Lieb-Thirring inequalities

Inequalities concerning the negative

eigenvalues of the

Schrödinger operator

(cf. also

Schrödinger equation)

on

,

.

With

denoting the negative eigenvalue(s) of

(if any), the Lieb–Thirring

inequalities state that for suitable

and constants

,

(a1)

with

.

When

,

the left-hand side is just the number of negative eigenvalues. Such an

inequality

(a1)

can hold if and only if

(a2)

The cases

,

,

,

,

were established by

E.H. Lieb

and

W.E. Thirring

[a14]

in connection with their proof of

stability of matter.

The case

,

,

was established by

T. Weidl

[a16].

The case

,

,

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

is in

[a6],

[a7].

Closely associated with the inequality

(a1)

is the

semi-classical approximation

for

,

which serves as a heuristic motivation for

(a1).

It is

(cf.

[a14]):

with

Indeed,

for all

,

whereas

(a1)

holds only for the range given in

(a2).

It is easy to prove (by considering

with

smooth and

)

that

An interesting, and mostly open

(1998)

problem is to determine the sharp

value

of the constant

,

especially to find those cases in which

.

M. Aizenman

and

Lieb

[a3]

proved that the ratio

is a monotonically non-increasing function of

.

Thus, if

for some

,

then

for all

.

The equality

was proved for

in

[a14]

and for

in

[a2]

by

A. Laptev

and

Weidl.

(See also

[a1].)

The following sharp constants are known:

,

all

,

[a14],

[a3],

[a2];

,

[a11].

There is strong support for the

conjecture

[a14]

that

(a3)

for

.

Instead of considering all the negative eigenvalues as in

(a1),

one can consider just

.

Then for

as in

(a2),

Clearly,

,

but equality can hold, as in the cases

and

for

.

Indeed, the conjecture in

(a3)

amounts to

for

.

The sharp value

(a3)

of

is obtained by solving a differential equation

[a14].

It has been conjectured that for

,

.

In any case,

B. Helffer

and

D. Robert

[a12]

showed that for all

and all

,

.

The sharp constant

,

,

is related to the sharp constant

in the

Sobolev inequality

(a4)

by

.

By a

"duality argument"

[a14],

the case

in

(a1)

can be converted into the following bound for the

Laplace operator,

.

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

be

any

orthonormal sequence (finite or infinite, cf. also

Orthonormal system)

in

such that

for all

.

Associated with this sequence is a

"density"

(a5)

Then, with

,

(a6)

This can be extended to

anti-symmetric functions

in

.

If

is such a function, one defines, for

,

Then, if

,

(a7)

Note that the choice

with

orthonormal reduces the general case

(a7)

to

(a6).

If the conjecture

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,

.

Inequalities of the type

(a7)

can be found for other powers of

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

by

in

.

Then an inequality similar to

(a1)

holds with

replaced by

(and with a different

,

of course). Likewise there is an analogue of

(a7)

with

replaced by

.

All proofs of

(a1)

(except

[a11]

and

[a16])

actually

proceed by finding an upper bound to

,

the number of eigenvalues of

that are below

.

Then, for

,

Assuming

(since

only raises the eigenvalues),

is most accessible via the positive semi-definite

Birman–Schwinger kernel

(cf.

[a4])

is an eigenvalue of

if and only if

is an eigenvalue of

.

Furthermore,

is

operator

that is monotone decreasing in

,

and hence

equals the number of eigenvalues of

that are greater than

.

An important generalization of

(a1)

is to replace

in

by

,

where

is some arbitrary vector field in

(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

,

as a kernel in

,

is pointwise greater than the absolute value of the kernel

.

There is another family of inequalities for orthonormal functions,

which is closely related to

(a1)

and to the CLR

bound

[a9].

As before, let

be

orthonormal functions in

and set

is a

Riesz potential

()

or a

Bessel potential

()

of

.

If

and

,

then

and

.

If

and

,

then for all

,

.

If

,

and

(including

),

then

.

Here,

,

,

are universal constants. Without the orthogonality,

would have to be replaced by

.

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 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 for " Comm. Math. Phys. , 178 1 (1996) pp. 135–146

Elliott H. Lieb

Copyright to this article is held by Elliott Lieb.