We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schr¨odinger operators and Schr¨odinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue ?N+1 in terms of the lower spectrum, bounds on ratios of means of eigenvalues, and universal monotonicity properties of eigenvalue moments, which imply sharp versions of Lieb-Thirring inequalities. In the case of a Schr¨odinger operator on an immersed manifold of dimension d, we derive a version of Reilly's inequality bounding the eigenvalue ?N+1 of the Laplace-Beltrami operator by a universal constant times h 2?N2/d.

In [

The original intent of the present article was to use similar methods to study the spectrum of periodic Schr¨odinger operators, a case that has been much less considered from the point of view of universal spectral bounds. One reason for the lack of attention to periodic problems is that multiplication by a coordinate function does not preserve a core of the domain of self-adjointness of H, so the simple algebraic relations in the canonical case for [H, G] and [G, [H, G]] are not valid. We have therefore sought an alternative whereby H is commuted with a family of auxiliary operators G not assumed to be self-adjoint. Indeed the case of greatest interest will be when G is the unitary operator of multiplication by exp(?iq · x). The second section of this article contains an abstract trace formula of this sort (see Theorem 2.2), which exhibits useful simplifications when G is unitary.

The identity that forms the point of departure for the later parts of the
article turned out to have the same algebraic form as one that applies to Schro¨dinger
operators on embedded manifolds with bounded mean curvature [

On a Hilbert space H we consider a self-adjoint operator H with domain of
definition DH , along with a second linear operator G subject to some conditions
relating to DH . In many of the examples to be discussed in this article the spectrum
of H consists entirely of eigenvalues ?j , and the corresponding eigenfunctions ?j
are chosen to form an orthonormal basis of the underlying Hilbert space H. (The
extension needed if H has a continuous spectrum is not difficult and has been
explicitly presented in the case where G is self-adjoint in [

[H, G]?j 2 + [H, G?]?j 2 (z ? ?j)(z ? ?k)(?k ? ?j ) | G?j, ?k |2 + | G??j , ?k |2 .

If G = G?, then (2.1) reduces to a key identity used in [

In preparation for the proof of Theorem 2.1 and the presentation of the general trace identity, we collect some straightforward algebraic identities. Let P be a spectral projector of the self-adjoint operator H and define the pair of operators A and B by

We recall the standard inclusions Ran (A?A) ? Ran (P ), Ran (AA?) ? Ran (1 ? P ), Ran (B?B) ? Ran (1 ? P ) and Ran (BB?) ? Ran (P ).

Theorem 2.2. Let H be a self-adjoint operator on H and P be a spectral projector of H. Let G be a linear operator with domain DG and adjoint G? defined on DG? such that G(DH ) ? DH ? DG and G?(DH ) ? DH ? DG? , respectively. Then (2.3) tr H2(G?[H, G] + G[H, G?])P

? tr H([H, G?][H, G] + [H, G][H, G?])P = tr HA?H2A ? HAH2A? + tr HBH2B? ? HB?H2B .

Proof. Using the projectors P and 1 ? P we write

tr H2(G?[H, G] + G[H, G?])P tr H2(G?P [H, G] + GP [H, G?])P + tr H2(G?(1 ? P )[H, G] + G(1 ? P )[H, G?])P , the first term of which can be computed as tr H2(G?P [H, G] + GP [H, G?])P = tr H2G?P (HG ? GH)P + H2GP (HG? ? G?H)P = tr H(HG? ? G?H)P (HG ? GH)P + H(HG ? GH)P (HG? ? G?H)P + tr HG?HP (HG ? GH)P + HGHP (HG? ? G?H)P .

The final term in this expression vanishes thanks to the cyclic property of the trace (viz., tr(AB) = tr(BA)), and therefore tr H2(G?P [H, G] + GP [H, G?])P = tr H([H, G?]P [H, G] + [H, G]P [H, G?])P . Adding and subtracting the expression

tr H([H, G?](1 ? P )[H, G] + [H, G](1 ? P )[H, G?])P , we see that the left side of (2.3) equals tr H2(G?(1 ? P )[H, G] + G(1 ? P )[H, G?])P ? tr H([H, G?](1 ? P )[H, G] + [H, G](1 ? P )[H, G?])P = tr HG?H(1 ? P )[H, G]P + HGH(1 ? P )[H, G?]P = tr HP G?(1 ? P )H(1 ? P )[H, G]P + HP G(1 ? P )H(1 ? P )[H, G?]P = tr HA?H[H, A] + HBH[H, B?] .

Since commutators are not affected by replacing H by H ? z, we have an immediate corollary: Corollary 2.3. Under the assumptions of Theorem 2.2, for all z ? R: tr (z ? H)2(G?[H, G] + G[H, G?])P

+ tr (z ? H)([H, G?][H, G] + [H, G][H, G?])P

Proof of Theorem 2.1. We may write the first trace in Corollary 2.3 in terms of second commutators by applying the following algebraic identity, which is a direct computation:

When (2.5) is multiplied by P and the trace is taken, the last term vanishes, and for the left side of (2.3) we obtain (2.6) tr H2(G?[H, G] + G[H, G?])P =

tr H2([G?, [H, G]] + [G, [H, G?]])P .

If the spectrum of H consists only of eigenvalues ?j , with an orthonormal basis of eigenfunctions {?j }, then the trace identity (2.6) and Corollary 2.3 imply (z ? ?j ) [H, G]?j , [H, G]?j + [H, G?]?j , [H, G?]?j

(z ? ?j )(z ? ?k)(?k ? ?j ) | G?j , ?k |2 + | G??j , ?k |2 ,

If H has a gap in its spectrum, then we consider the spectral projector P that separates the two parts of the spectrum. Then for all z ? [?, ?], Theorem 2.4. Let G, H satisfy the assumptions of Theorem 2.2. Suppose there exist constants ? < ? such that

HP ? ? < ? ? H(1 ? P ).

Remark 2.5. While this formula only makes sense if the spectrum of HP is discrete, it is not necessary for the whole spectrum of H to be discrete.

Proof of Theorem 2.4. We bound each term of the right side of (2.4). Since z ? [?, ?] and Ran (A?A) ? Ran (P ), with the cyclic property of traces we get tr (z ? H)A(z ? H)2A? ? (z ? ?) tr A(z ? H)2A? = (z ? ?)(z ? ?) tr (z ? H)2A?A ? (z ? ?)(z ? ?) tr (z ? H)A?A .

Since Ran (AA?) ? Ran (1 ? P ), we obtain similarly ?tr (z ? H)A?(z ? H)2A

? (z ? ?)(z ? ?) tr (z ? H)AA? , and therefore tr (z ? H)A(z ? H)2A? ? (z ? H)A?(z ? H)2A ? (z ? ?)(z ? ?) tr (z ? H)[A?, A] .

In the same manner, we estimate the second trace in (2.4), obtaining tr (z ? H)A(z ? H)2A? ? (z ? H)A?(z ? H)2A

+ tr (z ? H)B?(z ? H)2B ? (z ? H)B(z ? H)2B? ? (z ? ?)(z ? ?) tr (z ? H)[A?, A] + (z ? H)[B, B?] .

Comparing the coefficients of z2 in Corollary 2.3, we see that tr (G?[H, G] + G[H, G?])P

= tr ((z ? H)[A?, A] + (z ? H)[B, B?]) , which proves the theorem.

If we take P = PH<z, the spectral projector onto the spectrum below z, then we can rewrite (2.8) as follows: (2.9) tr (z ? H)2+(G?[H, G] + G[H, G?])

+ tr (z ? H)+([H, G?][H, G] + [H, G][H, G?]) ? 0,
where (z ? H)+ := (z ? H)PH<z. We extend this inequality to the class of
tracecontrollable functions f of [

1

In this section we suppose that H is of the form Theorem 2.6. Let G, H satisfy the assumptions of Theorem 2.2 and let f be as above. Then tr f ((z ? H)+)(G?[H, G] + G[H, G?]) 1 2 + tr f ((z ? H)+)([H, G?][H, G] + [H, G][H, G?]) ? 0.

3. On the eigenvalues of periodic Schro¨dinger operators and (3.4) or (3.5)

H = ?? + V (x)
and is defined as a self-adjoint operator on L2(?), where ? ? Rd is a bounded
domain and the boundary conditions are such that the multiplication operator
G = exp(?iq · x) satisfies the domain-mapping conditions of Theorem 2.1. This
situation arises in the Floquet decomposition of H when V (x) is a real, periodic,
bounded measurable function [

In Section 5 we shall introduce a semiclassical parameter ? proportional to the square of Planck?s constant and study

H? = ??? + V (x).

Although it is always possible to reset ? > 0 to 1 by a change of scale, we introduce H? in order to study the semiclassical limit ? ? 0. A further possible extension would be to introduce a magnetic field through the systematic replacement of ? by ? + iA(x); this entails only minor changes, because in the key identities the magnetic vector potential A(x) occurs only in commutators that vanish. In the interest of clarity we leave this generalization as an exercise for the interested reader.

The commutators appearing in Theorem 2.1 are easily calculated: [H, G] = exp(?iq · x) |q|2 + 2iq · ?

[G?, [H, G]] = [G, [H, G?]] = 2|q|2. With these facts in hand, Theorem 2.1 reads

(z ? ?j )(z ? ?k)(?k ? ?j) | G?j, ?k |2 + | G??j , ?k |2 2|q|2 = ?j?J (z ? ?j ) |q|4 + 4 q · ??j 2 (z ? ?j)(z ? ?k)(?k ? ?j )wjkq, where wkjq := 12 | exp(?iq · x)?j , ?k |2 + | exp(iq · x)?j , ?k |2 . Here we collect some properties of wkjq and associated ?sum rules? for the eigenvalues: Proposition 3.1. The quantities wkjq compose an infinite doubly stochastic matrix, i.e.,

a) k wkjq = j wkjq = 1, with the symmetries b) 0 ? wkjq = wjkq and c) wkj?q = wkjq.

Moreover, and in particular, for each j, k ?j = (?k ? ?j )wkjq = |q|2, k

?kwkjq ? |q|2; k

(?k ? ?j )2wkjq = |q|4 + 4 q · ??j 2. (3.6) and (3.9) to write (3.5) as

N j=1 Proof. Properties a)-c) are immediate from the definition of wkjq and the completeness relation of the eigenfunctions.

Choosing J = {?j }, Identity (3.6) results from taking the second derivative of (3.5) with respect to z. Formula (3.7) is just a reformulation of (3.6). For (3.8), set z = ?k, multiply (3.5) by wkjq , and then sum on k.

In the spirit of [

(z ? ?j )2 ? |q|2(z ? ?j ) ? 4(z ? ?j )Tqj = Hqj , Tqj := q · ??j 2 q 2 | | and

Hqj := k (z ? ?i)(z ? ?k)(?k ? ?j ) wqkj2q .

| | For J = {?1, . . . , ?N }, we sum in j, defining

1 ?N :=

N ?2N := 1 N j?N j?N ?j ?j2 (z ? ?j )2 = N (z2 ? 2?N z + ?2 )

N = |q|2

N j=1

N j=1 (z ? ?j ) + 4 (z ? ?j )Tqj + H, H :=

N

? where (3.10) (3.11) (3.13) Let g := d1

d=1 q2, and define the ?Riesz means?

R?(z) :=

(z ? ?j )?+.

j R2(z) ? gR1(z) +

(z ? ?j )Tj . 4 d j (In the case of periodicity with respect to basis vectors {q } that are not orthogonal, the factor Tj may be replaced by CTj for a constant C ? 1 determined by the geometry of the set {q }.) Because of (3.10), this inequality is equivalent to the statement that a certain quadratic polynomial, viz., (3.12) z2 ? 2 + 4 d

4 ?N + g ? d VN z + 1 + 4 d

4
?2N + g?N ? d ?VN ? 0
for all z ? [?N , ?N+1]. Here, in keeping with the notation for the averages of
eigenvalues and their squares, VN := N1 j?N Vj and ?VN := N1 j?N ?j Vj . Letting
z = ?N+1, we obtain a universal inequality on ?N+1 to be compared with a similar
result for the Dirichlet Laplacian in [

DN , where DN , defined as the discriminant of the quadratic polynomial in (3.12), is guaranteed to be ? 0.

For the problem of Schr¨odinger operators on bounded domains with Dirichlet
conditions we may let g ? 0, and if moreover V = 0, then (3.11) reduces to the
inequality of Yang for the Dirichlet Laplacian [

for periodic Schro¨dinger operators and Schro¨dinger operators on manifolds of bounded mean curvature

Inequality (3.11) is identical in form to a bound that applies for a suitable value
of g to Schro¨dinger operators on immersed manifolds of dimension d, according to
Corollary 4.3 of [

We shall refer to an operator H = ?? + V (x) or H? = ??? + V (x) as a
Schr¨odinger operator on a manifold M of bounded mean curvature when ? ? M
is a domain in a smooth closed manifold M immersed with finite mean curvature
h := d=1 ? in Rd+1, Dirichlet conditions being imposed on ?? if it is nonempty,
and the potential V (x) is a real, periodic, bounded measurable function. Setting
? = 1, by Corollary 4.3 of [

As in [

A Schro¨dinger operator H on a manifold of bounded mean curvature similarly satisfies (4.2) with ? := sup4(h)2 ? sup V . The differential inequality (4.2) is easily solved, and we have thus proved: Theorem 4.1. Let H = ?? + V (x) be a periodic Schro¨dinger operator with fundamental domain M or a Schro¨dinger operator on a bounded manifold M of bounded where mean curvature and finite volume. Then the function

R2(z) (z + ? )2+d/2 , with ? = g4d ? sup V or respectively sup4(h)2 ? sup V , is nondecreasing for all z real. Consequently,

R2(z) (z + ? )2+d/2 ? Lc2l,dVol(M ), where Lc?l,d = (4?)? d2 ?(?3(+3)d2 ) .

Corollary 4.2. For k ? j, the means of the eigenvalues of H satisfy

Proof. It suffices to prove the corollary assuming that ? = 0, as the effect on the eigenvalues of adding ? to z is equivalent to a systematic shift of ? in each eigenvalue. For any positive integer n we consider the function P2,n : [?n, ?) ? R+ defined by (4.6)

P2,n(z) :=
as in [

(zn0 ? ?j )2 = n j=1 (zn0)1?2? d2 n 1 + d4 j=1 (zn0 ? ?j ).

Since R2(?) = 2R1(?), it follows from (4.2) that

0 Consequently, for all ? ? zn, We note parenthetically that estimate (4.11) is asymptotically sharp since where Cd denotes the classical constant given by the Weyl limit,

n Cd = lim ?n( )? d2 = (2?)2Vol(Sd)?2/d.

n?? |?|

n

R1(?) ? 1 + d2 (zn0)? d2 ?1+ d2
and take the Legendre transformation on both sides, following standard calculations
to be found, e.g., in [

zn0 ?k ? (1 + d2 )n d2 w1+ d2 . which proves the theorem. (The simplification (4.5) is achieved with the upper bound in (4.9).)

and Schro¨dinger operators on manifolds

of bounded mean curvature

We next turn our attention to the one-parameter family of operators H? from
(3.2) in order to derive inequalities of Lieb-Thirring type for periodic Schro¨dinger
operators and for Schr¨odinger operators on domains of bounded mean curvature.
Some inequalities of Lieb-Thirring type appear in [

For the purposes of semiclassical analysis, we appeal to the Feynman-Hellman
theorem to note that
??j
(5.1) Tj = ??
(except at eigenvalue crossings; cf. [

Recalling that ?R2/?z = 2R1, we see that (5.2) can be regarded as a partial differential inequality. Letting U (z, ?) := ?d/2R2(z, ?), the inequality has the form (5.3) (5.4) for ? ? ?s.

?U gd ?U

,
?? ? 4 ?z
?U
?? ? 0;
U (?, z) ? U
?s, z +
gd
4 (? ? ?s)
which can be solved by changing to characteristic variables ? := ? ? g4d z, ? :=
? + g4d z, in terms of which
i.e., U decreases as ? increases while ? is fixed. In conclusion,
(4.13)
Hence for all k ? n (letting w approach k from below) we get
As ?s ? 0, the right side of (5.4) tends to Lc2l,d
V (x) ?
z + g4d ?
Since (see e.g. [

Ej(?)<z (5.5)

d lim ? 2 ??0+ (z ? Ej (?))? = Lc?l,d

M (V (x) ? z)?+d/2 dx, ? 2+d/2 ? dx.

M M 2 + ? + with Lc?l,d, the classical constant, given by

d Lc?l,d = (4?)? 2

?(? + 1) ?(? + d2 + 1) , we arrive at a sharp Lieb-Thirring inequality for R2: Theorem 5.1. For all ? > 0 the mapping

d ? ? ? 2 R2 z ? ?gd 4 is nonincreasing and therefore for all z ? R and all ? > 0 the following sharp Lieb-Thirring inequality holds:

R2(z, ?) ? ??d/2Lc2l,d

A similar monotonicity property can be proved for R?(z, ?) with ? > 2 (see also
[

d ? ? ? 2 R? is nonincreasing and therefore for all z ? R and all ? > 0 the following sharp Lieb-Thirring inequality holds:

R?(z, ?) ? ??d/2Lc?l,d ?

The conclusion of Theorem 5.1 also holds in the presence of vector potentials. In particular, in the periodic case, the operator ?(?i? + k)2 + V (x), with k ? Rd (more precisely, k in the dual lattice) being a constant vector, satisfies the LiebThirring inequality (5.8). Therefore, taking the average over a band, which we define by

1 ?j := ?j (k) dk,

Vol(M ) and using the convexity of the function ? ? (z ? ?)?+ we get the estimate (5.11) (z ? ?j )+ ? ??d/2Lc?l,d ? Then HU? = U [H, U ?], and we may rewrite the trace formula (2.4) as follows: tr (z ? H)2(HU + HU? )P

? tr (z ? H)(HU2 + HU HU2 ? )P (z ? ?j ) ( HU ?j , HU ?j + HU? ?j , HU? ?j ) (z ? ?j )(z ? ?k)(?k ? ?j ) | U ?j , ?k |2 + | U ??j , ?k |2 . ?j?J ?k?/J ?j?J ?k?/J (z ? H)?j 2 (2z ? U HU ? ? U ?HU )?j , ?j (z ? H)?j , ?j

(z ? U HU ?)?j 2 + (z ? U ?HU )?j 2 (z ? ?j )(z ? ?k)(?k ? ?j ) | U ?j , ?k |2 + | U ??j , ?k |2 . (6.5) Here, q and ? curvature vector h =

In Section 3 we used the auxiliary unitary operator U = e?iq·x to derive identities
for periodic Schr¨odinger operators. As another illustration we turn to the case
of Schr¨odinger operators on manifolds immersed in Rd+1, which was studied by
commutation with self-adjoint operators based on coordinate functions in [

Consider a Schro¨dinger operator H = ?? + V (x), where ?? denotes the
Laplace-Beltrami operator, on a domain ? in a smooth, orientable, d-dimensional
manifold M isometrically immersed in Rd+1. (Higher codimensions could also be
dealt with as in [

HU = ?2iq · ?

? iq · h + |q |2. are the tangential parts of q and the gradient, while the mean

? ?? n is parallel to the unit normal n. Using (6.3), the analogue of (3.5) is (6.6) |q |2

(z ? ?j ) |q |4 + 4 q · ? ?j 2 + q · h ?j 2 ?j?J = ?j?J for some positive quantities wjkq whose formal expression is identical to the ones in (3.5). To simplify this expression, one can sum as before for q taken from a frame of the form {q e?}, ? = 1, . . . , d + 1. (The same conclusion could alternatively be attained by fixing |q| and averaging over all directions.) The result, after a bit of calculation, is (6.7) ?j?J (z ? ?j )2 ? q2 = ?j?J In this identity it was convenient to assume a purely discrete spectrum, although in fact only J needs to be a discrete set if the sum over J c is replaced by the appropriate spectral integral.

The situation of greatest interest occurs when J = {?1, . . . , ?n} < inf J c, in which case the term on the right is nonpositive, and we may let q ? 0, yielding 4 h2 (6.8) R2(z) ? d (z ? ?j ) ?j , ?? + 4 ?j for all z, or, in the case where the spectrum is not purely discrete, z ? inf ?ess. As remarked in Section 4, this implies Inequality (4.1) and thereby Theorem 4.1, again in the case of a purely discrete spectrum. We observe that for the monotonicity part of Theorem 4.1 it is not necessary for the manifold M to be bounded or of finite volume, as long as the bottom part of the spectrum is discrete and z lies below that.

A basic question about the spectral geometry of immersed manifolds has to do
with extensions of the Reilly inequality [

For N > 1 the Reilly bound on ?N+1 can be improved in a form that grows as N 2/d, the power expected from the Weyl law: Theorem 6.1. Let a smooth, compact, d-dimensional manifold M, of finite volume and without boundary, be immersed in Rd+1. Let 0 = ?1 < ?2 ? · · · denote the eigenvalues of the Laplace-Beltrami operator on M. Then for each N , Proof. We start with (3.12) as adapted to this situation, viz., z2 ? where ? = hd2? and z ? (?k, ?k+1], and the corresponding specialization of the upper bound (3.13). In a standard fashion we use Cauchy-Schwarz to replace ?2k ? ?k2 and weaken (3.13) to

As was already noted in [

As a final application of commutation with unitaries, consider the integral operator H defined on L2(Rd) by

Rd and let U be the translation operator (U f )(p) = f (p?k) for some k ? Rd. This is in a sense dual to the situation given above, as the unitary operator of multiplication by e?iq·x corresponds to translation in the momenta by q. Then

(HU f )(p) = (T (p + k) ? T (p))f (p).

For simplicity we assume that the spectrum of H consists only of eigenvalues. Applying the trace formula (6.4) we get (z ? ?j )2

T (p + k) + T (p ? k) ? 2T (p) |?j (p)|2 dp .

One possibility to exploit this identity is to do a Taylor expansion about k = 0. We obtain the corresponding trace formula for a self adjoint operator G with G being the generator of the unitary group of translations in momentum space. Indeed, for C2 functions T (p) we have

T (p + k) + T (p ? k) ? 2T (p) = k(D(2)T )(p)k + O(|k|3) and

(T (p + k) ? T (p))2 + (T (p ? k) ? T (p))2 = 2 ?T (p)k 2 + O(|k|3), and therefore, after division by |k|2, (6.15) and and

T (p + k) + T (p ? k) ? 2T (p) = 2?k2, (T (p + k) ? T (p))2 + (T (p ? k) ? T (p))2 = 8?2(pk)2 + 2?2k4.

p2 + m2 ? m, then T (p)2 = p2 ? 2mT (p), and 2(pk)2 (T (p + k) ? T (p))2 + (T (p ? k) ? T (p))2 = p2 + m2 + O(|k|3).

For related work on this relativistic kinetic energy operator we refer to [

The study of the distribution of positive quadratic forms on vectors of integers is
sometimes referred to as the geometry of numbers (e.g., [

We note that d?=1 T?e?j = 2d + 8 ??j 2 = 2d + 8Tj . In the case of the Laplacian, Tj = ?j and ?j = 4?2 d?=1 n2?j . Exploiting the abstract gap formula of Theorem 2.7, after some simplification we find the following inequality: N j=1 d + 4

d (7.1)

P2,N (z) := ? d + 4 ?2 d N ? g?N and d + 2 g d + 2 g

d ?N + 2 ? DN ? ?N ? ?N+1 ? d ?N + 2 + DN .

As a first illustration we consider the case d = 1. Obviously, we want to choose g as small as possible and the best choice is the first nontrivial eigenvalue of the periodic Laplacian, i.e. g = 4?2. Let n be a natural number and set N := 2n + 1. Then ?N = n2 and ?N+1 = (n + 1)2, which means that there is a gap. We easily verify that in this case,

DN =

Consequently, the quadratic polynomials (in z) on the right and left sides of (7.1) coincide for these values of N . Since (see also (4.8)) d dz (z + ?2)?2? 12

N j=1

1 N (z ? ?j )2 = ? 2 j=1(z ? ?j )(z ? 5?j ? 4?2)(z + ?2)?3? 12 , we conclude that the nondecreasing function

z ? (z + ?2)?2? 12 R2(z) has critical points at the eigenvalues ?j . Therefore the positive shift gd/4 = ?2 in z cannot be replaced by any smaller shift without losing the monotonicity property.

Next consider the two-dimensional Laplacian with periodic boundary conditions
on the square Q = [0, 2?] × [0, 2?]. Its eigenvalues are m2 + n2, m, n ? Z with
corresponding eigenfunctions ?m,n(x) = (2?)?1 exp(imx + iny). The counting function
N = N (x) counts the number of lattice points inside the disc Dx of radius ?x
centered at the origin, a sharp estimate of which is known in the literature on
lattice points as the Gauss circle problem (see e.g. [

R(x) := #{(m, n) ? Z × Z : m2 + n2 ? x}, which counts the lattice points inside the disk Dx of radius x centered at the origin. Since f (m2 + n2) = R(x)f (x) ?

0
The bound of Theorem 4.1 reads as follows:
Defining, as in [

.

? x3 = 2?2(x),
R2(x) ? 3
which has to be compared with the standard asymptotic estimate [

Finally, we test the sharpness of our Lieb-Thirring inequalities for periodic Schr¨odinger operators. Consider the case of H(?) = ??? ? ? on [0, 2?] with periodic boundary conditions, where ? is some positive constant. Its eigenvalues are ?j = ?j2 ? ? with j ? Z. By Theorem 5.1, the function ? ? ?? is nondecreasing and therefore by (5.8) the following Lieb-Thirring inequality holds: ??

As in our first example we see that the shift ?/4 in z cannot be replaced by any smaller shift without losing the monotonicity property (choose ?/? = m2 for an integer m). The presence of the shift is due to the zero eigenvalue. Indeed, if ?/? < 1 we have ? 2 ? and without a shift this inequality clearly cannot be true.

We are grateful to the Centre Interfacultaire Bernoulli of the EPF Lausanne and the Mathematisches Forschungsinstitut Oberwolfach for their hospitality and to Lotfi Hermi and Rupert Frank for remarks and references.