Lieb-Thirring inequalities

Inequalities concerning the negative eigenvalues of the Schrödinger operator (cf. also Schrödinger equation)

$$H=-\mathrm{\Delta }+V(x)$$

on $L^2({\mathbf{R}}^n)$, $n\ge 1$. With $e_1\le e_2\le \cdots <0$ denoting the negative eigenvalue(s) of $H$ (if any), the Lieb–Thirring inequalities state that for suitable $\gamma \ge 0$ and constants $L_{\gamma ,n}$

$$\begin{array}{}\text{(a1)}& \sum _{j\ge 1}|e_j{|}^{\gamma }\le L_{\gamma ,n}{\int }_{{\mathbf{R}}^n}{V}_{-}(x{)}^{\gamma +n/2}dx\end{array}$$

with $V_{-}(x):=max\{-V(x),0\}$. When $\gamma =0$, the left-hand side is just the number of negative eigenvalues. Such an inequality (a1) can hold if and only if

$$\begin{array}{}\text{(a2)}& \{\begin{array}{ll}\gamma \ge \frac{1}{2}& \text{ for }n=1,\\ \gamma >0& \text{ for }n=2,\\ \gamma \ge 0& \text{ for }n\ge 3.\end{array}\end{array}$$

The cases $\gamma >1/2$, $n=1$, $\gamma >0$, $n\ge 2$, were established by E.H. Lieb and W.E. Thirring [a14] in connection with their proof of stability of matter. The case $\gamma =1/2$, $n=1$, was established by T. Weidl [a16]. The case $\gamma =0$, $n\ge 3$, was established independently by M. Cwikel [a15], Lieb [a6] and G.V. Rosenbljum [a10] by different methods and is known as the CLR bound; the smallest known value (as of 1998) for $L_{0,n}$ is in [a6], [a7]. Closely associated with the inequality (a1) is the semi-classical approximation for $\sum |e{|}^{\gamma }$, which serves as a heuristic motivation for (a1). It is (cf. [a14]):

$$\begin{array}{rcl}\sum _{j\ge 1}|e_j{|}^{\gamma }& \approx & (2\pi {)}^{-n}{\int }_{{\mathbf{R}}^n\times {\mathbf{R}}^n}[p^2+V(x){]}_{-}^{\gamma }dpdx\\ & =& L_{\gamma ,n}^c{\int }_{{\mathbf{R}}^n}{V}_{-}(x{)}^{\gamma +n/2}dx,\end{array}$$

with

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

Indeed, $L_{\gamma ,n}^c<\mathrm{\infty }$ for all $\gamma \ge 0$, whereas (a1) holds only for the range given in (a2). It is easy to prove (by considering $V(x)=\lambda W(x)$ with $W$ smooth and $\lambda \to \mathrm{\infty }$) that

$$L_{\gamma ,n}\ge L_{\gamma ,n}^c.$$

An interesting, and mostly open (1998) problem is to determine the sharp value of the constant $L_{\gamma ,n}$, especially to find those cases in which $L_{\gamma ,n}=L_{\gamma ,n}^c$. M. Aizenman and Lieb [a3] proved that the ratio $R_{\gamma ,n}=L_{\gamma ,n}/L_{\gamma ,n}^c$ is a monotonically non-increasing function of $\gamma$. Thus, if $R_{\mathrm{\Gamma },n}=1$ for some $\mathrm{\Gamma }$, then $L_{\gamma ,n}=L_{\gamma ,n}^c$ for all $\gamma \ge \mathrm{\Gamma }$. The equality $L_{\frac{3}{2},n}=L_{\frac{3}{2},n}^c$ was proved for $n=1$ in [a14] and for $n>1$ in [a2] by A. Laptev and Weidl. (See also [a1].)

The following sharp constants are known:

$L_{\gamma ,n}=L_{\gamma ,n}^c$, all $\gamma \ge 3/2$, [a14], [a3], [a2];

$L_{1/2,1}=1/2$, [a11].

There is strong support for the conjecture [a14] that

$$\begin{array}{}\text{(a3)}& L_{\gamma ,1}=\frac{1}{\sqrt{\pi }(\gamma -\frac{1}{2})}\frac{\mathrm{\Gamma }(\gamma +1)}{\mathrm{\Gamma }(\gamma +1/2)}{\left(\frac{\gamma -\frac{1}{2}}{\gamma +\frac{1}{2}}\right)}^{\gamma +1/2}\end{array}$$

for $1/2<\gamma <3/2$. Instead of considering all the negative eigenvalues as in (a1), one can consider just $e_1$. Then for $\gamma$ as in (a2),

$$|e_1{|}^{\gamma }\le L_{\gamma ,n}^1{\int }_{{\mathbf{R}}^n}{V}_{-}(x{)}^{\gamma +n/2}dx.$$

Clearly, $L_{\gamma ,n}^1\le L_{\gamma ,n}$, but equality can hold, as in the cases $\gamma =1/2$ and $3/2$ for $n=1$. Indeed, the conjecture in (a3) amounts to $L_{\gamma ,1}^1=L_{\gamma ,1}$ for $1/2<\gamma <3/2$. The sharp value (a3) of $L_{\gamma ,n}^1$ is obtained by solving a differential equation [a14].

It has been conjectured that for $n\ge 3$, $L_{0,n}=L_{0,n}^1$. In any case, B. Helffer and D. Robert [a12] showed that for all $n$ and all $\gamma <1$, $L_{\gamma ,n}>L_{\gamma ,n}^c$.

The sharp constant $L_{0,n}^1$, $n\ge 3$, is related to the sharp constant $S_n$ in the Sobolev inequality

$$\begin{array}{}\text{(a4)}& \Vert \mathrm{\nabla }f{\Vert }_{L^2({\mathbf{R}}^n)}\ge S_n\Vert f{\Vert }_{L^{2n/(n-2)}({\mathbf{R}}^n)}\end{array}$$

by $L_{0,n}^1=(S_n{)}^{-n}$. By a "duality argument" [a14], the case $\gamma =1$ in (a1) can be converted into the following bound for the Laplace operator, $\mathrm{\Delta }$. This bound is referred to as a Lieb–Thirring kinetic energy inequality and its most important application is to the stability of matter [a8], [a14].

Let $f_1,f_2,\dots$ be any orthonormal sequence (finite or infinite, cf. also Orthonormal system) in $L^2({\mathbf{R}}^n)$ such that $\mathrm{\nabla }{f}_j\in L^2({\mathbf{R}}^n)$ for all $j\ge 1$. Associated with this sequence is a "density"

$$\begin{array}{}\text{(a5)}& \rho (x)=\sum _{j\ge 1}|f_j(x){|}^2.\end{array}$$

Then, with $K_n:=n(2/L_{1,n}{)}^{2/n}(n+2{)}^{-1-2/n}$,

$$\begin{array}{}\text{(a6)}& \sum _{j\ge 1}{\int }_{{\mathbf{R}}^n}|\mathrm{\nabla }{f}_j(x){|}^{2}dx\ge K_n{\int }_{{\mathbf{R}}^n}\rho (x{)}^{1+2/n}dx.\end{array}$$

This can be extended to anti-symmetric functions in $L^2({\mathbf{R}}^{nN})$. If $\mathrm{\Phi }=\mathrm{\Phi }(x_1,\dots ,x_N)$ is such a function, one defines, for $x\in {\mathbf{R}}^n$, $$\rho (x)=N{\int }_{{\mathbf{R}}^{n(N-1)}}|\mathrm{\Phi }(x,x_2,\dots ,x_N){|}^{2}d{x}_2\dots d{x}_N.$$

Then, if ${\int }_{{\mathbf{R}}^{nN}}|\mathrm{\Phi }{|}^2=1$, $$\begin{array}{}\text{(a7)}& {\int }_{R^{nN}}|\mathrm{\nabla }\mathrm{\Phi }{|}^2\ge K_n{\int }_{{\mathbf{R}}^n}\rho (x{)}^{1+2/n}dx.\end{array}$$

Note that the choice $\mathrm{\Phi }=(N!{)}^{-1/2}\det f_j(x_k){|}_{j,k=1}^N$ with $f_j$ orthonormal reduces the general case (a7) to (a6). If the conjecture $L_{1,3}=L_{1,3}^c$ is correct, then the bound in (a7) equals the Thomas–Fermi kinetic energy Ansatz (cf. Thomas–Fermi theory), and hence it is a challenge to prove this conjecture. In the meantime, see [a7], [a5] for the best available constants to date (1998).

Of course, $\int (\mathrm{\nabla }f{)}^2=\int f(-\mathrm{\Delta }f)$. Inequalities of the type (a7) can be found for other powers of $-\mathrm{\Delta }$ than the first power. The first example of this kind, due to I. Daubechies [a13], and one of the most important physically, is to replace $-\mathrm{\Delta }$ by $\sqrt{-\mathrm{\Delta }}$ in $H$. Then an inequality similar to (a1) holds with $\gamma +n/2$ replaced by $\gamma +n$ (and with a different $L_{\gamma ,n_1}$, of course). Likewise there is an analogue of (a7) with $1+2/n$ replaced by $1+1/n$.

All proofs of (a1) (except [a11] and [a16]) actually proceed by finding an upper bound to $N_E(V)$, the number of eigenvalues of $H=-\mathrm{\Delta }+V(x)$ that are below $-E$. Then, for $\gamma >0$, $$\sum |e{|}^{\gamma }=\gamma {\int }_0^{\mathrm{\infty }}{N}_E(V)E^{\gamma -1}dE.$$

Assuming $V=-V_{-}$ (since $V_{+}$ only raises the eigenvalues), $N_E(V)$ is most accessible via the positive semi-definite Birman–Schwinger kernel (cf. [a4]) $$K_E(V)=\sqrt{V_{-}}(-\mathrm{\Delta }+E{)}^{-1}\sqrt{V_{-}}.$$

$e<0$ is an eigenvalue of $H$ if and only if $1$ is an eigenvalue of $K_{|e|}(V)$. Furthermore, $K_E(V)$ is operator that is monotone decreasing in $E$, and hence $N_E(V)$ equals the number of eigenvalues of $K_E(V)$ that are greater than $1$.

An important generalization of (a1) is to replace $-\mathrm{\Delta }$ in $H$ by $|i\mathrm{\nabla }+A(x){|}^2$, where $A(x)$ is some arbitrary vector field in ${\mathbf{R}}^n$ (called a magnetic vector potential). Then (a1) still holds, but it is not known if the sharp value of $L_{\gamma ,n}$ changes. What is known is that all presently (1998) known values of $L_{\gamma ,n}$ are unchanged. It is also known that $(-\mathrm{\Delta }+E{)}^{-1}$, as a kernel in ${\mathbf{R}}^n\times {\mathbf{R}}^n$, is pointwise greater than the absolute value of the kernel $(|i\mathrm{\nabla }+A{|}^2+E{)}^{-1}$. There is another family of inequalities for orthonormal functions, which is closely related to (a1) and to the CLR bound [a9]. As before, let $f_1,\dots ,f_N$ be $N$ orthonormal functions in $L^2({\mathbf{R}}^n)$ and set

$$u_j=(-\mathrm{\Delta }+m^2{)}^{-1/2}{f}_j,$$

$$\rho (x)=\sum _{j=1}^N|u_j(x){|}^2.$$

$u_j$ is a Riesz potential ($m=0$) or a Bessel potential ($m>0$) of $f_j$. If $n=1$ and $m>0$, then $\rho \in C^{0,1/2}({\mathbf{R}}^n)$ and $\Vert \rho {\Vert }_{L^{\mathrm{\infty }}(\mathbf{R})}\le L/m$. If $n=2$ and $m>0$, then for all $1\le p<\mathrm{\infty }$, $\Vert \rho {\Vert }_{L^p(R^2)}\le B_p{m}^{-2/p}{N}^{1/p}$. If $n\ge 3$, $p=n/(n-2)$ and $m\ge 0$ (including $m=0$), then $\Vert \rho {\Vert }_{L^p({\mathbf{R}}^n)}\le A_n{N}^{1/p}$. Here, $L$, $B_p$, $A_n$ are universal constants. Without the orthogonality, $N^{1/p}$ would have to be replaced by $N$.

Further generalizations are possible [a9].

References[edit]

Elliott H. Lieb

Copyright to this article is held by Elliott Lieb.