On the number of bound states for SchrSdinger operators with operator-valued potentials

Ark. Mat., 40 (2002), 73 87

@ 2002 by Institut Mittag-Leffier. All rights reserved

On the number of bound states for SchrSdinger operators with operator-valued potentials

Dirk Hundertmark

A b s t r a c t . Cwikel's bound is extended to an operator-valued setting. One application of this result is a semi-classical bound for the number of negative bound states for SchrSdinger operators with operator-valued potentials. We recover Cwikel's bound for the Lieb-Thirring constant Lo,3 which is far worse t h a n the best available by Lieb (for scalar potentials). However, it leads to a uniform bound (in the dimension d > 3 ) for the quotient

Lo,d/L~l,d,

where L~ia is the so-called classical constant. This gives some improvement in large dimensions.

1. I n t r o d u c t i o n

The Lieb Thirring inequalities bound certain moments of the negative eigen- values of a one-particle SchrSdinger operator by the corresponding classical phase space moment. More precisely, for "nice enough" potentials one has

(1) trc~cR~)(-• ~_ <_ ~ ~ (~2+v(~))~

a z ~ .

Here a n d in the following, (x)_ : i E(Ixl-x) is the negative part of a real n u m b e r or a selfadjoint operator. D o i n g the ~ integration explicitly with the help of scaling, the above inequality is equivalent to its m o r e often used form

(2) trL2(Rd)(--A+V) 7 _ <_L%d ~ ~ j _ a x ,

C L d with the classical where the Lieb Thirring constant LT,d is given by LT,d = 7,d 7,d

Lieb Thirring constant

(3) c~ i /R

L.r, d _ (2~) d a( 1_p2)~+ d p .

74 Dirk Hundertmark

This integral is, of course, explicitly given by a quotient of G a m m a functions, but we will have no need for this. The Lieb Thirring inequalities are valid as soon as the potential V is in

L "~+d/2

^(Rd).

These inequalities are important tools in the spectral theory of Schr6dinger operators and they are known to hold if and only if 7 > 89 if d = l , 7 > 0 , if d = 2 , and 7 > 0 , if d>3. The bound for the critical case 7 = 0 , that is, the bound for the number of negative eigenvalues of a Schr6dinger operator in three or more dimensions is the celebrated Cwikel Lieb Rozenblum bound [6], [17], [26]. Later, different proofs for this were given by Conlon and Li and Yau [5], [16]. The remaining case 7 = ! in 2 d = 1 was settled in [32]. The well-known Weyl asymptotic formula

lim t r ( - A + / ~ V ) ~ =

L~l,d [ V(x)~ +d/2 dx

/~ ~ cx:~ J R d

immediately gives the lower bound C~,d > 1. There are certain refined lower bounds [21], [9] for small values o f % In particulm', one always has

C.~,d>l

for 7 < 1 ; see [9].

In one dimension this even happens for 7 < 3, and in two dimensions, one always has C L 2 > I [21].

Depending on the dimension there are certain conjectures for the optimal value of the constants in these inequalities [20], [21]. One part of the conjectures on the Lieb Thirring constants is that, indeed, C~,d=l for d > 3 and moments 7 > 1. For the physically most important case 7 = 1, d = 3 this would imply, via a duality argument, that the kinetic energy of fermions is bounded below by the Thomas Fermi ansatz for the kinetic energy, which in turn has certain consequences for the energy of large quantum Coulomb systems [17], [20].

Laptev and Weidl [14] realized that an, at first glance, purely technical ex- tension of the Lieb-Thirring inequality from scalar to operator-valued potentials already suggested in [12] is a key in proving at least a part of the Lieb Thirring conjecture. It allowed them to show that

Cv,d=l

for all d E N as long as 7_> 3. To prove this they considered Schr6dinger operators of the form

--A|

^{on the}

Hilbert space

L2(R d, g),

where V now is an operator-valued potential with values

V(x)

in the set of bounded self-adjoint operators on the auxiliary Hilbert space g.

In this case the Lieb-Thirring inequalities (1) and (2) are modified to (4)

trL2(Rd,g)(--A| <_ ~ JR d ./Rdtrg(p2+g(X))~ dzdp,

or, again doing the ~ integral explicitly with the help of the spectral theorem and scaling,

(5)

trL2(aa,g ) ( - A @ l g + V ) ~ <

L%d JRfd

^trg

(V(x)2 +d/2) dx.

O n t h e n u m b e r of b o u n d s t a t e s for SchrSdinger o p e r a t o r s 75

Here we abused the notation slightly in using the same symbol for the constants as in the scalar case. But in the following, we will only consider the operator-valued case anyway. L a p t e v and Weidl realized t h a t this extension of the L i e b - T h i r r i n g inequality gives rise to the possibility of an inductive proof for

C3/2,d=1

as long as one has the a priori information

C3/2,1

= 1 for operator-valued potentials. This idea together with ideas in [11] was t h e n later used in [10] to prove improved bounds on

1 3.

C~,d

in the range ~ < 7 < ~, in particular, it was shown t h a t

Cl,d--<2

uniformly in d E N .

Unlike the scalar case, however, the range of p a r a m e t e r s 7 and d for which (4), or equivalently (5), holds is not known. T h e results in [10] only show t h a t these inequalities are true for 7_> 1 and all d c N . This shortcoming has to do with the way the L i e b - T h i r r i n g estimates are proven for operator-valued potentials: First, the estimate is shown to hold in one dimension. T h e n a suitable induction proof, using the one-dimensional result, is set up to prove the full result in all dimensions.

This turns out to give good estimates for the coefficients

Cv,d

in the L i e b - T h i r r i n g inequality, for example, they are independent of the dimension. However, m o m e n t s below ~ cannot be addressed with this method, since the a priori estimate fails 1 already for scalar potentials.

This led Ari L a p t e v [13], see also [15], to ask the question whether, in par- ticular, the Cwikel L i e b - R o z e n b l u m estimate holds for SchrSdinger operators with operator-valued potentials. In this note we answer his question affirmatively, t h a t is, the Lieb Thirring inequalities for operator-valued potentials are shown to hold also for 7 = 0 as long as d_>3 and then, by a monotonicity argument also for all 7 > 0 . More precisely, we want to show t h a t Cwikel's proof of the C w i k e ~ L i e b - R o z e n b l u m bound can be a d a p t e d to the operator-valued setting. However, the bound for

Co,d

is far from being optimal since we use Cwikel's approach. But, nevertheless, rea- soning similarly to L a p t e v and Weidl, any a priori bound on Co,3 implies the bound

Co,d-<Co,a

for d_>3, thus giving a

uniform

bound in the dimension, whereas the best available bound in the scalar case due to Lieb [17] grows like x / ~ , see [21].

2. S t a t e m e n t o f t h e r e s u l t s

Let ~ be a (separable) Hilbert space with n o r m I1" IIg, scalar p r o d u c t ( . , . ) g , and let l g be the identity operator on ~. We follow the convention t h a t scalar products are linear in the second component. Furthermore, B(~) is the Banach space of bounded operators equipped with the operator norm II "

I1~(~>

and/~(G) the (separable) ideal of the c o m p a c t operators on G. For a c o m p a c t o p e r a t o r A C ~ ( ~ ) , the singular values #,~(A), n C N , are the eigenvalues of

IAI :=(A'A) 1/2

arranged in

76 Dirk H u n d e r t m a r k

decreasing order counting multiplicity, A* is the adjoint of A. We let S q ( ~ ) denote the ideal of compact operators AC/C(G) whose singular values are q-summable, that is, ~ n e N

#n(A) q<~

In particular, SI(G) and 82(~) are the trace class and Hilbert Schmidt operators on G. We will often write B, /C, and

8q

if there is no ambiguity. Of course,

AE8 q

if and only if

trG(lAIq)=tr6((A*A) q/2)

< e c , where tr6 is the trace on 6.

T h e Hilbert space

L2(R d, ~)

is the space of all measurable functions r R d - + ~ such t h a t

O(3

and the Sobolev

space HI(R d, ~)

consists of all functions ~ E L 2 ( R d, G) with finite norm

d

2 2

I1~11~1 (Rd~):= E II0z~IIL~(R~ ~)+ IIr

l = l As in the scalar case, the quadratic form

d

h0(~, ~):= ~ II ;~IIL~<R~,~> 0

1 1

is closed

in L2(R d,

~) on the domain H ~

(R d, ~).

Naturally, this form corresponds to the Laplacian - A | G on

L2(R d, ~).

We let

Lq(Rd,13(U))

be the space of operator-valued functions f: Rd--+B(G) with finite norm

Ilfllq q = Ilfll~q(R~ B(~)):= fRd ]l/(X) llq(~)dx,

and

Lq(R d, 8~(~))

the space of operator-valued functions f whose norm

Ilfllq'~

= IlfllLq(Rd,S,-(~)):= ^{q / R}

trG(If(x)l~') q/~ dx

d

is finite. A potential is a function

VCLq(R d, B(~))

such that

V(x)

is a symmetric operator for almost every x c R d. If

q_>l for d = l , q > l for d = 2 , q_>89 f o r d > 3 ,

O n the n u m b e r of b o u n d s t a t e s for Schr6dinger o p e r a t o r s 77

one sees, using Sobolev embedding theorems as in the scalar case, that the real- valued quadratic tbrm

v[~,

~] := fR~ (~(~)'

v ( x ) ~ ( ~ ) ) ~ dx

is infinitesimally form-bounded with respect to h0. Hence the form sum

h[r ~] := h0[~, ~] +v[r ~]

is closed and semi-bounded from below on H 1 ( R d, G) and thus generates the self- adjoint operator

H = - A ~ I ~ + V

on

L2(R d, G)

by the Kato Lax-Lions Milgram Nelson theorem [22]. It is easy to see that any potential

VELq(Ra,B(6))

satisfying (6), ibr which

V(z)E]C(6)

for ahnost every

x E R d,

is relatively form compact with respect to h0. Hence by Weyl's theorem for such potentials, the negative eigenvalues E 0 < E 1_<E2<...<0 are at most a countable set with accumulation point zero and their eigenspaces are finite- dimensional. In particular, this is the case for potentials

V E L q ( R a, S~(G)).

Our first result is a generalized version of a basic observation of Laptev and Weidl: The two versions (4) and (5) of the Lieb-Thirring inequality give rise to two different monotonicity properties of

Cv,a

in d.

T h e o r e m 2.1. (Sub-nmltiplicativity of

C~/,d. ) If, for dimensions n and d - n , the Lieb Thirring inequality holds for operator-valued potentials then it also holds in dimension d. Moreover,

(7)

(8)

Remarks

2.2. (i) In the scalar case Aizenman and Lieb [1] showed that the

L cl

map

7~+C~,a= ~,a/L~, d

is decreasing. This monotonicity holds also in the general case, so, in fact, (8) implies (7). The monotonicity in 7 is most easily seen in the phase space picture: By scaling one has, for 7 >70->0,

fo ~ (s +t)~_ ~

^t~-~o-1

dt = (s)~ B(7-7o, 7o + 1),

where

B(a,/3)=f~

t ~ - l ( 1 - t ) ~-1 d~ is the Beta function. In other words, for each choice of 7>-7o_>0 there exists a positive measure # on R+ with

(s) ~ _ = f,~ ( s + t ) ~~ du(t).

+

78 Dirk H u n d e r t m a r k

Using this, the functional calculus, and the Fubini Tonelli theorem, we immediately get

//

trL2(Rd,G)(A| = trL2(Rd,@(A| ~ d#(t)

< ~ ~ tr~(~2+V(x)+t)7_ ~ dx d~ d#(t) - C'/~ trG(~2+V(x)+t)7_ ~ d#(t) dx d~

C"/o,d

-(2 y find fl d

(ii) Theorem 2.1 is a slight extension of a very nice observation of Laptev and Weidl [12], [14]. T h e y used it to show • , d = l as long as 7 > 3. Basically this follows immediately by induction and the above monotonieity from (7) for n = l once one knows that C3/2,1 =1. The beauty of this observation is t h a t this bound is well known in the scalar case [21] and Laptev and Weidl gave a proof for it in the general case. See also [2] for an elegant alternative proof which avoids the proof of Buslaev-Fadeev-Zhakarov type sum rules for matrix-valued potentials.

(iii) Using C~,d=I for 7_>~- and (8), we get the bound

G,d -< G,3

in d > 3 for all 7 > 0 . In particular, this implies a uniform bound (in d) ibr the constant in the Cwikel-Lieb Rozenblum bound as soon as such an estimate is es- tablished in dimension three for operator-valued potentials. Below we will recover Cwikel's bound C0,3<34=81, see Corollary 2.4. It is, already ibr scalar potentials, known, t h a t C 0 , 3 - > 8 / ~ >4.6188, [7], [21, equation (4.24)] (see also the discussion in [31, pp. 96 97]); in fact, it is conjectured to be the correct value [7], [21], [30]. In the scalar case Lieb's proof [17] of the CLR-bound gives by far the best estimate,

scalar

C~, 3 <6.87. However, Lieb's estimate grows like ~ for large dimensions [21, equation ( 5 . 5 ) ] . ( 1 ) While we get a quite large bound on Co,3 this at least furnishes the uniform bound C0,g_<81 for all d_>3. It would be nice to extend Lieb's or even Conlon's proof [5] of the CLR-bound to operator-valued potentials.

To state our second result, Cwikel's bound in the operator-valued case, we need

q d

some more notation: Lq(Rd,13(~)), the analog of the weak Lq-space L ~ ( R ), is given by all operator-valued functions 9: R a - + B ( ~ ) for which

Hgllq,w = IlgllLg~(Ra,g(~)) := sup tl{~ : IIg(~) I1~(~) > ~}11/q < ~ "

t>O

(1) Note a d d e d in proof, For a n excellent discussion of Lieb's m e t h o d see, for example, C h a p t e r 3.4 in R o e p s t o r f f [25].

O n t h e n u m b e r of b o u n d s t a t e s for Schr6dinger o p e r a t o r s 79 Here IBI is the d-dimensional Lebesgue measure of a Borel set B c R d. Note that II " [1~,~ is not a norm since it fails to obey the triangle inequality already for scalar g.

q d.

But, as in the scalar case, one can give a norm on L w ( R , B(U)) which is equivalent to II 9 II~q(rtd;~(6)). However, we will not need this.

By p we abbreviate the operator - i V and similarly to the scalar case we define the operator f(x)g(p) to be

--~ f(x)g(p)~(x) = f ( x ) (2%)d/2 d eiX~9(~)~(~) d~,

that is, f ( x ) g ( p ) = M f F - 1 M y - T with M f and My being the "multiplication" opera- tors by f ( x ) and g(~) and F the Fourier transform. A priori, f(x)g(p) is well-defined only for simple functions, but it will turn out to be a compact operator for rather general "functions" f and g. The extension of Cwikel's bound to the operator-valued case is the following result.

T h e o r e m 2.3. (Cwikel's bound, operator-valued case.) Let f and g be opera- tor-valued functions on an auxiliary H ilbert space G. Assume that f

E L q ( R d, S q ( ~ ) )

and gEL~(Ra,13(G)) for some q>2. Then f(x)g(p) is a compact operator on L 2 ( R d, G). In fact, it is in the weak operator ideal S q ( L 2 ( R d, ~)) and, moreove~5

(9) II f(x)g(P)I1~,,~ :=

sup nl/qp,,(f(x)g(p)) <_ Kq

[[flla,,~llgll~,,~,

where the constant

fs

is given by

1 q(

8 ~1--2/ql/ 2 ~l/q K q - - ( 2 7 c ) d / q 2 \ ~ _ 2 ]| ~ l + q _ - ~ ) .

As in the scalar case T h e o r e m 2.3 gives a bound for the number of negative eigenvalues of Schr6dinger operators with operator-valued potentials.

C o r o l l a r y 2.4. Let ~ be some auxiliary Hilbert space and V a potential in Ld/2(Iz~d, sd/2(~)). Then the o p e r a t o r - A Q 1 6 + V has a finite number N of nega- tive eigenvalues. Furthermore, we have the bound

with

that is, Co,d <_ (2Ir I ~ d ) d .

N < L o d f tr~(V(x)~/2)dx

- - , j R d

Lo,d <_ (2~ lfd)dL~ld,

80 D i r k H u n d e r t m a r k

Pro@

For completeness we explicitly derive the estimate for the number of neg- ative eigenvalues o f - A | fi'om Theorem 2.3. Replacing V with - ( V ) if nec- essary and using the min-max principle, we can assume V to be non-positive. Let N be the number of negative eigenvalues of - A | and put

Y:=IVIU2(IpI-I|

By the Birman-Schwinger principle [3], [28], [4], [29], [23] one has

l_<,~(Y).

But @->1{I l | has weak

Ld(Rd,13(~))-norm 1/d r d , rd

being the volume of the unit ball in R d. With Theorem 2.3 we arrive at

that is,

1 <_ Kd~-d , , , , ~/d ,,llIvl~/~lld,JV-~/d'

N _< K ~ a II IVl~/~ll~,d = (2rcKd)dL;1,d/R d trG(lV(x)l d/2) dx,

since

L;'d=rd/(2rc) d. []

Remark

2.5. Corollary 2.4 gives the a priori bound

Co,d<_(2rrKd) d

for d>3.

Using Theorem 2.1 and the fact t h a t

Cv,d=l

if ~ > ~ , [14], we know that

CO,d < -

min,~=3 ... d C0,,> Since the a priori bound given in Corollary 2.4 increases rather fast in the dimension, the best we can conclude is Co,d_< (27r K3) 3 =34 =81.

3. P r o o f of t h e s u b - m u l t i p l i c a t i v i t y of t h e L i e b - T h i r r i n g c o n s t a n t s We proceed very similarly to [14], but freeze the first

n<d

variables. Let x< = ( x l , ..., x , ) , x> = ( x n + l , ...,

Xd)

and ~<, ~> similarly defined. P u t

w(x<) := (-zx> + v ( x < , . ))_,

where A> is the Laplacian in the x> variables. Clearly, by assumption on V, W is a non-negative compact operator on

L2(Rd-n,G)

for almost all x < E R ~ and, moreover,

trL2(Re,6)

(--Aq-V) ~' <-- trL2(Rn,Le(Ra

",6)) (--A< -- W) ~_

(10)

^{<_ ~} f. . . . . LrL2(Rd r~,G)(~2<-W(x<))~ dx< d~<.

Since ( t - ( s ) _ ) _ = ( t + s ) _ for t > 0 , s e a , the spectral theorem gives trL2(Ra-=,g ) (~< --W(x<))~ = trL2(Rd-n,G)(~2< _ A> + V ( x < , 9 ))~_ 2

C'yd-n / R ,

JR trG(~2<

+~>+V(x<,x>)) dx> <>

2 ~/

<- (2~)d-~

^{~ n ~ ~}

This together with (10) and the Fubini-Tonelli theorem shows (7).

On the number of bound states for SchrSdinger operators 81 For the other inequality we use the more usual form (5) of the Lieb Thirring inequality. Again, freezing the first n coordinates and proceeding as before, we immediately get

(11)

L.y,d ~ L.y,nL.~+n/2,d n,

where L.y+n/2,d_ n e n t e r s now because in the first application of the Lieb Thirring inequality (5) the exponent is raised from 7 to 7 + i n . Using the definition (3) for the classical Lieb Thirring constant together with the Fubini Tonelli theorem and scaling, one easily sees

L~l,d

=

JR (IPl~-- 1> ~_ ^dp

_ Lcl rcl

=/R, (Ip<12- i)~ dp</R~_ (IP>12-- l)n_ +n/e

dp>- ~,n~+~/2,d_n.

This together with (11) proves (8) and thus T h e o r e m 2.1.

4. P r o o f o f C w i k e l ' s b o u n d

The proof of Theorem 2.3 follows closely Cwikel's original proof. We first need a criterion for

f(x)g(p)

to be a Hilbert Schmidt operator.

L e m m a 4.1.

Let f E L 2 ( R d,

S2(G))

and assume g obeys

IIg<"

>H~<~>

e L 2 ( R d ) 9

Then the operator f(x)g(p) is Hilbert Sehmidt and we have the estimate

1

IIf(x)g(P)ll~s --

(2~r) d /R~

/ R cz tr6(g*(~)f(x)* f(x)g(~) ) dx d~

_< ~ ~ t r a ( l f ( x ) l 2)

dx d IIg(~)ll~ d~.

Proof.

In the scalar case this is well known and is usually shown by noting that in this case

f(x)g(p)

is a convolution operator. Another proof is by changing the basis: Let ~ be the Fourier transform on

L2(R d, ~),

that is,

1 /R e-~'Xu(x) dx.

f ~ ( ~ ) : - (2~)d/2 d

82 D i r k H u n d e r t m a r k

Then the Hilbert Schmidt norms of

f(z)9(p)

and

M f F - 1 M g

are equal. The opera- tor

Mfy:-lMg

has "kernel"

(2~r)-a/2eiX'~f(x)g(~)

and thus by [24, Theorem VI.23]

or [8, Section III.9],

Ilf (x)9(p) II~s = Ilf (x)5-19(~) II~s

-(27r)d 1

JR a

JR atr~(gt~)*f(x)*ftx)g(~))dxd~

-(2~)d iRd iRd tr~(If(x)121g(r162

1

-< 7777 JR~ SR~ tr~/If(x)12)llg/~)ll5 d~d~ 1

1/. s

- (27r) d

~ tr~(If(x)12)dx

dllg(~)ll~d~" []

The first step rests on splitting the operator

f(x)g(p)

(which is a priori only defined on simple functions) into manageable pieces. Fix t > 0 , r > l and assume that f and 9 are non-negative, in particular, self-adjoint, operator-valued functions.

For a Borel subset B of R let

XB(I(x))

and

xB(g(~))

be the spectral projection operators of

f(x)

and g(~), respectively. By the functional calculus we have

f(x) -- Z f(~)~(,.,_l,~.~ (f(x)) = Z f,(x),

(12) *ez zcz

g(~) = ~ g(~)~<' ~,~-'1 (g(~)) = ~ g,(4),

1cZ l E Z

where fl (resp.,

gin)

are mutually orthogonal operators. We use this decomposition of f and g to split the operator

f(x)g(p)

into

(13)

f(x)9(p) = Bt +Ht

with

Bt

:= }-~4+.~<1

fl (x)g,~ (p), lit

:=~t+.~>1

fl (x)g,~ (p).

Note that this decompo- sition of

f(x)g(p)

is slightly different from the one used by Cwikel. We have the following result.

L e m m a 4.2.

Let f and 9 be non-negative operator-valued functions. If

q>2

and feLq(Rd, Sq(~)), 9eLq(Rd, B(G)) with Ilfllq,q=l and NgNq,~=l then

(a) Bt is a bounded operator with operator norm bounded by

r

IIBtlIL~(Rd,6) <--tl_ r 1;

(b)

Ht is a Hilbert-Schmidt operator with Hilbert Schmidt norm bounded by

1 1 + ~

IIHtll~s _<

(2~)dtq_~

On t h e n u m b e r of b o u n d s t a t e s for Schrgdinger o p e r a t o r s 83

Remarks

4.3. (i) Due to our choice of

Be

and

Ht

the bound in Lemma 4.2(b) is

independent

of r and in (a) it is easy to see that the choice r = 2 is optimal.

(ii) This lemma also shows that

f(x)9(p )

is a compact operator since it is the norm limit for t--+0 of the Hilbert Schmidt operators He.

Pro@

Part (a) follows completely Cwikel's original proof: Since the fz (resp., g-~) are orthogonal operators for different indices we get, for simple functions r and r say,

l + m < l

-< ~ < Z II ~-(" "~)f~-~(*)~ll211~-~v-~(v)~ll ~

s < l rnCZ

( m ~ Z X 2 , 1 / 2 / T m 2 , 1 / 2

-<E< II, ~ c*-'~)fs-,,~()vii2) ( E II >~(v)+l12)

s < l / \ m E Z

= ~ r ~ ~ r (~ "~)fs_~(x)~

~ r-mg,~(p)r

s < l HrncZ 2 m c Z 2

r

--< l _ r _ l t l l < l I1r

since ~ l c z

r-lfl (x)<_ tl G

and ~ , ~ e z

r-'~g',~

({)-< 1G- Thus

Bt

extends to a bounded operator on L 2 ( R a, G) with the given bound for its norm.

To prove part (b) observe that by Lemma 4.1 and the cyclicity of the trace, we have

t-.l- rr~,> 1

Assume for x, { c R d the operator inequality

(14)

fz(x)g.~(g)e f , ( x ) <~

(llg(g)Nf(x)x(t,~)(lIg(g)H

f ( x ) ) ) 2 = :

h(x, ~)2

/ + m > l

on the Hilbert space G. Note that the projection operator X(,,oo)(llg(r (on G) commutes with

f(x)

for all x, { E R d. Let ;~j (x) be the j t h ordered eigenvalue of

f(x),

and Ej ( a ) : = {{: II.q({)Iikj ({)>c t}. Each Ej has 2d dimensional Lebesgue measure

bEj(~)12d= ~ : Hg(~)H > dx < - - ~j(x)q dx,

d d - - o z q d

84 Dirk Hundertmark

since

1{911q,~=l

by assumption. Thus we see

1 /R J i tr6(h(x'~)2)dxd~

- (27c)d 2

IEj(max(a,t))]2dada

j = 0

- (2~r)d 2

IEj(t)l=dC~da+~2 IEj(c~)jzdc~dc~

j = 0

< 1 ( 1 + 2

j = 0

1 ( q _ ~ 2 2 )

- / z r~Tc'd'q-2 1+ )

since

Ej~_o f Aj(z) q dz=llfllq,q=l

by assumption. It remains to prove (14): Again, let

s=l+m

and note that the

g.m.({)=g({))i(,..rr~

1,.~rq(g({)) are orthogonal operators for different indices. As operators on ~,

/ + m > 1

E

f,(x)g~,(r = ~ ~ f~(z)g~ ~(r

l c Z s>2

= E fl (x)g(r z,oc)(g(~))ft (x)

l E Z

E iF/ (X) Ilg(~) ll~(7 -1

z,~) (llg(~) II)fl (x)

lCZ

= f(x)2]lg(~)ll~ E;~(~ z,~)([lg(~)ll)xt(~-l,rz](f(x))

ICZ

<- f(x) 2

Ilg(~) II~x(t,o~) (llg(() Ilf(x)) E

Xt(r~-z,~q (f(x))

lEZ

= / ( X ) 2 I[g(~)II~X(t,o~)([[g(~)[1 f ( x ) ) ,

where we used that

in the last inequality which proves (14) and hence the lemma. []

On the number of bound states for SchrSdinger operators 85 Given the above bounds the proof of T h e o r e m 2.3 is by now a s t a n d a r d inter- polation argument. We give this a r g u m e n t for the sake of completeness:

Proof of Theorem 2.3. First, without loss of generality assume t h a t f and 9 are non-negative operator-valued functions. Indeed, let b e be the Fourier transform and M f and My the operators of "multiplication" by f and g and note t h a t f(x)g(p) and M / . T - 1 M ~ have the same singular values. W i t h the polar decompositions f ( x ) = U l ( X ) l f ( x ) l and g(g)=lg*(g)lU~(g) in the Hilbert space ~ we have

Mf,~'-I M9 =

UI MIfI.T-1MIg. IU:~ ,

where Uj, j = l , 2, are fibered partial isometries in the space L 2 ( R d, Cj), ibr exmnple, (UI~)(x)=UI(X)~(x). Hence the singular values of f(x)g(p) are bounded by the singular values of Mlflbe 1Mig. I and [[[g*

IHq,~---IIgllq,~.*

^*

By one of the consequences of K y Fan's inequality [8] we have

~n(f(x)g(p)) = p n ( B t § < p l ( B t ) § <_

IIB~II+ ! IIH IIHN

r 1 / 2 \1/2 1

using L e m m a 4.2. Choosing t and r ( = 2 ) optimally gives

#,~(f(x)9(p)) <_ - -

2 ~l/q 1

1 q ( 8 kll--2/q 1~_~2 ) Tt 1/q

(2 )d/q 2 \ q - 2 / which proves T h e o r e m 2.3. []

Acknowledgment. It is a pleasure to t h a n k Ari L a p t e v and T i m o Weidl for stimulating discussions and Michael Sotomyak for comments and discussions on an earlier version of the paper.

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Received November 6, 2000 Dirk H u n d e r t m a r k

D e p a r t m e n t of Mathematics 253-37 California I n s t i t u t e of Technology Pasadena, CA 91125

U.S.A.

email: dirkh@caltech.edu