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volume 7, issue 3, article 80, 2006.

Received 21 June, 2005;

accepted 21 July, 2006.

Communicated by:S. Saitoh

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Journal of Inequalities in Pure and Applied Mathematics

ON THE SHARPENED HEISENBERG-WEYL INEQUALITY

JOHN MICHAEL RASSIAS

Pedagogical Department, E. E.

National and Capodistrian University of Athens Section of Mathematics and Informatics 4, Agamemnonos Str., Aghia Paraskevi Athens 15342, Greece.

EMail:jrassias@primedu.uoa.gr

URL:http://www.primedu.uoa.gr/∼jrassias/

c

2000Victoria University ISSN (electronic): 1443-5756 188-05

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On The Sharpened Heisenberg-Weyl Inequality

John Michael Rassias

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J. Ineq. Pure and Appl. Math. 7(3) Art. 80, 2006

http://jipam.vu.edu.au

Abstract

The well-known second order moment Heisenberg-Weyl inequality (or uncer- tainty relation) in Fourier Analysis states: Assume thatf:R→Cis a complex valued function of a random real variablex such thatf ∈ L²(R). Then the product of the second moment of the random realx for|f|²and the second moment of the random realξfor

fˆ

2

is at leastE_|f|² .

4π, wherefˆis the Fourier transform off, such thatfˆ(ξ) = R

Re^−2iπξxf(x)dx,f(x) = R

Re^2iπξxfˆ(ξ)dξ, andE_|f|²=R

R|f(x)|²dx.

This uncertainty relation is well-known in classical quantum mechanics. In 2004, the author generalized the afore-mentioned result to higher order mo- ments and in 2005, he investigated a Heisenberg-Weyl type inequality with- out Fourier transforms. In this paper, a sharpened form of this generalized Heisenberg-Weyl inequality is established in Fourier analysis. Afterwards, an open problem is proposed on some pertinent extremum principle.These results are useful in investigation of quantum mechanics.

2000 Mathematics Subject Classification:26, 33, 42, 60, 52.

Key words: Sharpened, Heisenberg-Weyl inequality, Gram determinant.

1. Introduction

The serious question of certainty in science was high-lighted by Heisenberg, in 1927, via his uncertainty principle [1]. He demonstrated the impossibility of specifying simultaneously the position and the speed (or the momentum) of an electron within an atom. In 1933, according to Wiener [7] a pair of transforms cannot both be very small. This uncertainty principle was stated in 1925 by Wiener, according to Wiener’s autobiography [8, p. 105-107], at a lecture in Göttingen. The following result of the Heisenberg-Weyl Inequality is credited to Pauli according to Weyl [6, p. 77, p. 393-394]. In 1928, according to Pauli [6] the less the uncertainty in |f|², the greater the uncertainty in

fˆ

2

, and conversely. This result does not actually appear in Heisenberg’s seminal paper [1] (in 1927). The following second order moment Heisenberg-Weyl inequality provides a precise quantitative formulation of the above-mentioned uncertainty principle according to W. Pauli.

1.1. Second Order Moment Heisenberg-Weyl Inequality ([3, 4, 5]):

For anyf ∈L²(R), f :R→C,such that kfk²₂ =

Z

R

|f(x)|²dx=E_|f|²,

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any fixed but arbitrary constants x_m, ξ_m ∈ R, and for the second order mo- ments

(µ₂)_|f|2 =σ_|f|² 2 = Z

R

(x−x_m)²|f(x)|²dx, (µ₂)|^f^ˆ|² =σ²

|^f^ˆ|² = Z

R

(ξ−ξ_m)²

f(ξ)ˆ

2

dξ, the second order moment Heisenberg-Weyl inequality

(H₁) σ_|f|² 2·σ²

|^f^ˆ|² ≥ kfk⁴₂ 16π²,

holds. Equality holds in (H₁) if and only if the generalized Gaussians f(x) =c₀exp (2πixξ_m) exp −c(x−x_m)²

hold for some constantsc₀ ∈Candc >0.

1.2. Fourth Order Moment Heisenberg-Weyl Inequality ([3, pp. 26-27]):

For anyf ∈ L²(R), f :R →C,such thatkfk²₂ = R

R|f(x)|²dx= E_|f_|², any fixed but arbitrary constantsx_m,ξ_m ∈R, and for the fourth order moments

(µ₄)_|f|² = Z

R

(x−x_m)⁴|f(x)|²dx and (µ₄)

|^f^ˆ|² = Z

R

(ξ−ξ_m)⁴

f(ξ)ˆ

2

dξ,

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the fourth order moment Heisenberg-Weyl inequality (H₂) (µ₄)_|f_|² ·(µ₄)

|^f^ˆ|² ≥ 1

64π⁴E_2,f² , holds, where

E2,f = 2 Z

R

h

(1−4π²ξ_m²x²_δ)|f(x)|²−x²_δ|f⁰(x)|²−4πξmx²_δIm(f(x)f⁰(x)) i

dx, with x_δ = x − x_m, ξ_δ = ξ −ξ_m, Im (·) is the imaginary part of (·), and

|E_2,f|<∞.

The “inequality” (H₂) holds, unlessf(x) = 0.

We note that if the ordinary differential equation of second order (ODE) f_α⁰⁰(x) = −2c₂x²_δf_α(x)

holds, with α = −2πξmi, fα(x) = e^αxf(x), and a constant c2 = ¹₂k₂² > 0, k₂ ∈ R and k₂ 6= 0,then “equality” in (H₂) seems to occur. However, the solution of this differential equation (ODE), given by the function

f(x) =p

|x_δ|e^2πixξ^m

c₂₀J−1/4

1 2|k₂|x²_δ

+c₂₁J_1/4 1

2|k₂|x²_δ

, in terms of the Bessel functionsJ±1/4 of the first kind of orders±1/4, leads to a contradiction, because this f /∈ L²(R). Furthermore, a limiting argument is required for this problem. For the proof of this inequality see [3].

It is open to investigate cases, where the integrand on the right-hand side of integral of E2,f will be nonnegative. For instance, for xm = ξm = 0, this

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integrand is: =|f(x)|²−x²|f⁰(x)|² (≥0).In 2004, we ([3,4]) generalized the Heisenberg-Weyl inequality and in 2005 we [5] investigated a Heisenberg-Weyl type inequality without Fourier transforms. In this paper, a sharpened form of this generalized Heisenberg-Weyl inequality is established in Fourier analysis.

We state our following two pertinent propositions. For their proofs see [3].

Proposition 1.1 (Generalized differential identity, [3]). Iff : R → Cis a complex valued function of a real variable x, 0 ≤ _k

2

is the greatest integer

≤ ^k₂,f^(j) = _dx^d^jjf, and(·)is the conjugate of(·), then (*) f(x)f^(k)(x) +f^(k)(x) ¯f(x)

= [^k2] X

i=0

(−1)ⁱ k k−i

k−i i

d^k−2i dx^k−2i

f⁽ⁱ⁾(x)

2,

holds for any fixed but arbitraryk ∈N={1,2, . . .}, such that0≤i≤_k

2

for i∈N0 ={0,1,2, . . .}.

Proposition 1.2 (Lagrange type differential identity, [3]). Iff : R→ Cis a complex valued function of a real variable x, andf_a = e^axf, wherea = −βi, with i = √

−1and β = 2πξm for any fixed but arbitrary real constantξm, as well as if

A_pk =p k

2

β^2(p−k), 0≤k ≤p, and

B_pkj =s_pkp k

p j

β^2p−j−k, 0≤k < j≤p,

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wheres_pk = (−1)^p−k(0≤k ≤p), then

(LD)

f_a^(p)

2 =

p

X

k=0

A_pk f^(k)

2+ 2 X

0≤khj≤p

B_pkjRe

r_pkjf^(k)f^(j) ,

holds for any fixed but arbitrary p∈ N⁰, where(·)is the conjugate of(·), and r_pkj = (−1)^p−^k+j² (0≤k < j ≤p), andRe (·)is the real part of(·).

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2. Sharpened Heisenberg-Weyl Inequality

We assume thatf :R→Cis a complex valued function of a real variablex(or absolutely continuous in [−a, a], a > 0), and w :R → Ra real valued weight function ofx, as well as x_m,ξ_m any fixed but arbitrary real constants. Denote f_a = e^axf, wherea = −2πξ_miwithi= √

−1, andfˆthe Fourier transform of f, such that

fˆ(ξ) = Z

R

e^−2iπξxf(x)dx and f(x) = Z

R

e^2iπξxfˆ(ξ)dξ.

Also we denote

(µ_2p)_w,|f|² = Z

R

w²(x) (x−x_m)^2p|f(x)|²dx, (µ_2p)

|^f^ˆ|² = Z

R

(ξ−ξ_m)^2p

f(ξ)ˆ

2

dξ

the 2p^th weighted moment ofx for|f|² with weight function w : R → R and the2p^th moment ofξfor

fˆ

2

, respectively. In addition, we denote

C_q = (−1)^q p p−q

p−q q

, if 0≤q≤hp 2

i

=the greatest integer≤ p 2

, I_ql = (−1)^p−2q

Z

R

w_p^(p−2q)(x)

f^(l)(x)

2dx, if 0≤l≤q ≤hp 2 i

,

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I_qkj = (−1)^p−2q Z

R

w_p^(p−2q)(x) Re

r_qkjf^(k)(x)f^(j)(x) dx,

if 0≤k < j ≤q≤hp 2 i

, wherer_qkj = (−1)^q−^k+j² ∈ {±1,±i}andw_p = (x−x_m)^pw. We assume that all these integrals exist. Finally we denote

D_q=

q

X

l=0

A_qlI_ql+ 2 X

0≤khj≤q

B_qkjI_qkj,

if|D_q|<∞holds for0≤q≤_p

2

, where

A_ql=q l

2

β^2(q−l), B_qkj =s_qkq k

q j

β^2q−j−k, with β = 2πξ_m, ands_qk = (−1)^q−k, and E_p,f = P[p/2]

q=0 C_qD_q, if|E_p,f| < ∞ holds forp∈N.

In addition, we assume the two conditions:

(2.1)

p−2q−1

X

r=0

(−1)^r lim

|x|→∞w^(r)_p (x)

f^(l)(x)

2(p−2q−r−1)

= 0,

for0≤l ≤q≤_p

2

, and

(2.2)

p−2q−1

X

r=0

(−1)^r lim

|x|→∞w^(r)_p (x)

Re

rqkjf^(k)(x)f^(j)(x)

(p−2q−r−1)

= 0,

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for0≤k < j ≤q≤_p

2

. Also,

|E_p,f^∗ |=q

E_p,f² + 4A²(≥ |E_p,f|), where A = kukx₀ − kvky₀, with L²−norm k·k² = R

R|·|², inner product (|u|,|v|) = R

R|u| |v|, and

u=w(x)x^p_δfα(x), v =f_α^(p)(x);

x0 = Z

R

|ν(x)| |h(x)|dx, y0 = Z

R

|u(x)| |h(x)|dx, as well as

h(x) = 1

√4

2π√

σe⁻¹⁴(^x−µσ )², whereµis the mean andσthe standard deviation, or

h(x) = 1

√4

nπ

sΓ ⁿ⁺¹₂

Γ ⁿ₂ · 1 1 + ^x_n²ⁿ⁺¹₄

, wheren∈N, and

kh(x)k² = Z

R

|h(x)|²dx = 1.

Theorem 2.1. Iff ∈L²(R)(or absolutely continuous in[−a, a],a >0), then (H_p^∗) ^2pq

(µ_2p)_w,|f|2 2p

q(µ_2p)

|^f^ˆ|² ≥ 1 2π√^p

2

p

q E_p,f^∗

,

holds for any fixed but arbitraryp∈N.

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Equality holds in (H_p^∗) iff v(x) =−2c_pu(x)holds for constantsc_p >0,and any fixed but arbitraryp ∈N;cp =k_p²/2>0,kp ∈Randkp 6= 0,p∈ N,and A = 0, or

h(x) = c_1pu(x) +c_2pv(x)

andx₀ = 0,ory₀ = 0,wherec_ip(i= 1,2)are constants andA² >0.

Proof. In fact, from the generalized Plancherel-Parseval-Rayleigh identity [3, (GPP)], and the fact that|e^ax|= 1asa=−2πξ_mi, one gets

M_p^∗ =M_p− 1 (2π)^2pA² (2.3)

= (µ_2p)_w,|f_|2 ·(µ_2p)|^f^ˆ|² − 1 (2π)^2pA²

= Z

R

w²(x) (x−x_m)^2p|f(x)|²dx

· Z

R

(ξ−ξ_m)^2p

fˆ(ξ)

2

dξ

− 1

(2π)^2pA²

= 1

(2π)^2p Z

R

w²(x) (x−x_m)^2p|f_a(x)|²dx

· Z

R

f_a^(p)(x)

2dx

−A²

= 1

(2π)^2p

kuk²kvk²−A² (2.4)

withu=w(x)x^p_δf_α(x), v =fα^(p)(x).

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From (2.3) – (2.4), the Cauchy-Schwarz inequality(|u|,|v|)≤ kuk kvkand the non-negativeness of the following Gram determinant [2]

0≤

kuk² (|u|,|v|) y₀ (|v|,|u|) kvk² x₀

y₀ x₀ 1

(2.5)

=kuk²kvk²−(|u|,|v|)²−

kuk²x²₀−2(|u|,|v|)x₀y₀+kvk²y²₀ , 0≤ kuk²kvk²−(|u|,|v|)²−A²

with

A=kukx₀ − kvky₀, x₀ = Z

R

|ν(x)| |h(x)|dx, y₀ = Z

R

|u(x)| |h(x)|dx, and

kh(x)k² = Z

R

|h(x)|²dx = 1, we find

M_p^∗ ≥ 1

(2π)^2p(|u|,|v|)² (2.6)

= 1

(2π)^2p Z

R

|u| |v|

2

= 1

(2π)^2p Z

R

w_p(x)f_a(x)f_a^(p)(x) dx

2

,

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wherew_p = (x−x_m)^pw, andf_a=e^axf. In general, ifkhk 6= 0, then one gets (u, v)² ≤ kuk²kvk²−R²,

whereR=A/khk=kukx− kvky, such thatx=x₀/khk, y =y₀/khk.

In this case,A has to be replaced by R in all the pertinent relations of this paper.

From (2.6) and the complex inequality, |ab| ≥ ¹₂ ab+ab

with a = w_p(x)f_a(x),b =fa^(p)(x), we get

(2.7) M_p^∗ = 1 (2π)^2p

1 2

Z

R

wp(x)

fα(x)fα^(p)(x) +f_α^(p)(x)fα(x)

dx 2

. From (2.7) and the generalized differential identity (*), one finds

(2.8) M_p^∗ ≥ 1 2^2(p+1)π^2p



 Z

R

w_p(x)





[p/2]

X

q=0

C_q d^p−2q dx^p−2q

f_a^(q)(x)

2



dx





2

.

From (2.8) and the Lagrange type differential identity (LD), we find

M_p^∗ ≥ 1 2^2(p+1)π^2p



 Z

R

w_p(x)





[p/2]

X

q=0

C_q d^p−2q dx^p−2q

q

X

l=0

A_ql

f^(l)(x)

2

+2 X

0≤khj≤q

B_qkjRe

r_qkjf^(k)(x)f^(j)(x)







dx





2

.

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From the generalized integral identity [3], the two conditions (2.1) – (2.2), and that all the integrals exist, one gets

Z

R

w_p(x) d^p−2q dx^p−2q

f^(l)(x)

2dx= (−1)^p−2q Z

R

w^(p−2q)_p (x)

f^(l)(x)

2dx=I_ql,

as well as Z

R

w_p(x) d^p−2q dx^p−2q Re

= (−1)^p−2q Z

R

w_p^(p−2q)(x)Re

=I_qkj. Thus we find

M_p^∗ ≥ 1 2^2(p+1)π^2p





[p/2]

X

q=0

Cq





q

X

l=0

AqlIql+ 2 X

0≤khj≤q

BqkjIqkj









2

= 1

2^2(p+1)π^2pE_p,f² , where Ep,f = P[p/2]

q=0 CqDq, if |Ep,f| < ∞ holds, or the sharpened moment uncertainty formula

2pp

Mp ≥ 1 2π√^p

2

p

q E_p,f^∗

≥ 1

2π√^p 2

p

q

|Ep,f|

, whereM_p =M_p^∗+ _(2π)¹2pA².

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We note that the corresponding Gram matrix to the above Gram determinant is positive definite if and only if the above Gram determinant is positive if and only if u, v, hare linearly independent. Besides, the equality in (2.5) holds if and only ifhis a linear combination of linearly independentuandv andu= 0 orv = 0, completing the proof of the above theorem.

Let

(m_2p)_|f|2 = Z

R

x^2p|f(x)|²dx

be the2p^thmoment ofxfor|f|² about the originx_m = 0, and (m_2p)|^f^ˆ|² =

Z

R

ξ^2p

f(ξ)ˆ

2

dξ the2p^th moment ofξfor

fˆ

2

about the originξ_m = 0. Denote ε_p,q = (−1)^p−q p

p−q · p!

(2q)!

p−q q

, ifp∈Nand0≤q≤_p

2

.

Corollary 2.2. Assume thatf : R → Cis a complex valued function of a real variablex,w= 1,x_m =ξ_m= 0, andfˆis the Fourier transform off, described in our theorem. Iff :R→C(or absolutely continuous in[−a, a], a >0), then the following inequality

(S_p) ^2pq

(m_2p)_|f|2 2p

q(m_2p)|^f^ˆ|² ≥ 1 2π√^p

2

p

v u u u t

[p/2]

X

q=0

ε_p,q(m_2q)|^f^(q)|²

2

+ 4A²,

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holds for any fixed but arbitraryp∈Nand0≤q ≤_p

2

, where

(m_2q)|^f^(q)|² = Z

R

x^2q

f^(q)(x)

2dx andAis analogous to the one in the above theorem.

We consider the extremum principle (via (9.33) on p. 51 of [3]):

(R) R(p)≥ 1

2π, p∈N for the corresponding “inequality”(H_p)[3, p. 22],p∈N.

Problem 1. Employing our Theorem 8.1 on p. 20 of [3], the Gaussian function, the Euler gamma functionΓ, and other related special functions, we established and explicitly proved the above extremum principle (R), where

R(p) = Γ p+¹₂

P[p/2]

q=0 (−1)^p−q_p−q^p ·_(2q)!^p!

p−q q

Γ_q

, with

Γ_q =

[q/2]

X

k=0

2^2k q 2k

2

Γ²

k+1 2

Γ

2q−2k+1 2

+ 2 X

0≤k≤j≤[q/2]

(−1)^k+j2^k+j q 2k

q 2j

×Γ

k+ 1 2

Γ

j+1

2

Γ

2q−k−j+ 1 2

,

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0 ≤ _q

2

is the greatest integer ≤ ^q₂ forq ∈ N∪ {0} = N⁰, p

q

= _q!(p−q)!^p! for p∈N,q∈N0 and0≤q≤p,p! = 1·2·3· · · · ·(p−1)·pand0! = 1, as well as

Γ

p+ 1 2

= 1

2^2p · (2p)!

p!

√π, p∈N and Γ 1

2

=√ π.

Furthermore, by employing computer techniques, this principle was verified forp= 1,2,3, . . .,32,33, as well. It now remains open to give a second explicit proof of verification for the extremum principle (R) using only special functions techniques and without applying our Heisenberg-Pauli-Weyl inequality [3].

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References

[1] W. HEISENBERG, Über den anschaulichen Inhalt der quantentheoretis- chen Kinematic und Mechanik, Zeit. Physik 43, 172 (1927); The Physi- cal Principles of the Quantum Theory (Dover, New York, 1949; The Univ.

Chicago Press, 1930).

[2] G. MINGZHE, On the Heisenberg’s inequality, J. Math. Anal. Appl., 234 (1999), 727–734.

[3] J.M. RASSIAS, On the Heisenberg-Pauli-Weyl inequality, J. Inequ. Pure

& Appl. Math., 5 (2004), Art. 4. [ONLINE:http://jipam.vu.edu.

au/article.php?sid=356].

[4] J.M. RASSIAS, On the Heisenberg-Weyl inequality, J. Inequ. Pure &

Appl. Math., 6 (2005), Art. 11. [ONLINE: http://jipam.vu.edu.

au/article.php?sid=480].

[5] J.M. RASSIAS, On the refined Heisenberg-Weyl type inequality, J. Inequ.

Pure & Appl. Math., 6 (2005), Art. 45. [ONLINE:http://jipam.vu.

edu.au/article.php?sid=514].

[6] H. WEYL, Gruppentheorie und Quantenmechanik (S. Hirzel, Leipzig, 1928; and Dover edition, New York, 1950).

[7] N. WIENER, The Fourier Integral and Certain of its Applications (Cam- bridge, 1933).

[8] N. WIENER, I am a Mathematician (MIT Press, Cambridge, 1956).