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Inequalities concerning the negative
 
Inequalities concerning the negative
  
Line 9: Line 17:
 
[[Schrödinger equation|Schrödinger equation]])
 
[[Schrödinger equation|Schrödinger equation]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l1201001.png" /></td> </tr></table>
+
\begin{equation*} H = - \Delta + V ( x ) \end{equation*}
  
 
on
 
on
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l1201002.png" />,
+
$L ^ { 2 } ( \mathbf{R} ^ { n } )$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l1201003.png" />.
+
$n \geq 1$.
  
 
With
 
With
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l1201004.png" />
+
$e _ { 1 } \leq e _ { 2 } \leq \ldots &lt; 0$
  
 
denoting the negative eigenvalue(s) of
 
denoting the negative eigenvalue(s) of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l1201005.png" />
+
$H$
  
 
(if any), the Lieb–Thirring
 
(if any), the Lieb–Thirring
Line 29: Line 37:
 
inequalities state that for suitable
 
inequalities state that for suitable
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l1201006.png" />
+
$\gamma \geq 0$
  
 
and constants
 
and constants
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l1201007.png" />,
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l1201007.png"/>,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l1201008.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\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
 
with
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l1201009.png" />.
+
$V _ { - } ( x ) : = \operatorname { max } \{ - V ( x ) , 0 \}$.
  
 
When
 
When
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010010.png" />,
+
$\gamma = 0$,
  
 
the left-hand side is just the number of negative eigenvalues. Such an
 
the left-hand side is just the number of negative eigenvalues. Such an
Line 53: Line 61:
 
can hold if and only if
 
can hold if and only if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\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
 
The cases
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010012.png" />,
+
$\gamma &gt; 1 / 2$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010013.png" />,
+
$n = 1$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010014.png" />,
+
$\gamma &gt; 0$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010015.png" />,
+
$n \geq 2$,
  
 
were established by
 
were established by
Line 81: Line 89:
 
The case
 
The case
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010016.png" />,
+
$\gamma = 1 / 2$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010017.png" />,
+
$n = 1$,
  
 
was established by
 
was established by
Line 93: Line 101:
 
The case
 
The case
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010018.png" />,
+
$\gamma = 0$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010019.png" />,
+
$n \geq 3$,
  
 
was established independently by
 
was established independently by
Line 125: Line 133:
 
for
 
for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010020.png" />
+
$L _ { 0 , n }$
  
 
is in
 
is in
Line 143: Line 151:
 
for
 
for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010021.png" />,
+
$\sum | e | ^ { \gamma }$,
  
 
which serves as a heuristic motivation for
 
which serves as a heuristic motivation for
Line 155: Line 163:
 
[[#References|[a14]]]):
 
[[#References|[a14]]]):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010022.png" /></td> </tr></table>
+
\begin{equation*} \sum _ { j \geq 1 } | e | ^ { \gamma } \approx \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010023.png" /></td> </tr></table>
+
\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*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010024.png" /></td> </tr></table>
+
\begin{equation*} = L _ { \gamma , n } ^ { c } \int _ { {\bf R} ^ { n } } V _ { - } ( x ) ^ { \gamma + n / 2 } d x, \end{equation*}
  
 
with
 
with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010025.png" /></td> </tr></table>
+
\begin{equation*} L _ { \gamma , n } ^ { c } = 2 ^ { - n } \pi ^ { - n / 2 } \frac { \Gamma ( \gamma + 1 ) } { \Gamma ( \gamma + 1 + n / 2 ) }. \end{equation*}
  
 
Indeed,
 
Indeed,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010026.png" />
+
$L _ { \gamma , n } ^ { c } &lt; \infty$
  
 
for all
 
for all
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010027.png" />,
+
$\gamma \geq 0$,
  
 
whereas
 
whereas
Line 183: Line 191:
 
It is easy to prove (by considering
 
It is easy to prove (by considering
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010028.png" />
+
$V ( x ) = \lambda W ( x )$
  
 
with
 
with
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010029.png" />
+
$W$
  
 
smooth and
 
smooth and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010030.png" />)
+
$\lambda \rightarrow \infty$)
  
 
that
 
that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010031.png" /></td> </tr></table>
+
\begin{equation*} L _ { \gamma , n } \geq L _ { \gamma , n } ^ { c }. \end{equation*}
  
 
An interesting, and mostly open
 
An interesting, and mostly open
Line 207: Line 215:
 
of the constant
 
of the constant
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010032.png" />,
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010032.png"/>,
  
 
especially to find those cases in which
 
especially to find those cases in which
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010033.png" />.
+
$L _ { \gamma , n } = L _ { \gamma , n } ^ { c }$.
  
 
M. Aizenman
 
M. Aizenman
Line 223: Line 231:
 
proved that the ratio
 
proved that the ratio
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010034.png" />
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010034.png"/>
  
 
is a monotonically non-increasing function of
 
is a monotonically non-increasing function of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010035.png" />.
+
$\gamma$.
  
 
Thus, if
 
Thus, if
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010036.png" />
+
$R _ { \Gamma , n } = 1$
  
 
for some
 
for some
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010037.png" />,
+
$\Gamma$,
  
 
then
 
then
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010038.png" />
+
$L _ { \gamma , n } = L _ { \gamma , n } ^ { c }$
  
 
for all
 
for all
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010039.png" />.
+
$\gamma \geq \Gamma$.
  
 
The equality
 
The equality
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010040.png" />
+
$L _ { \frac { 3 } { 2 } ,\, n } = L _ { \frac { 3 } { 2 } ,\, n } ^ { c }$
  
 
was proved for
 
was proved for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010041.png" />
+
$n = 1$
  
 
in
 
in
Line 259: Line 267:
 
and for
 
and for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010042.png" />
+
$n &gt; 1$
  
 
in
 
in
Line 279: Line 287:
 
The following sharp constants are known:
 
The following sharp constants are known:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010043.png" />,
+
$L _ { \gamma , n } = L _ { \gamma , n } ^ { c }$,
  
 
all
 
all
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010044.png" />,
+
$\gamma \geq 3 / 2$,
  
 
[[#References|[a14]]],
 
[[#References|[a14]]],
Line 291: Line 299:
 
[[#References|[a2]]];
 
[[#References|[a2]]];
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010045.png" />,
+
$L _ { 1 / 2,1 } = 1 / 2$,
  
 
[[#References|[a11]]].
 
[[#References|[a11]]].
Line 303: Line 311:
 
that
 
that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\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
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010047.png" />.
+
$1 / 2 &lt; \gamma &lt; 3 / 2$.
  
 
Instead of considering all the negative eigenvalues as in
 
Instead of considering all the negative eigenvalues as in
Line 315: Line 323:
 
one can consider just
 
one can consider just
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010048.png" />.
+
$e_1$.
  
 
Then for
 
Then for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010049.png" />
+
$\gamma$
  
 
as in
 
as in
Line 325: Line 333:
 
(a2),
 
(a2),
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010050.png" /></td> </tr></table>
+
\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,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010051.png" />,
+
$L _ { \gamma , n } ^ { 1 } \leq L _ { \gamma ,n  }$,
  
 
but equality can hold, as in the cases
 
but equality can hold, as in the cases
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010052.png" />
+
$\gamma = 1 / 2$
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010053.png" />
+
$3 / 2$
  
 
for
 
for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010054.png" />.
+
$n = 1$.
  
 
Indeed, the conjecture in
 
Indeed, the conjecture in
Line 349: Line 357:
 
amounts to
 
amounts to
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010055.png" />
+
$L _ { \gamma , 1 } ^ { 1 } = L _ { \gamma , 1 }$
  
 
for
 
for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010056.png" />.
+
$1 / 2 &lt; \gamma &lt; 3 / 2$.
  
 
The sharp value
 
The sharp value
Line 361: Line 369:
 
of
 
of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010057.png" />
+
$L _ { \gamma , n} ^ { 1 }$
  
 
is obtained by solving a differential equation
 
is obtained by solving a differential equation
Line 369: Line 377:
 
It has been conjectured that for
 
It has been conjectured that for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010058.png" />,
+
$n \geq 3$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010059.png" />.
+
$L _ { 0 ,\, n } = L _ { 0 ,\, n } ^ { 1 }$.
  
 
In any case,
 
In any case,
Line 385: Line 393:
 
showed that for all
 
showed that for all
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010060.png" />
+
$n$
  
 
and all
 
and all
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010061.png" />,
+
$\gamma &lt; 1$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010062.png" />.
+
$L _ { \gamma , n } &gt; L _ { \gamma , n } ^ { c }$.
  
 
The sharp constant
 
The sharp constant
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010063.png" />,
+
$L _ { 0 , n } ^ { 1 }$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010064.png" />,
+
$n \geq 3$,
  
 
is related to the sharp constant
 
is related to the sharp constant
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010065.png" />
+
$S _ { n }$
  
 
in the
 
in the
Line 407: Line 415:
 
Sobolev inequality
 
Sobolev inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010066.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
\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
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010067.png" />.
+
$L _ { 0 , n } ^ { 1 } = ( S _ { n } ) ^ { - n }$.
  
 
By a
 
By a
Line 421: Line 429:
 
the case
 
the case
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010068.png" />
+
$\gamma = 1$
  
 
in
 
in
Line 431: Line 439:
 
[[Laplace operator|Laplace operator]],
 
[[Laplace operator|Laplace operator]],
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010069.png" />.
+
$\Delta$.
  
 
This bound is
 
This bound is
Line 449: Line 457:
 
Let
 
Let
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010070.png" />
+
$f _ { 1 } , f _ { 2 } , \ldots$
  
 
be
 
be
Line 461: Line 469:
 
in
 
in
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010071.png" />
+
$L ^ { 2 } ( \mathbf{R} ^ { n } )$
  
 
such that
 
such that
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010072.png" />
+
$\nabla f _ { j } \in L ^ { 2 } ( \mathbf{R} ^ { n } )$
  
 
for all
 
for all
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010073.png" />.
+
$j \geq 1$.
  
 
Associated with this sequence is a
 
Associated with this sequence is a
Line 475: Line 483:
 
"density"  
 
"density"  
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010074.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
+
\begin{equation} \tag{a5} \rho ( x ) = \sum _ { j \geq 1 } | f _ { j } ( x ) | ^ { 2 }. \end{equation}
  
 
Then, with
 
Then, with
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010075.png" />,
+
$K _ { n } : = n ( 2 / L _ { 1 , n } ) ^ { 2 / n } ( n + 2 ) ^ { - 1 - 2 / n }$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010076.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
+
\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
Line 489: Line 497:
 
in
 
in
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010077.png" />.
+
$L ^ { 2 } ( \mathbf{R} ^ { n N } )$.
  
 
If
 
If
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010078.png" />
+
$\Phi = \Phi ( x _ { 1 } , \dots , x _ { N } )$
  
 
is such a function, one defines, for
 
is such a function, one defines, for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010079.png" />,
+
$x \in \mathbf{R} ^ { n }$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010080.png" /></td> </tr></table>
+
\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
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010081.png" />,
+
$\int _ { \mathbf{R} ^ { n N } } | \Phi | ^ { 2 } = 1$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010082.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
+
\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
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010083.png" />
+
$\Phi = ( N ! ) ^ { - 1 / 2 } \operatorname { det } f _ { j } ( x _ { k } ) | _ { j , k = 1 } ^ { N }$
  
 
with
 
with
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010084.png" />
+
$f_j$
  
 
orthonormal reduces the general case
 
orthonormal reduces the general case
Line 525: Line 533:
 
If the conjecture
 
If the conjecture
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010085.png" />
+
$L _ { 1,3 } = L _ { 1,3 } ^ { c }$
  
 
is correct, then the bound in
 
is correct, then the bound in
Line 553: Line 561:
 
Of course,
 
Of course,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010086.png" />.
+
$\int ( \nabla f ) ^ { 2 } = \int f ( - \Delta f )$.
  
 
Inequalities of the type
 
Inequalities of the type
Line 561: Line 569:
 
can be found for other powers of
 
can be found for other powers of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010087.png" />
+
$- \Delta$
  
 
than the first power. The first example of this kind, due to
 
than the first power. The first example of this kind, due to
Line 571: Line 579:
 
and one of the most important physically, is to replace
 
and one of the most important physically, is to replace
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010088.png" />
+
$- \Delta$
  
 
by
 
by
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010089.png" />
+
$\sqrt { - \Delta }$
  
 
in
 
in
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010090.png" />.
+
$H$.
  
 
Then an inequality similar to
 
Then an inequality similar to
Line 587: Line 595:
 
holds with
 
holds with
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010091.png" />
+
$\gamma + n / 2$
  
 
replaced by
 
replaced by
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010092.png" />
+
$\gamma + n$
  
 
(and with a different
 
(and with a different
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010093.png" />,
+
$L _ { \gamma  , n _ { 1 }}$,
  
 
of course). Likewise there is an analogue of
 
of course). Likewise there is an analogue of
Line 603: Line 611:
 
with
 
with
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010094.png" />
+
$1 + 2 / n$
  
 
replaced by
 
replaced by
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010095.png" />.
+
$1 + 1 / n$.
  
 
All proofs of
 
All proofs of
Line 625: Line 633:
 
proceed by finding an upper bound to
 
proceed by finding an upper bound to
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010096.png" />,
+
$N _ { E } ( V )$,
  
 
the number of eigenvalues of
 
the number of eigenvalues of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010097.png" />
+
$H = - \Delta + V ( x )$
  
 
that are below
 
that are below
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010098.png" />.
+
$- E$.
  
 
Then, for
 
Then, for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l12010099.png" />,
+
$\gamma &gt; 0$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100100.png" /></td> </tr></table>
+
\begin{equation*} \sum | e | ^ { \gamma } = \gamma \int _ { 0 } ^ { \infty } N _ { E } ( V ) E ^ { \gamma - 1 } d E. \end{equation*}
  
 
Assuming
 
Assuming
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100101.png" />
+
$V = - V _ { - }$
  
 
(since
 
(since
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100102.png" />
+
$V _ { + }$
  
 
only raises the eigenvalues),
 
only raises the eigenvalues),
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100103.png" />
+
$N _ { E } ( V )$
  
 
is most accessible via the positive semi-definite
 
is most accessible via the positive semi-definite
Line 661: Line 669:
 
[[#References|[a4]]])
 
[[#References|[a4]]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100104.png" /></td> </tr></table>
+
\begin{equation*} K _ { E } ( V ) = \sqrt { V _ { - } } ( - \Delta + E ) ^ { - 1 } \sqrt { V _ { - } }. \end{equation*}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100105.png" />
+
$e &lt; 0$
  
 
is an eigenvalue of
 
is an eigenvalue of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100106.png" />
+
$H$
  
 
if and only if
 
if and only if
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100107.png" />
+
$1$
  
 
is an eigenvalue of
 
is an eigenvalue of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100108.png" />.
+
$K _ { |e| } ( V )$.
  
 
Furthermore,
 
Furthermore,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100109.png" />
+
$K _ { E } ( V )$
  
 
is
 
is
Line 687: Line 695:
 
that is monotone decreasing in
 
that is monotone decreasing in
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100110.png" />,
+
$E$,
  
 
and hence
 
and hence
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100111.png" />
+
$N _ { E } ( V )$
  
 
equals the number of eigenvalues of
 
equals the number of eigenvalues of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100112.png" />
+
$K _ { E } ( V )$
  
 
that are greater than
 
that are greater than
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100113.png" />.
+
$1$.
  
 
An important generalization of
 
An important generalization of
Line 707: Line 715:
 
is to replace
 
is to replace
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100114.png" />
+
$- \Delta$
  
 
in
 
in
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100115.png" />
+
$H$
  
 
by
 
by
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100116.png" />,
+
$| i \nabla + A ( x ) | ^ { 2 }$,
  
 
where
 
where
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100117.png" />
+
$A ( x )$
  
 
is some arbitrary vector field in
 
is some arbitrary vector field in
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100118.png" />
+
${\bf R} ^ { n }$
  
 
(called a
 
(called a
Line 735: Line 743:
 
still holds, but it is not known if the sharp value of
 
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" />
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100119.png"/>
  
 
changes. What is known is that all
 
changes. What is known is that all
Line 745: Line 753:
 
known values of
 
known values of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100120.png" />
+
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100120.png"/>
  
 
are unchanged. It is also known that
 
are unchanged. It is also known that
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100121.png" />,
+
$( - \Delta + E ) ^ { - 1 }$,
  
 
as a kernel in
 
as a kernel in
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100122.png" />,
+
$\mathbf{R} ^ { n } \times \mathbf{R} ^ { n }$,
  
 
is pointwise greater than the absolute value of the kernel
 
is pointwise greater than the absolute value of the kernel
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100123.png" />.
+
$( | i \nabla + A | ^ { 2 } + E ) ^ { - 1 }$.
  
 
There is another family of inequalities for orthonormal functions,
 
There is another family of inequalities for orthonormal functions,
Line 773: Line 781:
 
As before, let
 
As before, let
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100124.png" />
+
$f _ { 1 } , \dots , f _ { N }$
  
 
be
 
be
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100125.png" />
+
$N$
  
 
orthonormal functions in
 
orthonormal functions in
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100126.png" />
+
$L ^ { 2 } ( \mathbf{R} ^ { n } )$
  
 
and set
 
and set
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100127.png" /></td> </tr></table>
+
\begin{equation*} u _ { j } = ( - \Delta + m ^ { 2 } ) ^ { - 1 / 2 } f _ { j }, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100128.png" /></td> </tr></table>
+
\begin{equation*} \rho ( x ) = \sum _ { j = 1 } ^ { N } | u _ { j } ( x ) | ^ { 2 }. \end{equation*}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100129.png" />
+
$u _ { j }$
  
 
is a
 
is a
Line 795: Line 803:
 
[[Riesz potential|Riesz potential]]
 
[[Riesz potential|Riesz potential]]
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100130.png" />)
+
($m = 0$)
  
 
or a
 
or a
Line 801: Line 809:
 
[[Bessel potential|Bessel potential]]
 
[[Bessel potential|Bessel potential]]
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100131.png" />)
+
($m &gt; 0$)
  
 
of
 
of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100132.png" />.
+
$f_j$.
  
 
If
 
If
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100133.png" />
+
$n = 1$
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100134.png" />,
+
$m &gt; 0$,
  
 
then
 
then
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100135.png" />
+
$\rho \in C ^ { 0,1 / 2 } ( \mathbf{R} ^ { n } )$
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100136.png" />.
+
$\| \rho \| _ { L^\infty ( {\bf R} )} \leq L / m$.
  
 
If
 
If
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100137.png" />
+
$n = 2$
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100138.png" />,
+
$m &gt; 0$,
  
 
then for all
 
then for all
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100139.png" />,
+
$1 \leq p &lt; \infty$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100140.png" />.
+
$\| \rho \| _ { L ^ { p } ( R ^ { 2 } ) } \leq B _ { p } m ^ { - 2 / p } N ^ { 1 / p }$.
  
 
If
 
If
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100141.png" />,
+
$n \geq 3$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100142.png" />
+
$p = n / ( n - 2 )$
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100143.png" />
+
$m \geq 0$
  
 
(including
 
(including
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100144.png" />),
+
$m = 0$),
  
 
then
 
then
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100145.png" />.
+
$\| \rho \| _ { L ^ { p } ( \mathbf{R} ^ { n } ) } \leq A _ { n } N ^ { 1 / p }$.
  
 
Here,
 
Here,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100146.png" />,
+
$L$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100147.png" />,
+
$B _ { p }$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100148.png" />
+
$A _ { n }$
  
 
are universal constants. Without the orthogonality,
 
are universal constants. Without the orthogonality,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100149.png" />
+
$N ^ { 1 / p }$
  
 
would have to be replaced by
 
would have to be replaced by
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100150.png" />.
+
$N$.
  
 
Further generalizations are
 
Further generalizations are
Line 878: Line 886:
  
 
====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,  
Line 888: Line 896:
 
''Preprint''
 
''Preprint''
  
(1999)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">
+
(1999)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">
  
 
A. Laptev,  
 
A. Laptev,  
Line 898: Line 906:
 
''Acta Math.''
 
''Acta Math.''
  
(in press 1999)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">
+
(in press 1999)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">
  
 
M.A. Aizenman,  
 
M.A. Aizenman,  
Line 912: Line 920:
 
(1978)
 
(1978)
  
pp. 427–429</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">
+
pp. 427–429</td></tr><tr><td valign="top">[a4]</td> <td valign="top">
  
 
B. Simon,  
 
B. Simon,  
Line 924: Line 932:
 
, Acad. Press
 
, Acad. Press
  
(1979)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">
+
(1979)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">
  
 
Ph. Blanchard,  
 
Ph. Blanchard,  
Line 938: Line 946:
 
(1996)
 
(1996)
  
pp. 503–547</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">
+
pp. 503–547</td></tr><tr><td valign="top">[a6]</td> <td valign="top">
  
 
E.H. Lieb,  
 
E.H. Lieb,  
Line 954: Line 962:
 
(1980)
 
(1980)
  
pp. 241–251</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">
+
pp. 241–251</td></tr><tr><td valign="top">[a7]</td> <td valign="top">
  
 
E.H. Lieb,  
 
E.H. Lieb,  
Line 966: Line 974:
 
(1984)
 
(1984)
  
pp. 473–480</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">
+
pp. 473–480</td></tr><tr><td valign="top">[a8]</td> <td valign="top">
  
 
E.H. Lieb,  
 
E.H. Lieb,  
Line 986: Line 994:
 
(1989)
 
(1989)
  
pp. 371–382</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">
+
pp. 371–382</td></tr><tr><td valign="top">[a9]</td> <td valign="top">
  
 
E.H. Lieb,  
 
E.H. Lieb,  
Line 992: Line 1,000:
 
"An
 
"An
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100151.png" /> bound for the Riesz and Bessel potentials of orthonormal functions"
+
$L ^ { p }$ bound for the Riesz and Bessel potentials of orthonormal functions"
  
 
''J. Funct. Anal.''
 
''J. Funct. Anal.''
Line 1,000: Line 1,008:
 
(1983)
 
(1983)
  
pp. 159–165</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">
+
pp. 159–165</td></tr><tr><td valign="top">[a10]</td> <td valign="top">
  
 
G.V. Rosenbljum,  
 
G.V. Rosenbljum,  
Line 1,014: Line 1,022:
 
pp. 1012–1015
 
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">
+
((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,  
 
D. Hundertmark,  
Line 1,030: Line 1,038:
 
(1998)
 
(1998)
  
pp. 719–731</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">
+
pp. 719–731</td></tr><tr><td valign="top">[a12]</td> <td valign="top">
  
 
B. Helffer,  
 
B. Helffer,  
Line 1,044: Line 1,052:
 
(1990)
 
(1990)
  
pp. 139–147</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">
+
pp. 139–147</td></tr><tr><td valign="top">[a13]</td> <td valign="top">
  
 
I. Daubechies,  
 
I. Daubechies,  
Line 1,056: Line 1,064:
 
(1983)
 
(1983)
  
pp. 511–520</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">
+
pp. 511–520</td></tr><tr><td valign="top">[a14]</td> <td valign="top">
  
 
E.H. Lieb,  
 
E.H. Lieb,  
Line 1,078: Line 1,086:
 
pp. 269–303
 
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">
+
((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,  
 
M. Cwikel,  
Line 1,090: Line 1,098:
 
(1977)
 
(1977)
  
pp. 93–100</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">
+
pp. 93–100</td></tr><tr><td valign="top">[a16]</td> <td valign="top">
  
 
T. Weidl,  
 
T. Weidl,  
Line 1,096: Line 1,104:
 
"On the Lieb–Thirring constants
 
"On the Lieb–Thirring constants
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100152.png" /> for
+
$L_{ \gamma , 1}$ for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l120/l120100/l120100153.png" />"
+
$\gamma \geq 1 / 2$"
  
 
''Comm. Math. Phys.''
 
''Comm. Math. Phys.''
Line 1,108: Line 1,116:
 
(1996)
 
(1996)
  
pp. 135–146</TD></TR></table>
+
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.

Revision as of 17:01, 1 July 2020

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

$e _ { 1 } \leq e _ { 2 } \leq \ldots < 0$

denoting the negative eigenvalue(s) of

$H$

(if any), the Lieb–Thirring

inequalities state that for suitable

$\gamma \geq 0$

and constants

,

\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 } } & { \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{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,

$L _ { \gamma , n } ^ { c } < \infty$

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

,

especially to find those cases in which

$L _ { \gamma , n } = L _ { \gamma , n } ^ { c }$.

M. Aizenman

and

Lieb

[a3]

proved that the ratio

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

changes. What is known is that all

presently

(1998)

known values of

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=50401