Lieb-Thirring inequalities

Inequalities concerning the negative

eigenvalues of the

Schrödinger operator

(cf. also

\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

,

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

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

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

The cases

$\gamma > 1 / 2$,

$n = 1$,

$\gamma > 0$,

$n \geq 2$,

were established by

E.H. Lieb

and

W.E. Thirring

in connection with their proof of

stability of matter.

The case

$\gamma = 1 / 2$,

$n = 1$,

was established by

T. Weidl

The case

$\gamma = 0$,

$n \geq 3$,

was established independently by

M. Cwikel

Lieb

and

G.V. Rosenbljum

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

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

and for

$n > 1$

in

by

A. Laptev

and

Weidl.

[a1].)

The following sharp constants are known:

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

all

$\gamma \geq 3 / 2$,

[a3],

[a2];

$L _ { 1 / 2,1 } = 1 / 2$,

There is strong support for the

conjecture

that

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

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

It has been conjectured that for

$n \geq 3$,

$L _ { 0 ,\, n } = L _ { 0 ,\, n } ^ { 1 }$.

In any case,

B. Helffer

and

D. Robert

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

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

by

$L _ { 0 , n } ^ { 1 } = ( S _ { n } ) ^ { - n }$.

By a

"duality argument"

the case

$\gamma = 1$

in

(a1)

can be converted into the following bound for the

$\Delta$.

This bound is

referred to as a

Lieb–Thirring kinetic energy inequality

and its most important application is to the

stability of matter

[a8],

Let

$f _ { 1 } , f _ { 2 } , \ldots$

be

any

orthonormal sequence (finite or infinite, cf. also

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"

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

Then, with

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

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

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

$\int _ { \mathbf{R} ^ { n N } } | \Phi | ^ { 2 } = 1$,

$$\tag{a7} \int _ { R ^ { n N } } | \nabla \Phi | ^ { 2 } \geq K _ { n } \int _ { {\bf R} ^ { n } } \rho ( x ) ^ { 1 + 2 / n } d x.$$

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.

and hence it is a challenge to prove this conjecture. In the meantime,

see

[a7],

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

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

and

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

($m = 0$)

or a

($m > 0$)

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