View source for Lieb-Thirring inequalities

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

$$
\begin{equation*} H = - \Delta + V ( x ) \end{equation*}
$$

on $L ^ { 2 } ( \mathbf{R} ^ { n } )$, $n \geq 1$. With $e _ { 1 } \leq e _ { 2 } \leq \ldots &lt; 0$ denoting the negative eigenvalue(s) of $H$
(if any), the Lieb–Thirring inequalities state that for suitable $\gamma \geq 0$ and constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l1201007.png"/>,

$$
\begin{equation} \tag{a1} \sum _ { j \geq 1 } | e _ { j } | ^ { \gamma } \leq L _ { \gamma , n } \int _ { \mathbf{R} ^ { n } } V _ { - } ( x ) ^ { \gamma + n / 2 } d x \end{equation}
$$

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

$$
\begin{equation} \tag{a2} \left\{ \begin{array} { l l } { \gamma \geq \frac { 1 } { 2 } } &amp; { \text { for } n= 1, } \\ { \gamma &gt; 0 } &amp; { \text { for }n = 2, } \\ { \gamma \geq 0 } &amp; { \text { for } n\geq 3. } \end{array} \right. \end{equation}
$$

The cases $\gamma &gt; 1 / 2$, $n = 1$, $\gamma &gt; 0$, $n \geq 2$, were established by E.H. Lieb and W.E. Thirring [[#References|[a14]]] in connection with their proof of stability of matter. The case $\gamma = 1 / 2$, $n = 1$, was established by T. Weidl [[#References|[a16]]]. The case $\gamma = 0$, $n \geq 3$, was established independently by M. Cwikel [[#References|[a15]]], Lieb [[#References|[a6]]] and G.V. Rosenbljum [[#References|[a10]]] by different methods and is known as the CLR bound; the smallest known value (as of 1998) for $L _ { 0 , n }$ is in [[#References|[a6]]], [[#References|[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. [[#References|[a14]]]):

$$
\begin{equation*} \sum _ { j \geq 1 } | e | ^ { \gamma } \approx \end{equation*}
$$

$$
\begin{equation*} \approx ( 2 \pi ) ^ { - n } \int _ { {\bf R} ^ { n } \times {\bf R} ^ { n } } [ p ^ { 2 } + V ( x ) ] _ { - } ^ { \gamma } d p d x = \end{equation*}
$$

$$
\begin{equation*} = L _ { \gamma , n } ^ { c } \int _ { {\bf R} ^ { n } } V _ { - } ( x ) ^ { \gamma + n / 2 } d x, \end{equation*}
$$

with

$$
\begin{equation*} L _ { \gamma , n } ^ { c } = 2 ^ { - n } \pi ^ { - n / 2 } \frac { \Gamma ( \gamma + 1 ) } { \Gamma ( \gamma + 1 + n / 2 ) }. \end{equation*}
$$

Indeed, $L _ { \gamma , n } ^ { c } &lt; \infty$ for all $\gamma \geq 0$, 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 $\lambda \rightarrow \infty$) that

$$
\begin{equation*} L _ { \gamma , n } \geq L _ { \gamma , n } ^ { c }. \end{equation*}
$$

An interesting, and mostly open (1998) problem is to determine the sharp value of the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010032.png"/>, especially to find those cases in which $L _ { \gamma , n } = L _ { \gamma , n } ^ { c }$. M. Aizenman and Lieb [[#References|[a3]]] proved that the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010034.png"/> is a monotonically non-increasing function of $\gamma$. Thus, if $R _ { \Gamma , n } = 1$ for some $\Gamma$, then $L _ { \gamma , n } = L _ { \gamma , n } ^ { c }$ for all $\gamma \geq \Gamma$. The equality $L _ { \frac { 3 } { 2 } ,\, n } = L _ { \frac { 3 } { 2 } ,\, n } ^ { c }$ was proved for $n = 1$ in [[#References|[a14]]] and for $n &gt; 1$ in [[#References|[a2]]] by A. Laptev and Weidl. (See also [[#References|[a1]]].)

The following sharp constants are known:

$L _ { \gamma , n } = L _ { \gamma , n } ^ { c }$, all $\gamma \geq 3 / 2$, [[#References|[a14]]], [[#References|[a3]]], [[#References|[a2]]];

$L _ { 1 / 2,1 } = 1 / 2$, [[#References|[a11]]].

There is strong support for the conjecture [[#References|[a14]]] that

$$
\begin{equation} \tag{a3} L _ { \gamma , 1 } = \frac { 1 } { \sqrt { \pi } ( \gamma - \frac { 1 } { 2 } ) } \frac { \Gamma ( \gamma + 1 ) } { \Gamma ( \gamma + 1 / 2 ) } \left( \frac { \gamma - \frac { 1 } { 2 } } { \gamma + \frac { 1 } { 2 } } \right) ^ { \gamma + 1 / 2 } \end{equation}
$$

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

$$
\begin{equation*} | e _ { 1 } | ^ { \gamma } \leq L _ { \gamma , n } ^ { 1 } \int _ { \mathbf{R} ^ { n } } V _ { - } ( x ) ^ { \gamma + n / 2 } d x. \end{equation*}
$$

Clearly, $L _ { \gamma , n } ^ { 1 } \leq L _ { \gamma ,n  }$, 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 $1 / 2 &lt; \gamma &lt; 3 / 2$. The sharp value (a3) of $L _ { \gamma , n} ^ { 1 }$ is obtained by solving a differential equation [[#References|[a14]]].

It has been conjectured that for $n \geq 3$, $L _ { 0 ,\, n } = L _ { 0 ,\, n } ^ { 1 }$. In any case, B. Helffer and D. Robert [[#References|[a12]]] showed that for all $n$ and all $\gamma &lt; 1$, $L _ { \gamma , n } &gt; L _ { \gamma , n } ^ { c }$.

The sharp constant $L _ { 0 , n } ^ { 1 }$, $n \geq 3$, is related to the sharp constant $S _ { n }$ in the Sobolev inequality

$$
\begin{equation} \tag{a4} \| \nabla f \| _ {{ L } ^ 2   ( \mathbf{R} ^ { n } ) } \geq S _ { n } \| f \| _ { L ^{ 2 n / ( n - 2 ) } ( \mathbf{R} ^ { n } ) } \end{equation}
$$

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

Let $f _ { 1 } , f _ { 2 } , \ldots$ be any orthonormal sequence (finite or infinite, cf. also [[Orthonormal system|Orthonormal system]]) in $L ^ { 2 } ( \mathbf{R} ^ { n } )$ such that $\nabla f _ { j } \in L ^ { 2 } ( \mathbf{R} ^ { n } )$ for all $j \geq 1$. Associated with this sequence is a "density" 

$$
\begin{equation} \tag{a5} \rho ( x ) = \sum _ { j \geq 1 } | f _ { j } ( x ) | ^ { 2 }. \end{equation}
$$

Then, with $K _ { n } : = n ( 2 / L _ { 1 , n } ) ^ { 2 / n } ( n + 2 ) ^ { - 1 - 2 / n }$,

$$
\begin{equation} \tag{a6} \sum _ { j \geq 1 } \int _ { \mathbf{R} ^ { n } } | \nabla f _ { j } ( x ) | ^ { 2 } d x \geq K _ { n } \int _ { \mathbf{R} ^ { n } } \rho ( x ) ^ { 1 + 2 / n } d x. \end{equation}
$$

This can be extended to anti-symmetric functions in $L ^ { 2 } ( \mathbf{R} ^ { n N } )$. If $\Phi = \Phi ( x _ { 1 } , \dots , x _ { N } )$ is such a function, one defines, for $x \in \mathbf{R} ^ { n }$,
$$
\begin{equation*} \rho ( x ) = N \int _ { \mathbf{R} ^ { n ( N - 1 ) } } | \Phi ( x , x _ { 2 } , \ldots , x _ { N } ) | ^ { 2 } d x _ { 2 } \ldots d x _ { N }. \end{equation*}
$$

Then, if $\int _ { \mathbf{R} ^ { n N } } | \Phi | ^ { 2 } = 1$,
$$
\begin{equation} \tag{a7} \int _ { R ^ { n N } } | \nabla \Phi | ^ { 2 } \geq K _ { n } \int _ { {\bf R} ^ { n } } \rho ( x ) ^ { 1 + 2 / n } d x. \end{equation}
$$

Note that the choice $\Phi = ( N ! ) ^ { - 1 / 2 } \operatorname { det } f _ { j } ( x _ { k } ) | _ { j , k = 1 } ^ { N }$ 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|Thomas–Fermi theory]]), and hence it is a challenge to prove this conjecture. In the meantime, see [[#References|[a7]]], [[#References|[a5]]] for the best available constants to date (1998).

Of course, $\int ( \nabla f ) ^ { 2 } = \int f ( - \Delta f )$. Inequalities of the type (a7) can be found for other powers of $- \Delta$ than the first power. The first example of this kind, due to I. Daubechies [[#References|[a13]]], and one of the most important physically, is to replace $- \Delta$ by $\sqrt { - \Delta }$ 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 [[#References|[a11]]] and [[#References|[a16]]]) actually proceed by finding an upper bound to $N _ { E } ( V )$, the number of eigenvalues of $H = - \Delta + V ( x )$ that are below $- E$. Then, for $\gamma &gt; 0$,
$$
\begin{equation*} \sum | e | ^ { \gamma } = \gamma \int _ { 0 } ^ { \infty } N _ { E } ( V ) E ^ { \gamma - 1 } d E. \end{equation*}
$$

Assuming $V = - V _ { - }$ (since $V _ { + }$ only raises the eigenvalues), $N _ { E } ( V )$ is most accessible via the positive semi-definite Birman–Schwinger kernel (cf. [[#References|[a4]]])
$$
\begin{equation*} K _ { E } ( V ) = \sqrt { V _ { - } } ( - \Delta + E ) ^ { - 1 } \sqrt { V _ { - } }. \end{equation*}
$$

$e &lt; 0$ 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 $- \Delta$ in $H$ by $| i \nabla + A ( x ) | ^ { 2 }$, where $A ( x )$ is some arbitrary vector field in ${\bf R} ^ { n }$ (called a magnetic vector potential). Then (a1) still holds, but it is not known if the sharp value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100119.png"/> changes. What is known is that all presently (1998) known values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100120.png"/> are unchanged. It is also known that $( - \Delta + E ) ^ { - 1 }$, as a kernel in $\mathbf{R} ^ { n } \times \mathbf{R} ^ { n }$, is pointwise greater than the absolute value of the kernel $( | i \nabla + A | ^ { 2 } + E ) ^ { - 1 }$. There is another family of inequalities for orthonormal functions, which is closely related to (a1) and to the CLR bound [[#References|[a9]]]. As before, let $f _ { 1 } , \dots , f _ { N }$ be $N$ orthonormal functions in $L ^ { 2 } ( \mathbf{R} ^ { n } )$ and set

$$
\begin{equation*} u _ { j } = ( - \Delta + m ^ { 2 } ) ^ { - 1 / 2 } f _ { j }, \end{equation*}
$$

$$
\begin{equation*} \rho ( x ) = \sum _ { j = 1 } ^ { N } | u _ { j } ( x ) | ^ { 2 }. \end{equation*}
$$

$u _ { j }$ is a [[Riesz potential|Riesz potential]] ($m = 0$) or a [[Bessel potential|Bessel potential]] ($m &gt; 0$) of $f_j$. If $n = 1$ and $m &gt; 0$, then $\rho \in C ^ { 0,1 / 2 } ( \mathbf{R} ^ { n } )$ and $\| \rho \| _ { L^\infty ( {\bf R} )} \leq L / m$. If $n = 2$ and $m &gt; 0$, then for all $1 \leq p &lt; \infty$, $\| \rho \| _ { L ^ { p } ( R ^ { 2 } ) } \leq B _ { p } m ^ { - 2 / p } N ^ { 1 / p }$. If $n \geq 3$, $p = n / ( n - 2 )$ and $m \geq 0$ (including $m = 0$), then $\| \rho \| _ { L ^ { p } ( \mathbf{R} ^ { n } ) } \leq A _ { n } N ^ { 1 / p }$. 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 [[#References|[a9]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">

R. Benguria,  M. Loss,  "A simple proof of a theorem of Laptev and Weidl" ''Preprint'' (1999)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">

A. Laptev,  T. Weidl,  "Sharp Lieb–Thirring inequalities in high dimensions" ''Acta Math.'' (in press 1999)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">

M.A. Aizenman,  E.H. Lieb,  "On semiclassical bounds for eigenvalues of Schrödinger operators" ''Phys. Lett.'' , '''66A''' (1978) pp. 427–429</td></tr><tr><td valign="top">[a4]</td> <td valign="top">

B. Simon,  "Functional integration and quantum physics" , ''Pure Appl. Math.'' , '''86''' , Acad. Press (1979)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">

Ph. Blanchard,  J. Stubbe,  "Bound states for Schrödinger Hamiltonians: phase space methods and applications" ''Rev. Math. Phys.'' , '''8''' (1996) pp. 503–547</td></tr><tr><td valign="top">[a6]</td> <td valign="top">

E.H. Lieb,  "The numbers of bound states of one-body Schrödinger operators and the Weyl problem" , ''Geometry of the Laplace Operator (Honolulu, 1979)'' , ''Proc. Symp. Pure Math.'' , '''36''' , Amer. Math. Soc. (1980) pp. 241–251</td></tr><tr><td valign="top">[a7]</td> <td valign="top">

E.H. Lieb,  "On characteristic exponents in turbulence" ''Comm. Math. Phys.'' , '''92''' (1984) pp. 473–480</td></tr><tr><td valign="top">[a8]</td> <td valign="top">

E.H. Lieb,  "Kinetic energy bounds and their applications to the stability of matter" H. Holden (ed.) A. Jensen (ed.) , ''Schrödinger Operators (Proc. Nordic Summer School, 1988)'' , ''Lecture Notes Physics'' , '''345''' , Springer (1989) pp. 371–382</td></tr><tr><td valign="top">[a9]</td> <td valign="top">

E.H. Lieb,  "An $L ^ { p }$ bound for the Riesz and Bessel potentials of orthonormal functions" ''J. Funct. Anal.'' , '''51''' (1983) pp. 159–165</td></tr><tr><td valign="top">[a10]</td> <td valign="top">

G.V. Rosenbljum,  "Distribution of the discrete spectrum of singular differential operators" ''Dokl. Akad. Nauk SSSR'' , '''202''' (1972) pp. 1012–1015 ((The details are given in: Izv. Vyss. Uchebn. Zaved. Mat. 164 (1976), 75-86 (English transl.: Soviet Math. (Izv. VUZ) 20 (1976), 63-71)))</td></tr><tr><td valign="top">[a11]</td> <td valign="top">

D. Hundertmark,  E.H. Lieb,  L.E. Thomas,  "A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator" ''Adv. Theor. Math. Phys.'' , '''2''' (1998) pp. 719–731</td></tr><tr><td valign="top">[a12]</td> <td valign="top">

B. Helffer,  D. Robert,  "Riesz means of bound states and semi-classical limit connected with a Lieb–Thirring conjecture, II" ''Ann. Inst. H. Poincaré Phys. Th.'' , '''53''' (1990) pp. 139–147</td></tr><tr><td valign="top">[a13]</td> <td valign="top">

I. Daubechies,  "An uncertainty principle for fermions with generalized kinetic energy" ''Comm. Math. Phys.'' , '''90''' (1983) pp. 511–520</td></tr><tr><td valign="top">[a14]</td> <td valign="top">

E.H. Lieb,  W. Thirring,  "Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities" E. Lieb (ed.) B. Simon (ed.) A. Wightman (ed.) , ''Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann'' , Princeton Univ. Press (1976) pp. 269–303 ((See also: W. Thirring (ed.), The stability of matter: from the atoms to stars, Selecta of E.H. Lieb, Springer, 1977))</td></tr><tr><td valign="top">[a15]</td> <td valign="top">

M. Cwikel,  "Weak type estimates for singular values and the number of bound states of Schrödinger operators" ''Ann. Math.'' , '''106''' (1977) pp. 93–100</td></tr><tr><td valign="top">[a16]</td> <td valign="top">

T. Weidl,  "On the Lieb–Thirring constants $L_{ \gamma , 1}$ for $\gamma \geq 1 / 2$" ''Comm. Math. Phys.'' , '''178''' :  1 (1996) pp. 135–146</td></tr></table>

''Elliott H. Lieb''

Copyright to this article is held by Elliott Lieb.