The well-known second order moment Heisenberg-Weyl inequality (or uncertainty relation) in Fourier Analysis states: Assume thatf : R → Cis a complex valued function of a random real variablexsuch that f ∈ L2(R)

http://jipam.vu.edu.au/

Volume 7, Issue 3, Article 80, 2006

ON THE SHARPENED HEISENBERG-WEYL INEQUALITY

JOHN MICHAEL RASSIAS PEDAGOGICALDEPARTMENT, E. E.

NATIONAL ANDCAPODISTRIANUNIVERSITY OFATHENS

SECTION OFMATHEMATICS ANDINFORMATICS

4, AGAMEMNONOSSTR., AGHIAPARASKEVI

ATHENS15342, GREECE

[email protected]

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

Received 21 June, 2005; accepted 21 July, 2006 Communicated by S. Saitoh

ABSTRACT. The well-known second order moment Heisenberg-Weyl inequality (or uncertainty relation) in Fourier Analysis states: Assume thatf : R → Cis a complex valued function of a random real variablexsuch that f ∈ L²(R). Then the product of the second moment of the random realxfor |f|²and the second moment of the random realξfor

fˆ

2

is at least E_|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 moments and in 2005, he 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. Afterwards, an open problem is proposed on some pertinent extremum principle.These results are useful in investigation of quantum mechanics.

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

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

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,

ISSN (electronic): 1443-5756

c 2006 Victoria University. All rights reserved.

188-05

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,

any fixed but arbitrary constantsxm,ξm ∈R, and for the second order moments (µ₂)_|f|² =σ²_|f|2 =

Z

R

(x−x_m)²|f(x)|²dx, (µ₂)|^fˆ|² =σ²

|^fˆ|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

|^fˆ|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 any f ∈ L²(R), f :R →C,such thatkfk²₂ =R

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

(µ₄)_|f|² = Z

R

(x−x_m)⁴|f(x)|²dx and (µ4)|^fˆ|² =

Z

R

(ξ−ξm)⁴

fˆ(ξ)

2

dξ, the fourth order moment Heisenberg-Weyl inequality

(H₂) (µ₄)_|f|² ·(µ₄)

|^fˆ|² ≥ 1

64π⁴E_2,f² , holds, where

E_2,f = 2 Z

R

h(1−4π²ξ_m²x²_δ)|f(x)|²−x²_δ|f⁰(x)|²−4πξ_mx²_δIm(f(x)f⁰(x))i dx, withx_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, withα = −2πξ_mi, f_α(x) = e^αxf(x), and a constantc₂ = 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^2πixξm

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 thisf /∈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 E_2,f will be nonnegative. For instance, for x_m = ξ_m = 0, this 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 pa- per, 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]). If f : R → C is a complex valued function of a real variable x, 0 ≤ _k

2

is the greatest integer ≤ ^k₂, f^(j) = _dx^djjf, 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 arbitrary k ∈ N = {1,2, . . .}, such that 0 ≤ 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 variablex, andf_a =e^axf, wherea=−βi, withi=√

−1andβ = 2πξ_mfor any fixed but arbitrary real constantξ_m, 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, 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 arbitraryp∈N0, where(·)is the conjugate of(·), andr_pkj = (−1)^p−k+j2 (0≤k < j≤p), andRe (·)is the real part of(·).

2. SHARPENEDHEISENBERG-WEYL INEQUALITY

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

−1, andfˆthe Fourier transform off, 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, (µ2p)|^fˆ|² =

Z

R

(ξ−ξm)^2p

fˆ(ξ)

2

dξ

the2p^th weighted moment ofxfor|f|² with weight functionw :R → Rand 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

, Iqkj = (−1)^p−2q

Z

R

w_p^(p−2q)(x) Re

rqkjf^(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, withβ = 2πξm, andsqk= (−1)^q−k, andEp,f =P[p/2]

q=0 CqDq, if|Ep,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

r_qkjf^(k)(x)f^(j)(x)(p−2q−r−1)

= 0,

for0≤k < j ≤q≤_p

2

. Also,

|E_p,f^∗ |= q

E_p,f² + 4A²(≥ |E_p,f|), whereA=kukx₀− kvky₀, withL²−normk·k² =R

R|·|², inner product(|u|,|v|) =R

R|u| |v|, and

u=w(x)x^p_δf_α(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−µ_σ )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)|^fˆ|² ≥ 1 2π√^p

2

p

q E_p,f^∗

,

holds for any fixed but arbitraryp∈N.

Equality holds in (H_p^∗) iff v(x) = −2c_pu(x) holds for constantsc_p > 0,and any fixed but arbitraryp∈N;c_p =k_p²/2>0,k_p ∈Randk_p 6= 0,p∈N,andA= 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^∗ =Mp− 1 (2π)^2pA² (2.3)

= (µ_2p)_w,|f|² ·(µ_2p)|^fˆ|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_δfα(x), v =fα^(p)(x).

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

, 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,Ahas to be replaced byRin 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

.

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

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

= (−1)^p−2q Z

R

w_p^(p−2q)(x)Re

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

=Iqkj.

Thus we find

M_p^∗ ≥ 1 2^2(p+1)π^2p





[p/2]

X

q=0

C_q





q

X

l=0

A_qlI_ql+ 2 X

0≤khj≤q

B_qkjI_qkj









2

= 1

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

q=0 C_qD_q, if|E_p,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.

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

Let

(m_2p)_|f|² = Z

R

x^2p|f(x)|²dx be the2p^th moment ofxfor|f|² about the originxm = 0, and

(m_2p)

|^fˆ|² = Z

R

ξ^2p

f(ξ)ˆ

2

dξ the2p^th moment ofξfor

fˆ

2

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

p−q · p!

(2q)!

p−q q

, ifp∈Nand0≤q≤_p

2

.

Corollary 2.2. Assume that f : R → C is a complex valued function of a real variable x, w = 1, x_m = ξ_m = 0, and fˆis the Fourier transform of f, described in our theorem. If f :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)

|^fˆ|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,

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 2.1. 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

, 0 ≤ _q

2

is the greatest integer ≤ ₂^q forq ∈ N∪ {0} = N⁰, p

q

= _q!(p−q)!^p! forp ∈ 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 ex- tremum principle (R) using only special functions techniques and without applying our Heisenberg- Pauli-Weyl inequality [3].

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

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