Generalized Bargmann inequalities

Vol. 19 (1984)

REPORTS ON MATHEMATICAL

GENERALIZED

BARGMANN

No. 2

PHYSICS

INEQUALITIES

P. EXNER Laboratory

of Theoretical

Physics, JINR, Dubna, USSR

(Received September 29, 1979; Revised April 15, 1980)

An extensive set of inequalities is derived for ($, - A+) in L’(F) Bargmann inequalities as a special case.

which contains

the

Some years ago Bargmann proved validity of a set of inequalities [l] which includes the conventional uncertainty principle relations in L’(F), n > 2, as well as for example the known relation

pj(X)12d3X 2 i

s

a

i”-211j(x)12d3x, l+kC,m(R3).

(*I

Inequalities of this set can be applied in at least two directions. The relation (e) is useful in self-adjointness proofs (cf. [2]), Sec. X.2) and its generalizations can also serve the same purpose [3]. On the other hand, some of the Bargmann inequalities give e.g. the (exact) lower bounds for energies of orbital series of the hydrogen-like atom in n dimensions, n >, 2 [l]. Another inequality from the set under consideration analogous to (*), can be also used for energy estimates [4]. Recently Sachrajda, Weldon and Blankenbecler (SWB) have derived formally inequalities which generalize the Bargmann-type inequalities in the one-dimensional case [5]; the purpose of their paper was again to give some lower bounds for Schrodinger operators. We shall use here their idea to prove validity of a set of inequalities which will contain the Bargmann inequalities as well as the exact version of the SWB-result as special cases. There are also other ways of utilizing this idea [7 - 91; one of them has been already applied to a problem of physical interest [S]. None of them, however, give “I-dependent” estimates, which is the most remarkable feature of the Bargmann inequalities. The paper is concluded with several simple examples intended mainly as an illustration of some of the main properties of the method; further applications will be discussed elsewhere.

P491

250

P.EXNER

Let us first introduce the notation. We shall deal with the Hilbert space A?” = L’(R”), elements of R” and their distance from origin will be denoted by x (x 1,..., x,) and r, respectively. The operators V = (V1,..., V,) and A = V 2 are defined Ziventionally ([2], Sec. 1X.7; [6], Chap. 7): (vjjlc/)(x)

=

i(kj$(k))(x)

=

~ti)(x)

=

(~24wNx)

=

(-

jCFi

(la)

’ (kjFn+))Cx)9

(lb)

P,‘(~2w)Nx) n

domains; here k = (k, ,..., k,), k2 = c kj, and F,, denotes the n-

with the appropriate

j=l

dimensional

Fourier-Plancherel

operator.

PROPOSITION 1. Let $ED( - A) and assume that g is a realfunction on R”, g$ # 0, g, Vg are Lebesgue measurable and such that 1lgcp11-c co, /gr-‘xVq II< 03, and IlVgr-‘xcp II < CC for IlQr-‘xjq II < 00, j = l,...,n, Il(~Q~-‘x)cpII< a, cpE { $1 u Y(R”). Then W, - A$) 2 t Proof

Since Ir-‘xjl

($,-A$) The last expression

Ils$ II-2(tW-1xVg + (n- W’s)\l/)2.

< 1 and $ED(-

A), we have

= JlV+ Ii2 G i

IlVjill/1)’Z i

j=l

j=l

can be estimated

(2)

Ilr-‘xjVj$))2

E Ilr-lxVll/

)12.

(3)

by the Schwartz inequality: (4)

The operator

of multiplication

= {II/E%?g$EX”), analogously symmetric on D( - A), thus

by g (denoted also by g) is self-adjoint on D(g) Further, each Vj is skewr - ‘xj are Hermitean.

2Re(r-‘xV$,g$)

= (gr-‘xV$,ll/)

- (Vr-‘xgrl/,tj).

(5)

For any rp~9’(R~) we have Vr-‘xgcp = (Vr-‘xg)cp + r-‘xgVcp. Since 9(R”) is dense in A?” and all V;s are closed, there exists a sequence {(Pi} c Y( R”) to any $ E D( - A), q+ -+ t+Q,Vq+ + V$. The multiplication operators r-‘xjg and Vr-‘xjg are self-adjoint, thus also closed. Due to the assumptions {Vj$) u Y(R”) c D(r-‘xjg), j = l,...,n and ($} u Y’(R”) c D(Vr-‘xg) so that r-‘xgV’cp, + r-‘xgV$, (Vr-‘xg)cp, + (Vr-‘xg)$. The same argument applied to the operators rplxig gives rplxjgcpk + r-‘xjg$; using once more the closedness of VJs we obtain Vr-‘xgcp, + Vr-‘xgtj, and therefore Vr-‘xg$

Now the relations (3)-(6) desired result. n

together

= (Vrplxg)+

with Vr-‘xg

+ r-‘xgV$. = r-‘xVg

‘ (6) + (n - I)r-‘g

prove the

GENERALIZED

BARGMANN INEQUALITIES

251

Remarks: 1) The inequality (2) will be denoted by B(n;g) or B(n;g,O). Especially for n > 2, g(x) = rpfl, p 2 - 2, we obtain the inequalities formally identical with (C,) of [l]. Assumptions of the proved proposition are, however, more restrictive than those of [l]; that is the price we pay for the more general g. Later we shall give weaker assumptions related to a special class of functions g (Corollary 1 to Proposition 4). 2) The inequalities under consideration can be presented also in the form

llV$ II23

‘..,

then tj~D(-

however, more suitable operators.

d) may be replaced by @ED(V) = ;\, D(VJ. The form (2) is, i= 1 for estimating the kinetic energy terms of the Schrodinger

3) The condition /IVgr -lx(p I/ < cc isashorthandforgr-‘xj~~ED(Vj),j general it has to be verified with the help of definition (la).

= l,...,n;in

The main difficulty of the presented proof was to establish relation (6). This can be done in a simpler way if the action of c7is explicitly known. Such a situation occurs if n = 1; then D(P) consists of all functions $EL,‘(R) which are absolutely continuous in any finite interval of R and the derivatives $‘EL~(R). This domain is conventionally denoted by AC[R], the action of V on it being V $ = +‘. Analogously D( - d) = AC2 [R] G {$EAC[R]:$‘EAC[R]} and -A+ = -II/“. With these prerequisites we can prove the exact version of the SWB-inequalities: PROPOSITION 2. Let $ E AC’[R] and g: R -+ R be Lebesgue measurable such that g’ exists a.e. in R, 0 < Ilg$ 11< GO,g$EAC[R]and Ilg’$ II < co. Then

($9 - $“) 2 $ IW II- 2okI’lc/)2. Proof: In the same way as above we obtain ($, - $“) 2 Ilg$ (I-’ (Re(ll/‘,gtj))’ 2Re(ti’&)

(3 and

= (sll/‘M - ((gN’,$);

the last equality requires g$‘E L2(R) apart from the assumptions explicitly stated. Both functions g, $ are differentiable a.e. inR (I)’ is even continuous), thus the same holds for g and (g$)’ = g’$ + gt,V; furthermore, gtj’ as the difference of L2-vectors belongs to L’(R). Substitution of the last equality into (8) gives (7). n Bargmann has shown that for g(x) = r c+l inequalities stronger than (2) hold on eigenspaces of the “angular momentum”. We shall deduce an analogous result for spherically symmetric g’s. Let us start with some auxiliary statements borrowed from [l]. The harmonic polynomials zI are defined by the relations XVZI = IZ,,

AZ,=@

(9)

they are related to standard spherical harmonics x = Z, t S,‘, Si being a sphere in R” of radius r, by Z,(x) = r’I;(r-‘x). Further {yl,):! 1 will denote on orthonormal basis for the spherical harmonics with given 1:

252

P. EXNER

(LI;t,)s = 4 X,(x) X&W b”(X)= 4t’Y

t,

t’ = 1,...,sl, where do’, is the Euclidean measure on S;, do, z da,‘; and Zr,(x) = r’I;,(r-lx). Finally, let (.,.)l be the inner product in L’(R+,r”dr), fit = n - 1 + 21. PROPOSITION 3. (a) The following #”

= ~@A?;,.X; l=O

where the projection

E, refers

decomposition

= L2(R+,rS’dr)@E,L2(S,l,da,), to the subspace

analogously each SF; can be decomposed = Et?+?

is valid for n > 2: (10)

of L2(Si,do,)

spanned

by {&)fL1;

into a direct sum of orthogonal subspaces %yt

= {VW(~)

=f(r)Z,,(x), llfllt < ~4. (b) $E Xyt, $(x) = f(r) Z,,(x), belongs to D(V) ifand only iff is absolutely continuous on R, and Ilf’ IIt< 00; then IIV$

(4

If \I/EW),

II2= llf IIf.

(11)

th e same is true for each Et,$ and /IV+ iI2= c IIVElt$ 1j2.

lJ Let us further consider a real-valued function = g’(r), g’being a function on R,. We shall assume finite interval of) R, so that Q” exists a.e. in R,. $E %‘y,, $(x) =f(r) Zt,(x), the following relations

llslcl II =

g on R”, depending on r only:g(x) 4 to be absolutely continuous in (any One can easily verify that for each hold

Mf 1117 lb--‘s$ II = lb-l4f IL, (12)

IltvS)$ II2ZZj$I PROPOSITION 4. Assume IC/E&‘~D(V), inequality

i

IICvjis)ti

is absolutely

II2= llC7f IIf* continuous

n > 2, such that 0 < Ilg$ II< co, Ilr-‘g$

in R,.

7hen for

IIv$ II22 t Ils$ ll-2(vU’ + Plr-‘gM)2 (denoted by B(n;g,l))

holds, where g’ = r_lxVg,

any

II < cc and II(Pg)$ II2-C CC the

(13)

g’(x) = g”‘(r).

Proof: (a) Let us first take r+G E &Z, I&X) = f(r) Z,,(x). With the help of (1 l), (12) and the Schwartz inequality we obtain

IV@ II2IlslcIII2= I!f’IIT IMfIIf 3 (Mf’~8h)2; all these expressions

make sense due to our assumptions.

The functions J#

(14) are

GENERALIZED

absolutely

continuous,

= jfmG(r)f(r)rs’d d

thus

also

absolutely

continuous

j@(r) a

all three integrals here are absolutely

Ilr-lJfjl,

< cc means

convergent

2 fll

r , c, = lim g’r)If(r)(

*+O

hand,

Zf

+ Blr-‘g’(r))If(v)12rs’dr.

+ 0, p -+ cc so that there exist finite limits c0 = limg’r)If(r)j On the other

and

by parts:

- ~~~~(r)~(r)r%ra

to the assumptions

253

INEQUALITIES

is

,f@’

Y may be integrated

Ig = [&r,lj-(r>I’/‘]: According

BARGMANN

for CI 2 p-1

r .

I'm

that the integral

~y’(r)f(r)12rP’-1dr

is

0

absolutely convergent which is impossible that i is real-valued we obtain

unless co = c, = 0. Using further the fact

ReCf’,$Jr = 312 + i jf’(r)tj(r)for”dr

= - f(J(g

0

+ &r-1g7f)l,

i.e. ReV’Zfl, Substituting

= - t($,(s’

+ PIr-‘s)$).

into (14), we get (13) for the considered

(15)

1+9.

II/ = 2 tilt, $Jx) =J;,(r)ZJx). The operator of t=1 multiplication by g is of the form 40 I,, i.e. gtilt, glt, II/ are orthogonal for t # t’ and IIs+ II < a implies llg$ll II < ~0, t = L.., sI. Analogously, we obtain lIr-lgtjlt I/ < 00, II(c7g)t,blt/I < cc and with the help of Proposition 3(c) also tjr,~D(V) for t = l,...,s,; thus the relation (13) may be used for each $lt. For G, = g’ + /$r - ‘g we have G,$,, E S;,, and therefore (b) Further

we take

$E%‘;,

tWW)

= i i Ch@hJ t= 1

Finally, the Hiilder inequality

d 2 [ ll~h, II2II&, 11’1~. t=1

gives the desired result:

Remarks. 1) The inequalities B(n;r”+ ‘,1), p 3 - 2, it 3 2, coincide with (Cl,,) of Bargmann. A more detailed analysis given in [l] shows that in this case $ E Z’; n D(V) and Ilg+ I( < cc is sufficient, or even only $ES?; n D(V) if - 2 6 p < 1 (except for

sll/ #O).

254

P.EXNER 2) One has fll = n - 1 + 21, thus the inequalities

(13) are stronger

than (2) if

(Il/,r_ ‘9$) > 0. COROLLARY 1.Let the assumptions of Proposition 4 be valid with replacement of &’ by IF’ and D(V) by D( - V). If($,r-lg $) 2 0, then the inequality (2) holds. The proof is essentially the same as part (b) of the preceding proof, the only thing to add is the inequality

hW,ti) = ~~MtX%+ht) G f~WJv,h). 1,t

One can obtain also a result analogous

1-t

to Proposition

.

2 for 2

= L2(R,):

COROLLARY 2. Let g:R+ + R be absolutely continuous, $EAC’[R+], +‘(O) = 0, and 0 < Ilg$ Jj< 00, /jg’Ic/1)< co; then the inequality (7) holds.

$(O) = 0

or

The proof reproduces essentially part (a) of the preceding proof with PI = 0; an additional assumption is needed for performing integration by parts. n Let us finally mention several simple examples concerning energy estimates: (i) The authors of Ref. 4 have suggested a “quasiclassical method” for obtaining lower bounds for one-dimensional Schrodinger operators. They replace the estimating term which results from (7) simply by the minimum of the function +(g’(x)/g(x))’ + V(x), I/ being the potential considered. Baumgartner has noticed that this is not correct, however, a simple counterexample can be given without reference to the method of Ref. 9. Assume a square-well potential of width 2a and depth V, > (4a2)- l, the mass being f. The method of [4] with g(x) =x yields ($,H$) > (4x)-’ = 1.The ground state energy E, = t2am2 - V,, + V(x) > 0.25ae2 - V, = ESWB, ~\I,!III where 5 satisfies the equation (5 tg<)2 = V,a2 - t2. If e.g. V,a2 = 0.29286, then 5 = 0.48 so that E, = 0.2304ap2 - V, < ESWB. (ii) Examples showing that in some cases one obtains better estimates via functions g which are not powerlike can be found in Ref. 4. It is probably worth to discuss at least the simplest one in view of the exact formulation given by Proposition 2. Let us find a lower bound for the first excited state of the harmonic oscillator, H = - (d2/dx2) + x2 on D(H) c L2(R). The eigenvectors and eigenvalues are abbreviated conventionally as Gi and Ei, i = 0,1,2 ,..., respectively; we denote X0 = ($,,} ) and S, = {$1,$2,...}iin. The operator H, = H ) A?,, is e.s.a. on S, so that E, = info(H,) = inf(ll/,H$):$E&,, I!$ jJ = 11. We choose g:g(x) = x - x-i; each $ES,, is of the form (alx + . .. + a,x”) x xexp( - i.?). thus the assumptions of Proposition 3 are fulfilled and we obtain from(7) ($,Hti)

2 a(1 + y)2(z - 2 + y)-’ + z =.f(y,z)

(16)

for every unit I++E&,where y E (+,x-‘rl/), z G ($,x2e). Further one has to minimize the functionfin the region yz > 1 allowed by the Schwartz inequality: a simple calculation

GENERALIZED

BARGMANN

INEQUALITIES

255

yields min {f(y,z):y z > l} =f(5 - 22,~) = 3 so that E, 2 3, which is, of course, the best possible estimate. Let us stress that the restriction to X0 is substantial here since Proposition 2 should not be applied to the ground state eigenvector tiO together with the given g. As to the powerlike g’s, let us take e.g. g(x) = x which gives via ($:H$)

3 (42))’

+ z

(17)

the best possible estimate E, 3 1 for the ground state energy. Now, one can easily find a vector $EZO, say $(x) = UXX~_~,~)with suitably chosen constants (xM denotes the characteristic function of the set M), such that the r.h.s of(17) equals 1, providing thus a poor estimate E 1 3 1. These considerations can be also extended to the higher excited levels as well as to some other polynomial potentials, but we shall not pursue this line here. (iii) The most remarkable feature of the inequalities B(n;g,[) is their l-dependence. Assume e.g. the n-dimensional harmonic oscillator, y13 2, perturbed by a central potential I/such that H = - A + rz + V(r) is e.s.a., say on Y(R”). Then Proposition 4 with g(r) = r gives for any unit vector $E Y(W) n ST; the estimate (ti,Hti)

3 $(m + 202(rl/,rz$)-’

+ ($,r’$)

+ ($,W)$).

(18)

In particular, if I/= 0 we get for energies of the “I-series” the lower bound E, 3 II + 21; if e.g. n = 3 then this estimate is saturated by the eigenvector $01 having the principal quantum number zero. REFERENCES V.: He/r. Phy.\.A(,11145. ( 1971) 249. [iI Bargman, PI Reed, M., Simon, B.: Methods qf Modern Mathetnatiuil Physics II Fourier-Analysis, Self-adjointness, Academic Press, New York 1975. 131Kalf, H., Walter, J.: 1. Func. Anal. 10, (1972) 114. r41 Blankenbecler, R., Golderberg, M.L., Simon, B.: Ann. of Phys. 108, (1977) 69. [51 Sachrajda. C.T., Weldon, H.A., Blankenbecler, R.: Phys. Rev. D 17, (1978) 507. [61 Blank, J., Exner, P.: Selected Topics of Mathematical Physics IV(Czech), SPN, Prague 1980. [71 Faris, W.: J. Math. Phys. 19, (1978) 461. M., Hoffman-Ostenhof, T., Thirring W.: J. Phys. B : Atom. Mol. Phys. 11, (1978) PI Hoffmann-Ostenhof, L571.

r91 Baumgartner,

B.: J. Phys. A : Math. Gem 12, (1979) 459.