backslash gamma,n\}^\{\backslash mathrm\{cl\}\}$ was also shown to hold for all $n$ when $\backslash gamma\backslash ge\; 3/2$ by A. Laptev and T. Weidl.{{cite journal | last1=Laptev | first1=Ari | last2=Weidl | first2=Timo | title=Sharp Lieb-Thirring inequalities in high dimensions | journal=Acta Mathematica | volume=184 | issue=1 | year=2000 | doi=10.1007/bf02392782 | pages=87–111|doi-access=free| arxiv=math-ph/9903007 }} For $\backslash gamma=1/2,\backslash ,n=1$ D. Hundertmark, E. H. Lieb and L. E. Thomas {{cite journal | last1=Hundertmark | first1=Dirk | last2=Lieb | first2=Elliott H. | last3=Thomas | first3=Lawrence E. | title=A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator | journal=Advances in Theoretical and Mathematical Physics | volume=2 | issue=4 | year=1998 | doi=10.4310/atmp.1998.v2.n4.a2 | pages=719–731| doi-access=free }} proved that the best constant is given by $L\_\{1/2,1\}=2L\_\{1/2,1\}^\{\backslash mathrm\{cl\}\}=1/2$.

On the other hand, it is known that

In the former case Lieb and Thirring conjectured that the sharp constant is given by

:$$

L_{\gamma,1}=2L^\mathrm{cl}_{\gamma,1}\left(\frac{\gamma-\frac12}{\gamma+\frac12}\right)^{\gamma-\frac12}.

The best known value for the physical relevant constant $L\_\{1,3\}$ is $1.456\; L\_\{1,3\}^\backslash mathrm\{cl\}$ {{cite journal | last1=Frank | first1=Rupert | last2=Hundertmark | first2=Dirk | last3=Jex | first3=Michal | last4=Nam | first4=Phan Thành | title=The Lieb-Thirring inequality revisited | journal=Journal of the European Mathematical Society | year=2021 | doi=10.4171/JEMS/1062 |volume=10|issue=4| pages=2583–2600|doi-access=free| arxiv=1808.09017 }} and the smallest known constant in the Cwikel–Lieb–Rosenbljum inequality is $6.869L\_\{0,3\}^\backslash mathrm\{cl\}$. A complete survey of the presently best known values for $L\_\{\backslash gamma,n\}$ can be found in the literature.{{cite journal |last=Laptev |first=Ari |title=Spectral inequalities for Partial Differential Equations and their applications |journal=AMS/IP Studies in Advanced Mathematics |volume=51 |pages=629–643}}

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

:$$

\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 $1\backslash le\; i,j\backslash le\; N$, the one-body density is defined as

:$$

\rho_\psi(x)

=N\int_{\Reals^{(N-1)n}}|\psi(x,x_2\dots,x_N)|^2

\mathrm{d}^n x_2\cdots\mathrm{d}^n x_{N},\, x\in\Reals^n.

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

{{NumBlk|:|$$

\sum_{i=1}^N \int_{\Reals^n}|\nabla_i\psi|^2\mathrm{d}^n x_i\ge K_n\int_{\Reals^n}{\rho_\psi(x)^{1+\frac 2n}}\mathrm{d}^n x

|{{EquationRef|2}}}}

where the sharp constant $K\_n$ is defined via

:$$

\left(\left(1+\frac2n\right)K_n\right)^{1+\frac n2}\left(\left(1+\frac n2\right)L_{1,n}\right)^{1+\frac2n}=1\,.

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

:$$

\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 $1\backslash le\; i,j\backslash le\; N$, the constant $K\_n$ has to be replaced by $K\_n/N^\{2/n\}$. Inequality ({{EquationNote|2}}) describes the minimum kinetic energy necessary to achieve a given density $\backslash rho\_\backslash psi$ with $N$ particles in $n$ dimensions. If $L\_\{1,3\}=L^\backslash mathrm\{cl\}\_\{1,3\}$ was proven to hold, the right-hand side of ({{EquationNote|2}}) for $n=3$ would be precisely the kinetic energy term in Thomas–Fermi theory.

The inequality can be compared to the Sobolev inequality. M. Rumin{{cite journal | last=Rumin | first=Michel | title=Balanced distribution-energy inequalities and related entropy bounds | journal=Duke Mathematical Journal | volume=160 | issue=3 | year=2011 | doi=10.1215/00127094-1444305 | pages=567–597|mr=2852369| arxiv=1008.1674 | s2cid=638691 }} derived the kinetic energy inequality ({{EquationNote|2}}) (with a smaller constant) directly without the use of the Lieb–Thirring inequality.

(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 Hamiltonian under consideration describes a system of $N$ particles with $q$ spin states and $M$ fixed nuclei at locations $R\_j\backslash in\backslash Reals^3$ with charges $Z\_j>0$. 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 $\backslash psi$ is antisymmetric), then the kinetic energy inequality ({{EquationNote|2}}) holds with the constant $K\_n/q^\{2/n\}$ (not $K\_n/N^\{2/n\}$). This is a crucial ingredient in the proof of stability of matter for a system of fermions. It ensures that the ground state energy $E\_\{N,M\}(Z\_1,\backslash dots,Z\_M)$ of the system can be bounded from below by a constant depending only on the maximum of the nuclei charges, $Z\_\{\backslash max\}$, times the number of particles,

:$$

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 be bosons (i.e. the wave function $\backslash psi$ is symmetric), then the kinetic energy inequality ({{EquationNote|2}}) holds only with the constant $K\_n/N^\{2/n\}$ and for the ground state energy only a bound of the form $-CN^\{5/3\}$ holds. Since the power $5/3$ 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 $-\backslash Delta=-\backslash nabla^2$ is replaced by $(\backslash mathrm\{i\}\backslash nabla+A(x))^2$, where $A(x)$ is a magnetic field vector potential in $\backslash Reals^n,$ the Lieb–Thirring inequality ({{EquationNote|1}}) remains true. The proof of this statement uses the diamagnetic inequality. Although all presently known constants $L\_\{\backslash gamma,n\}$ 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 $-\backslash Delta$. In particular for the operator $\backslash sqrt\{-\backslash Delta\}$, a Lieb–Thirring inequality similar to ({{EquationNote|1}}) holds with a different constant $L\_\{\backslash gamma,n\}$ and with the power on the right-hand side replaced by $\backslash gamma+n$. Analogously a kinetic inequality similar to ({{EquationNote|2}}) holds, with $1+2/n$ replaced by $1+1/n$, which can be used to prove stability of matter for the relativistic Schrödinger operator under additional assumptions on the charges $Z\_k$.{{cite journal | last1=Frank | first1=Rupert L. | last2=Lieb | first2=Elliott H. | last3=Seiringer | first3=Robert |author-link3=Robert Seiringer| title=Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators | journal=Journal of the American Mathematical Society | volume=21 | issue=4 | date=10 October 2007 | doi=10.1090/s0894-0347-07-00582-6 | pages=925–950|doi-access=free| url=https://authors.library.caltech.edu/77817/2/S0894-0347-07-00582-6.pdf }}

In essence, the Lieb–Thirring inequality ({{EquationNote|1}}) gives an upper bound on the distances of the eigenvalues $\backslash lambda\_j$ to the essential spectrum $[0,\backslash infty)$ in terms of the perturbation $V$. Similar inequalities can be proved for Jacobi operators.{{cite journal | last1=Hundertmark | first1=Dirk | last2=Simon | first2=Barry |author-link2=Barry Simon| title=Lieb–Thirring Inequalities for Jacobi Matrices | journal=Journal of Approximation Theory | volume=118 | issue=1 | year=2002 | doi=10.1006/jath.2002.3704 | pages=106–130|doi-access=free| arxiv=math-ph/0112027 }}

{{reflist}}

• {{cite book |author1=Lieb, E.H. |author2=Seiringer, R.| title=The stability of matter in quantum mechanics | year=2010 | edition=1st | publisher=Cambridge University Press |location=Cambridge | isbn=9780521191180}}
• {{cite book|last1=Hundertmark|first1=D.|chapter=Some bound state problems in quantum mechanics|series=Proceedings of Symposia in Pure Mathematics |title=Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday |date=2007|volume=76|pages=463–496|publisher=American Mathematical Society|location=Providence, RI|bibcode=2007stmp.conf..463H|isbn=978-0-8218-3783-2|editor1=Fritz Gesztesy|editor2=Percy Deift|editor3=Cherie Galvez|editor4=Peter Perry|editor5=Wilhelm Schlag}}

{{DEFAULTSORT:Lieb-Thirring inequality}}

Category:Inequalities (mathematics)