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Inequalities concerning the negative
+
Inequalities concerning the negative eigenvalues of the Schrödinger operator (cf. also [[Schrödinger equation|Schrödinger equation]])
eigenvalues of the
 
Schrödinger operator
 
(cf. also
 
[[Schrödinger equation|Schrödinger equation]])
 
  
 
$$
 
$$
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$$
 
$$
  
on
+
on $L ^ { 2 } ( \mathbf{R} ^ { n } )$, $n \geq 1$. With $e _ { 1 } \leq e _ { 2 } \leq \cdots < 0$ denoting the negative eigenvalue(s) of $H$
$L ^ { 2 } ( \mathbf{R} ^ { n } )$, $n \geq 1$.
+
(if any), the Lieb–Thirring inequalities state that for suitable $\gamma \geq 0$ and constants $L_{\gamma,n}$
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"/>,
 
  
 
$$
 
$$
Line 33: Line 20:
 
$$
 
$$
  
with
+
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
$V _ { - } ( x ) : = \operatorname { max } \{ - V ( x ) , 0 \}$.
+
 
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}
 
\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*}
 
 
$$
 
$$
  
$$
+
The cases $\gamma > 1 / 2$, $n = 1$, $\gamma > 0$, $n \geq 2$, 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*} \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*}
+
\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
 
with
 +
 
$$
 
$$
 
\begin{equation*} L _ { \gamma , n } ^ { c } = 2 ^ { - n } \pi ^ { - n / 2 } \frac { \Gamma ( \gamma + 1 ) } { \Gamma ( \gamma + 1 + n / 2 ) }. \end{equation*}
 
\begin{equation*} L _ { \gamma , n } ^ { c } = 2 ^ { - n } \pi ^ { - n / 2 } \frac { \Gamma ( \gamma + 1 ) } { \Gamma ( \gamma + 1 + n / 2 ) }. \end{equation*}
 
$$
 
$$
Indeed,
 
  
$L _ { \gamma , n } ^ { c } &lt; \infty$
+
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
 
 
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*}
 
\begin{equation*} L _ { \gamma , n } \geq L _ { \gamma , n } ^ { c }. \end{equation*}
 +
$$
  
An interesting, and mostly open
+
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 [[#References|[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 [[#References|[a14]]] and for $n > 1$ in [[#References|[a2]]] by A. Laptev and Weidl. (See also [[#References|[a1]]].)
 
 
(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:
 
The following sharp constants are known:
  
$L _ { \gamma , n } = L _ { \gamma , n } ^ { c }$,
+
$L _ { \gamma , n } = L _ { \gamma , n } ^ { c }$, all $\gamma \geq 3 / 2$, [[#References|[a14]]], [[#References|[a3]]], [[#References|[a2]]];
  
all
+
$L _ { 1 / 2,1 } = 1 / 2$, [[#References|[a11]]].
  
$\gamma \geq 3 / 2$,
+
There is strong support for the conjecture [[#References|[a14]]] that
 
 
[[#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}
 
\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
+
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),
 
 
$1 / 2 &lt; \gamma &lt; 3 / 2$.
 
 
 
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*}
 
\begin{equation*} | e _ { 1 } | ^ { \gamma } \leq L _ { \gamma , n } ^ { 1 } \int _ { \mathbf{R} ^ { n } } V _ { - } ( x ) ^ { \gamma + n / 2 } d x. \end{equation*}
 +
$$
  
Clearly,
+
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 [[#References|[a14]]].
  
$L _ { \gamma , n } ^ { 1 } \leq L _ { \gamma ,n }$,
+
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 $\gamma < 1$, $L _ { \gamma , n } > L _ { \gamma , n } ^ { c }$.
  
but equality can hold, as in the cases
+
The sharp constant $L _ { 0 , n } ^ { 1 }$, $n \geq 3$, is related to the sharp constant $S _ { n }$ in the Sobolev inequality
 
 
$\gamma = 1 / 2$
 
 
 
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}
 
\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
+
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]]].
  
$L _ { 0 , n } ^ { 1 } = ( S _ { n } ) ^ { - n }$.
+
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"  
 
 
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}
 
\begin{equation} \tag{a5} \rho ( x ) = \sum _ { j \geq 1 } | f _ { j } ( x ) | ^ { 2 }. \end{equation}
 +
$$
  
Then, with
+
Then, with $K _ { n } : = n ( 2 / L _ { 1 , n } ) ^ { 2 / n } ( n + 2 ) ^ { - 1 - 2 / n }$,
 
 
$K _ { n } : = n ( 2 / L _ { 1 , n } ) ^ { 2 / n } ( n + 2 ) ^ { - 1 - 2 / n }$,
 
  
 +
$$
 
\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}
 
\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
+
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 $x \in \mathbf{R} ^ { n }$,
 
+
$$
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*}
 
\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
+
Then, if $\int _ { \mathbf{R} ^ { n N } } | \Phi | ^ { 2 } = 1$,
 
+
$$
$\int _ { \mathbf{R} ^ { n N } } | \Phi | ^ { 2 } = 1$,
 
 
 
 
\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}
 
\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
+
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).
  
$\Phi = ( N ! ) ^ { - 1 / 2 } \operatorname { det } f _ { j } ( x _ { k } ) | _ { j , k = 1 } ^ { N }$
+
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 $1 + 1 / n$.
 
 
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
 
 
 
$\gamma &gt; 0$,
 
  
 +
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 $\gamma > 0$,
 +
$$
 
\begin{equation*} \sum | e | ^ { \gamma } = \gamma \int _ { 0 } ^ { \infty } N _ { E } ( V ) E ^ { \gamma - 1 } d E. \end{equation*}
 
\begin{equation*} \sum | e | ^ { \gamma } = \gamma \int _ { 0 } ^ { \infty } N _ { E } ( V ) E ^ { \gamma - 1 } d E. \end{equation*}
 +
$$
  
Assuming
+
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]]])
 
+
$$
$V = - V _ { - }$
 
 
 
(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*}
 
\begin{equation*} K _ { E } ( V ) = \sqrt { V _ { - } } ( - \Delta + E ) ^ { - 1 } \sqrt { V _ { - } }. \end{equation*}
 +
$$
  
$e &lt; 0$
+
$e < 0$ 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 $1$.
  
is an eigenvalue of
+
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 [[#References|[a9]]]. As before, let $f _ { 1 } , \dots , f _ { N }$ be $N$ orthonormal functions in $L ^ { 2 } ( \mathbf{R} ^ { n } )$ and set
 
 
$H$
 
 
 
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*} 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*}
 
\begin{equation*} \rho ( x ) = \sum _ { j = 1 } ^ { N } | u _ { j } ( x ) | ^ { 2 }. \end{equation*}
 +
$$
  
$u _ { j }$
+
$u _ { j }$ is a [[Riesz potential|Riesz potential]] ($m = 0$) or a [[Bessel potential|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 $1 \leq p &lt; \infty$, $\| \rho \| _ { L ^ { p } ( R ^ { 2 } ) } \leq B _ { p } m ^ { - 2 / p } N ^ { 1 / p }$. 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 $N$.
 
 
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,
 
 
 
$L$,
 
  
$B _ { p }$,
+
Further generalizations are possible [[#References|[a9]]].
 
 
$A _ { n }$
 
 
 
are universal constants. Without the orthogonality,
 
 
 
$N ^ { 1 / p }$
 
 
 
would have to be replaced by
 
 
 
$N$.
 
 
 
Further generalizations are
 
 
 
possible
 
 
 
[[#References|[a9]]].
 
  
 
====References====
 
====References====
 
<table><tr><td valign="top">[a1]</td> <td valign="top">
 
<table><tr><td valign="top">[a1]</td> <td valign="top">
  
R. Benguria,  
+
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">
 
 
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,  
+
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">
  
"Weak type estimates for singular values and the number of bound states of Schrödinger operators"
+
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">
  
''Ann. Math.''
+
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">
  
, '''106'''
+
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">
  
(1977)
+
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">
  
pp. 93–100</td></tr><tr><td valign="top">[a16]</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">
  
T. Weidl,  
+
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">
  
"On the Lieb–Thirring constants
+
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">
  
$L_{ \gamma , 1}$ for
+
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">
  
$\gamma \geq 1 / 2$"
+
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">
  
''Comm. Math. Phys.''
+
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">
  
, '''178'''
+
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">
  
: 1
+
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">
  
(1996)
+
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">
  
pp. 135–146</td></tr></table>
+
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''
 
''Elliott H. Lieb''
  
 
Copyright to this article is held by Elliott Lieb.
 
Copyright to this article is held by Elliott Lieb.

Latest revision as of 19:26, 26 March 2023

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

[a1] R. Benguria, M. Loss, "A simple proof of a theorem of Laptev and Weidl" Preprint (1999)
[a2] A. Laptev, T. Weidl, "Sharp Lieb–Thirring inequalities in high dimensions" Acta Math. (in press 1999)
[a3] M.A. Aizenman, E.H. Lieb, "On semiclassical bounds for eigenvalues of Schrödinger operators" Phys. Lett. , 66A (1978) pp. 427–429
[a4] B. Simon, "Functional integration and quantum physics" , Pure Appl. Math. , 86 , Acad. Press (1979)
[a5] Ph. Blanchard, J. Stubbe, "Bound states for Schrödinger Hamiltonians: phase space methods and applications" Rev. Math. Phys. , 8 (1996) pp. 503–547
[a6] 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
[a7] E.H. Lieb, "On characteristic exponents in turbulence" Comm. Math. Phys. , 92 (1984) pp. 473–480
[a8] 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
[a9] E.H. Lieb, "An $L ^ { p }$ bound for the Riesz and Bessel potentials of orthonormal functions" J. Funct. Anal. , 51 (1983) pp. 159–165
[a10] 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)))
[a11] 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
[a12] 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
[a13] I. Daubechies, "An uncertainty principle for fermions with generalized kinetic energy" Comm. Math. Phys. , 90 (1983) pp. 511–520
[a14] 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))
[a15] M. Cwikel, "Weak type estimates for singular values and the number of bound states of Schrödinger operators" Ann. Math. , 106 (1977) pp. 93–100
[a16] T. Weidl, "On the Lieb–Thirring constants $L_{ \gamma , 1}$ for $\gamma \geq 1 / 2$" Comm. Math. Phys. , 178 : 1 (1996) pp. 135–146

Elliott H. Lieb

Copyright to this article is held by Elliott Lieb.

How to Cite This Entry:
Lieb-Thirring inequalities. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lieb-Thirring_inequalities&oldid=52484