Lieb-Thirring inequalities

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

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

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

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

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

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

The cases $\gamma > 1 / 2$, $n = 1$, $\gamma > 0$, $n \geq 2$, 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 $\gamma = 0$, $n \geq 3$, 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]):

$$ \begin{eqnarray*} \sum _ { j \geq 1 } | e_j | ^ { \gamma } &\approx& ( 2 \pi ) ^ { - n } \int _ { {\bf R} ^ { n } \times {\bf R} ^ { n } } [ p ^ { 2 } + V ( x ) ] _ { - } ^ { \gamma } d p d x \\ &=& L _ { \gamma , n } ^ { c } \int _ { {\bf R} ^ { n } } V _ { - } ( x ) ^ { \gamma + n / 2 } d x, \end{eqnarray*} $$

with

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

Indeed, $L _ { \gamma , n } ^ { c } < \infty$ 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 $L_{\gamma, n}$, 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 $R _ { \Gamma , n } = 1$ 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 [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 $\gamma \geq 3 / 2$, [a14], [a3], [a2];

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

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

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

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 $1 / 2 < \gamma < 3 / 2$. The sharp value (a3) of $L _ { \gamma , n} ^ { 1 }$ is obtained by solving a differential equation [a14].

It has been conjectured that for $n \geq 3$, $L _ { 0 ,\, n } = L _ { 0 ,\, n } ^ { 1 }$. 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 $L _ { 0 , n } ^ { 1 }$, $n \geq 3$, 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" [a14], the case $\gamma = 1$ in (a1) can be converted into the following bound for the 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 [a8], [a14].

Let $f _ { 1 } , f _ { 2 } , \ldots$ be any orthonormal sequence (finite or infinite, cf. also 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), 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, $\int ( \nabla f ) ^ { 2 } = \int f ( - \Delta f )$. 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 [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 [a11] and [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 $\gamma > 0$, $$ \begin{equation*} \sum | e | ^ { \gamma } = \gamma \int _ { 0 } ^ { \infty } N _ { E } ( V ) E ^ { \gamma - 1 } d E. \end{equation*} $$

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

$e < 0$ 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 $- \Delta$ 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 $L_{\gamma,n}$ changes. What is known is that all presently (1998) known values of $L_{\gamma,n}$ are unchanged. It is also known that $( - \Delta + E ) ^ { - 1 }$, 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 [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 ($m = 0$) 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 $\| \rho \| _ { L^\infty ( {\bf R} )} \leq L / m$. If $n = 2$ and $m > 0$, then for all $1 \leq p < \infty$, $\| \rho \| _ { L ^ { p } ( R ^ { 2 } ) } \leq B _ { p } m ^ { - 2 / p } N ^ { 1 / p }$. 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 [a9].

References[edit]

Elliott H. Lieb

Copyright to this article is held by Elliott Lieb.