Raymond Aschheim, Carlos Castro Perelman, Klee Irwin (2016)

Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the *asymptotic* (very large *N*) region. The ordinates λn are the positive imaginary parts of the nontrivial zeta zeros in the critical line : *sn = 1/2 + iλn* . The latter results are consistent with the validity of the Bohr-Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly non-trivial functional form of the potential is found via the Bohr-Sommerfeld quantization formula using the full-fledged Riemann-von Mangoldt counting formula (without any truncations) for the number *N (E)* of zeroes in the critical strip with imaginary part greater than *0* and less than or equal to *E*.

### About Klee Irwin

**Klee Irwin is an author, researcher and entrepreneur who now dedicates the majority of his time to Quantum Gravity Research (QGR), a non-profit research institute that he founded in 2009. The mission of the organization is to discover the geometric first-principles unification of space, time, matter, energy, information, and consciousness. **

**As the Director of QGR, Klee manages a dedicated team of mathematicians and physicists in developing emergence theory to replace the current disparate and conflicting physics theories. Since 2009, the team has published numerous papers and journal articles analyzing the fundamentals of physics. **

**Klee is also the founder and owner of Irwin Naturals, an award-winning global natural supplement company providing alternative health and healing products sold in thousands of retailers across the globe including Whole Foods, Vitamin Shoppe, Costco, RiteAid, WalMart, CVS, GNC and many others. Irwin Naturals is a long time supporter of Vitamin Angels, which aims to provide lifesaving vitamins to mothers and children at risk of malnutrition thereby reducing preventable illness, blindness, and death and creating healthier communities.**

**Outside of his work in physics, Klee is active in supporting students, scientists, educators, and founders in their aim toward discovering solutions to activate positive change in the world. He has supported and invested in a wide range of people, causes and companies including Change.org, Upworthy, Donors Choose, Moon Express, Mayasil, the X PRIZE Foundation, and Singularity University where he is an Associate Founder.**

Energy Spectrum coincides with the

Riemann Zeta Zeroes

Raymond Aschheim∗, Carlos Castro Perelman†, Klee Irwin‡

Quantum Gravity Research, Topanga, CA. 90290 USA

August 2016

Abstract

Inspired by the Hilbert-Polya proposal to prove the Riemann Hypoth-

esis we have studied the Schroedinger QM equation involving a highly

non-trivial potential, and whose self-adjoint Hamiltonian operator has for

its energy spectrum one which approaches the imaginary parts of the zeta

zeroes only in the asymptotic (very large N) region. The ordinates λn are

the positive imaginary parts of the nontrivial zeta zeros in the critical line

: sn =

1

2

+ iλn. The latter results are consistent with the validity of the

Bohr-Sommerfeld semi-classical quantization condition. It is shown how

one may modify the parameters which define the potential, and fine tune

its values, such that the energy spectrum of the (modified) Hamiltonian

matches not only the first two zeroes but the other consecutive zeroes.

The highly non-trivial functional form of the potential is found via the

Bohr-Sommerfeld quantization formula using the full-fledged Riemann-

von Mangoldt counting formula (without any truncations) for the number

N(E) of zeroes in the critical strip with imaginary part greater than 0 and

less than or equal to E.

Keywords: Hilbert-Polya conjecture, Quantum Mechanics, Riemann Hypothe-

sis, Quasicrystals.

1 Introduction

Riemann’s outstanding hypothesis [1] that the non-trivial complex zeroes of the

zeta-function ζ(s) must be of the form sn = 1/2± iλn, is one of most important

∗raymond@quantumgravityresearch.org

†perelmanc@hotmail.com

‡klee@quantumgravityresearch.org

1

open problems in pure mathematics. The zeta-function has a relation with the

number of prime numbers less than a given quantity and the zeroes of zeta are

deeply connected with the distribution of primes [1]. References [2] are devoted

to the mathematical properties of the zeta-function.

The RH (Riemann Hypothesis) has also been studied from the point of view

of mathematics and physics by [12], [22], [6], [11], [23], [24], [8], [14], [26], [28],

[30] among many others. We refer to the website devoted to the interplay of

Number Theory and Physics [20] for an extensive list of articles related to the

RH.

A novel physical interpretation of the location of the nontrivial Riemann zeta

zeroes which corresponds to the presence of tachyonic-resonances/tachyonic-

condensates in bosonic string theory was found in [27] : if there were zeroes

outside the critical line violating the RH these zeroes do not correspond to any

poles of the string scattering amplitude.

The spectral properties of the λn’s are associated with the random statistical

fluctuations of the energy levels (quantum chaos) of a classical chaotic system

[6]. Montgomery [25] has shown that the two-level correlation function of the

distribution of the λn’s coincides with the expression obtained by Dyson with

the help of Random Matrices corresponding to a Gaussian unitary ensemble.

Extending the results by [28], [29], we were able to construct one-dimensional

operators HA = D2D1 and HB = D1D2 in [26] comprised of logarithmic deriva-

tives (d/dlnt) and potential terms V (t) involving the Gauss-Jacobi theta series.

The Hamiltonians HA = D2D1 and HB = D1D2 had a continuous family of

eigenfunctions Ψs(t) = t

−s+keV (t) with s being complex-valued and k real such

that

HA Ψs(t) = s(1− s)Ψs(t). HB Ψs(

1

t

) = s(1− s)Ψs(

1

t

).

(1.1)

Due to the relation Ψs(1/t) = Ψ1−s(t) which was based on the modular prop-

erties of the Gauss-Jacobi series we were able to show that the eigenvalues

Es = s(1 − s) are real so that s = real (location of the trivial zeroes), and/or

s = 12 ± iλ (critical line). Furthermore, we showed that the orthogonality con-

ditions

〈 Ψ 1

2+2m

(t) | Ψsn(t) 〉 = 0 ⇔ ζ(sn) = 0; sn =

1

2

±iλn, m = 1, 2, 3, ..... (1.3)

were consistent with the Riemann Hypothesis.

As described by [15], there is considerable circumstantial evidence for the

existence of a spectral interpretation of the Riemann zeta zeros [6], [7]. There is

a scattering theory interpretation of the Riemann zeta zeros arising from work of

[9] concerning the Laplacian acting on the modular surface. Lagarias observed

in [10] that there is a natural candidate for a Hilbert-Polya operator, using

the framework of the de Branges Hilbert spaces of entire functions, provided

that the Riemann hypothesis holds. This interpretation leads to a possible

connection with Schroedinger operators on a half-line. Lagarias [15] studied the

2

Schroedinger Operator with a Morse (exponential) potential on the right half-

line and obtained information on the location of zeros of the Whittaker function

Wκ,µ(x) for fixed real parameters κ, x with x > 0, viewed as an entire function

of the complex variable µ. In this case all zeroes lie on the imaginary axis, with

the possible exception, if κ > 0. The Whittaker functions and the right half-line

are essential ingredients in this present work.

We shall explore further the Hilbert-Polya proposal [3] to generate the zeros

in the critical line by constructing an operator 12+iH (H is a self-adjoint operator

with real eigenvalues) whose spectrum is given by the nontrivial zeta zeroes

sn =

1

2 + iλn in the critical line. We should note that the operator

1

2 + iH does

not capture zeros off the critical line in case the Riemann Hypothesis is false.

Meyer [8] gave an unconditional formulation of an operator on a more general

Banach space whose eigenvalues detect all zeta zeros, including those that are

off the critical line if the Riemann hypothesis fails.

The self-adjoint operator H described here corresponds to the Hamiltonian

associated with the Schroedinger QM equation involving a highly non-trivial

(fragmented, “fractal” like) see-saw aperiodic potential. In [26] we studied a

modified Dirac operator involving a potential related to the number counting

function of zeta zeroes and left the Schroedinger operator case for a future

project that we undertake in this work. The functional form of the potential

is found in section 2 via the Bohr-Sommerfeld quantization formula using the

full-fledged Riemann-von Mangoldt counting formula (without truncations) for

the number N(E) of zeroes in the critical strip with imaginary part greater than

0 and less than or equal to E.

The sought-after single-valued potential is given by a saw potential com-

prised of an infinite number of finite size tree-like branches. Because it is very

difficult to derive the analytical expressions for each one of these branches, in sec-

tion 3 we approximate these infinite branches of the saw potential by a hierarchy

of branches whose analytical expressions are of the form Vk = (Mkx+Nk)

−2+λk

and which allows to solve exactly the Schroedinger equation in each one of the in-

finite number of intervals associated with the (approximate) potential branches.

In the first interval (n = 1) the wave function is given in terms of Bessel

functions, while the wave functions in the following intervals (n = 2, 3, · · ·) are

Ψn(x,E) = an,E φn(x,E) + bn,E χ(x,E). The numerical amplitude coefficients

an,E , bn,E are energy-dependent and φn(x,E), χn(x,E) are given in terms of the

Whittaker functions (which can also be rewritten in terms of modified Bessel

functions).

If one matches the values of the wave functions and their derivatives at the

boundaries of the intervals, and imposes a relationship among the numerical

coefficients of the form aN,E = 0, in order to have a vanishing wave function at

x =∞, one obtains a discrete energy spectrum that approaches the zeta zeroes

only in the asymptotic (very large N) region. The latter results are consistent

with the validity of the Bohr-Sommerfeld semi-classical quantization condition.

On the other hand, we find that when the energy spectrum En = λn, n =

2, 3, · · · , coincides exactly with the positive imaginary parts of the nontrivial

zeta zeroes in the critical line (except for the first one λ1), it leads to an,λn =

3

∞; bn,λn = 0, which does not mean that the wave functions collapse to zero

or blow up (as we shall show), and to an,λm

6= 0; bn,λm

6= 0 when m

6= n. It

is shown at the end of section 3 how one may modify the parameters which

define the potential, and fine tune its values, such that the energy spectrum of

the (modified) Hamiltonian matches not only the first two zeroes but the other

consecutive zeroes. After the concluding remarks on quasicrystals we display

many figures and a table with numerical values to support our results.

2 Riemann Hypothesis and Bohr-Sommerfeld Quan-

tization

Inspired by the work of [4], [5], we begin with the Schroedinger equation

{ −h̄

2

2m

∂2

∂x2

+ V (x) } Ψ = E Ψ;

h̄2 = 2m = 1.

(2.1)

with the provision that the potential is symmetric V (−x) = V (x). We shall

fix the physical units so thath̄ = 2m = 1 ⇒ p =

√

E − V , and write the

Bohr-Sommerfeld quantization condition

∮

pdx = 2π(n+ 12 )h̄ as follows

2

π

∫ xE

∞

√

E − V dx = 2

π

∫ E

V0

√

E − V dx

dV

dV = ( N(E)−N(V0) ) +

1

2

.

(2.2)

we shall see below that V0 = V (x =∞) = E1 and why the cycle path begins at

−xE , then it goes to −∞→∞→ xE and back. The choice of the ± signs under

the square root ±

√

E − V is dictated by the signs of dx/dV in each interval.

For example, dx/dV < 0 in the

∫ xE

∞

integration so one must choose the minus

sign −

√

E − V in order to retrieve a positive number. Eq-(2.2) is 14 of the full

cycle of the integral

∮

pdx in the Bohr-Sommerfeld formula.

A differentiation of eq-(2.2) w.r.t to E using the most general Leibniz formula

for differentiation of a definite integral when the upper b(E) and lower b(E)

limits are functions of a parameter E :

d

dE

∫ b(E)

a(E)

f(V ;E) dV =

∫ b(E)

a(E)

(

∂f(V ;E)

∂E

) dV +

f(V = b(E) ;E) (

d b(E)

dE

) − f(V = a(E) ;E) (d a(E)

dE

).

(2.3)

leads to

2

π

d

dE

∫ E

Vo

√

E − V dx

dV

dV =

1

π

∫ E

Vo

1

√

E − V

dx

dV

dV =

dN(E)

dE

(2.4)

4

if dx/dV is not singular at V = E. The above equation belongs to the family

of Abel’s integral equations corresponding to α = 12 and associated with the

unknown function f(V ) ≡ (dx/dV )

1√

π

J 1/2 [ dx

dV

] =

1√

π

1

Γ(1/2)

∫ E

V0

(dx/dV )

(E − V )1/2

dV =

dN(E)

dE

(2.5)

Abel’s integral equation is basically the action of a fractional derivative operator

J α [32] , for the particular value α = 12 , on the unknown function f(V ) =

(dx/dV ) . Inverting the action of the fractional derivative operator (fractional

anti-derivative) yields the solution for

dx

dV

=

√

π

1

Γ(1/2)

d

dV

∫ V

V0

dN(E)

dE

1

(V − E)1/2

dE.

(2.6)

where the average level counting N (E) is the Riemann-von Mangoldt formula

given below 1 . Hence, one has the solution

x(V )− x(V0) =

√

π

1

Γ(1/2)

∫ V

V0

dN (E)

dE

1

(V − E)1/2

dE

(2.7)

Let us write the functional form for N (E) to be given by the Riemann-von

Mangoldt formula which is valid for E ≥ 1

NRvM (E) =

E

2π

[ log(

E

2π

) −1 ] + 7

8

+

1

π

arg [ ζ(

1

2

+ iE) ] +

1

π

∆(E). (2.8)

where the (infinitely many times) strongly oscillating function is given by the

argument of the zeta function evaluated in the critical line

S(E) =

1

π

arg [ ζ(

1

2

+ iE) ] = lim→0

1

π

Im log [ ζ(1

2

+ iE + ) ].

(2.9)

the argument of ζ( 12 + iE) is obtained by the continuous extension of arg ζ(s)

along the broken line starting at the point s = 2 + i 0 and then going to the

point s = 2 + iE and then to s = 12 + iE. If E coincides with the imaginary

part of a zeta zero, then

S(En) = lim→0

1

2

[S(En + ) + S(En − )].

(2.10)

An extensive analysis of the behaviour of S(E) can be found in [31]. In par-

ticular the property that S(E) is a piecewise smooth function with discontinuities

at the ordinates En of the complex zeroes of ζ(sn =

1

2 + iEn) = 0. When E

1We will show that we can use the solutions to Abel’s integral equation despite that N(E)

turns out to be discontinuous and non-differentiable at a discrete number of points E = En =

λn

5

passes through a point of discontinuity, En, the function S(E) makes a jump

equal to the sum of multiplicities of the zeta zeroes at that point. The zeros

found so far in the critical line are simple [36]. In every interval of continuity

(E,E′), where En < E < E

′ < En+1, S(E) is monotonically decreasing with

derivatives given by

S′(E) = − 1

2π

log (

E

2π

) + O(E−2); S′′(E) = − 1

2πE

+ O(E−3). (2.11)

The most salient feature of these properties is that the derivative S′(E) blows

up at the location of the zeta zeroes En due the discontinuity (jump) of S(E) at

En. Also, the strongly oscillatory behaviour of S(E) forces the potential V (x)

to be a multi-valued function of x.

The expression for ∆(E) is [31]

∆(E) =

E

4

log (1+

1

4E2

) +

1

4

arctan (

1

2E

) − E

2

∫ ∞

0

ρ(u) du

(u+ 1/4)2 + (E/2)2

.

(2.12)

with ρ(u) = 12 − {u}, where {u} is the fractional part of u and which can be

written as u− [u], where [u] is the integer part of u. In this way one can perform

the integral involving [u] in the numerator by partitioning the [0,∞] interval in

intervals of unit length : [0, 1], [1, 2], [2, 3], .....[n, n+ 1], ....

The graph of the N(E) level counting function is displayed in fig-1. The

derivative dN(E)

dE

is given by a Dirac-comb of the form

dN(E)

dE

=

∞∑

n=1

δ(E − En)

(2.13)

After taking the derivative of the Bohr-Sommerfeld quantization formula

eq-(2.2), upon using the Leibniz rule (2.3) and the solution eq-(2.6) to Abel’s

integral equation, leads to

dx

dV

=

√

π

1

Γ(1/2)

d

dV

∫ V

V0

dN(E)

dE

1

(V − E)1/2

dE =

− 1

2

∫ V

V0

( ∑

En

1

(V − E)3/2

)

dE +

∑

En

1

(V − V )1/2

=

− 1

2

∑

En

(V − En)3/2

+

∑

En

1

(V − V )1/2

(2.14)

The last terms δ(V − En)

1

(V−V )1/2 are 0 when En

6= V (due to the fact that

δ(V − En) = 0 goes to zero faster than the denominator), and are equal to

∞ when En = V . Consequently one learns that dx/dV has poles only at the

locations V = En. If we take E > En for all values of n, then dx/dV will be

6

finite at V = E and such that eq-(2.4) will remain unmodified after using the

generalized Leibniz rule.

From eq- (2.14) we can also deduce the exact expression for x = x(V )

x(V ) =

√

π

1

Γ(1/2)

∫ V

V0

dN (E)

dE

1

(V − E)1/2

dE =

∑

En

(V − En)1/2

(2.15)

from which one can infer that V0 = E1 ⇒ x(V0) = x(E1) =∞ so that the lowest

point V0 is consistent with the following expression

x(V ) − x(V0) =

∑

En

(V − En)1/2

−

1

(V0 − E1)1/2

(2.16)

One learns that there are many different branches of the function x(V ). In

the V -interval [E1, E2] one has

x(V ) =

1

(V − E1)1/2

⇒ V1(x) =

1

x2

+ E1

(2.17a)

where the domain of V1(x) is

x ∈ (−∞,−x1] ∪ [x1,∞), x1 =

1

√

E2 − E1

(2.17b)

In the V -interval [E2, E3] one has

x(V ) =

1

(V − E1)1/2

+

1

(V − E2)1/2

(2.18)

and so forth, in the V -interval [En, En+1] one has

x(V ) =

1

(V − E1)1/2

+

1

(V − E2)1/2

+

. . . +

1

(V − En)1/2

(2.19)

Inverting these functions is a highly nontrivial task. For example, inverting

(2.18) to determine V (x) requires solving a polynomial equation of quartic de-

gree in V . Despite that dx

dV diverges due to the presence of the poles whenever

V = En, we shall show that the integral (2.2) is finite due to an explicit cancel-

lation of these poles in the Bohr-Sommerfled formula. Because there are many

different branches of the function x(V ) one hast to split up the V -integral (2.2)

into different V -intervals (beginning with V0 = E1) as follows

2

π

∫ E

V0

√

E − V dx

dV

dV =

2

π

∫ E1

E1

√

E − V dx

dV

dV +

2

π

∫ E2

E1

√

E − V

(

− 1

2

1

(V − E1)3/2

+ δ(V − E1)

1

(V − V )1/2

)

dV +

7

2

π

∫ E3

E2

√

E − V

(

− 1

2

(

1

(V − E1)3/2

+

1

(V − E2)3/2

) + δ(V − E2)

1

(V − V )1/2

)

dV + . . .

+

2

π

∫ E

EN

√

E − V (−1

2

)

(

1

(V − E1)3/2

+

1

(V − E2)3/2

+

. . . +

1

(V − EN )3/2

)

dV +

2

π

∫ E

EN

√

E − V δ(V − EN )

1

(V − V )1/2

dV

(2.20)

After performing the integrals (2.20) for all values of n = 1, 2, · · · , N , with

∫ √

E − V

(V − En)3/2

dV = 2 arctan(

√

E − V

√

V − En

) − 2

√

E − V

√

V − En

(2.21)

one learns the following facts :

(i) There is an exact cancellation of all the poles

− 2

π

∑

n

(E − En)1/2

(En − En)1/2

+

2

π

∑

n

(E − En)1/2

(En − En)1/2

= 0

(2.22)

(ii) The contribution from the upper limits of the integrals

∫ Em+1

Em

cancel

out most of the contributions from the lower limits of the integrals of the next

interval

∫ Em+2

Em+1

(iii) The contribution from the upper limit of the last integral

∫ E

EN

is iden-

tically zero

2 arctan(

√

E − E

√

E − EN

) − 2

√

E − E

√

E − EN

= 0

(2.23)

(iv) Leaving for the net contribution to the integrals (2.20), when E > En

for all n = 1, 2, · · · , N , the following sum

2

π

n=N∑

n=1

arctan(

√

E − En

√

En − En

) =

2

π

π

2

N = N ; E > En

(2.24)

where we have set N(V0) = N(E1) =

1

2 in eq-(2.2) so that −N(V0) +

1

2 = 0.

Choosing N(V0) = N(E1) =

1

2 is tantamount of taking the average between 0

and 1 which are the number of (positive) energy levels less than or equal to E1,

respectively.

Therefore, we can safely conclude that the exact expressions for N(E) and

dN(E)

dE

to be used in eqs-(2.2, 2.14) are indeed consistent with the solutions to

the Abel integral equation (2.6) for the parameter α = 1/2. This consistency

occurs even if N(E) is discontinuous at the location of each energy level En,

and the derivatives dN(E)

dE

are given by the Dirac-comb expression (2.13); i.e.

the derivatives are singular at En.

8

3 The Construction of a Single-Valued Potential

Self-Adjoint Hamiltonian

In order to construct a Hamiltonian operator which is self-adjoint one also needs

to specify the domain in the x-axis over which the Hamiltonian is defined.

In particular the eigenfunctions must have compact support and be square-

integrable to ensure that inner products are well-defined. An operator A is

defined not only by its action, but also by its domain DA, the space of (square

integrable) functions on which it acts [33].

If A = A† and DA† = DA the

operator is said to be self-adjoint [33].

We note that the first branch of the potential is not defined in the interval

[−x1, x1] with x1 =

1

√

E2−E1

. In order to solve this problem we shall erect an

infinite potential barrier at ±x1 so that the wave-functions Ψ(x) evaluated

inside the interval [−x1, x1] are zero and the quantum particle never reaches

the interior region of the interval [−x1, x1]. The Hamiltonian is self adjoint in

this case because

∫ ∞

−∞

Ψ∗

d2Ψ

dx2

dx =

∫ −x1

−∞

Ψ∗

d2Ψ

dx2

dx+

∫ x1

−x1

Ψ∗

d2Ψ

dx2

dx+

∫ ∞

x1

Ψ∗

d2Ψ

dx2

dx =

∫ −x1

−∞

Ψ∗

d2Ψ

dx2

dx +

∫ ∞

x1

Ψ∗

d2Ψ

dx2

dx

(3.1)

Performing an integration by parts twice and taking into account that Ψ(x),Ψ∗(x)

vanish at x = ±∞ and x = ±x1, the integral (3.1) becomes

∫ −x1

−∞

d2Ψ∗

dx2

Ψ dx +

∫ ∞

x1

d2Ψ∗

dx2

Ψ dx =

∫ ∞

−∞

d2Ψ∗

dx2

Ψ dx

(3.2)

Hence, the equality of eq-(3.1) and eq-(3.2) reveals that the Hamiltonian in

eq-(2.1) is self adjoint. Similar conclusions apply when one has a potential

defined on a half-line [x1,∞) if the wave functions obey the boundary conditions

Ψ(x1) = 0; (dΨ/dx)(x1)

6=∞ and Ψ(x)→ 0, (dΨ(x)/dx)→ 0 as x→∞.

The remaining step before solving the Schroedinger equation is to extract a

single-valued potential comprised of finite tree-like branches that are defined

over a sequence of finite intervals in the x-axis. See figures. The first branch

of the potential was already defined above in eqs-(2.17a, 2.17b). The second

branch of the potential is defined in (−∞,−x2] ∪ [x2,∞) where

x2 =

1

√

E3 − E1

+

1

√

E3 − E2

(3.3)

The third branch of the potential is defined in the interval (−∞,−x3]∪ [x3,∞)

where

x3 =

1

√

E4 − E1

+

1

√

E4 − E2

+

1

√

E4 − E3

(3.4)

9

and so forth. In order to extract a single-valued potential we may select the

appropriate finite tree-branches as follows :

In the interval [ x1, x2 ] we choose the first branch of the saw potential

and label it by V1(x) such that V1(x1) = E2 = λ2. In the interval [ x2, x3 ]

we choose the second branch of the saw potential and label it by V2(x) such

that V2(x2) = E3 = λ3. In the interval [ x3, x4 ] we choose the third branch

of the saw potential and label it by V3(x), such that V3(x3) = E4 = λ4, and so

forth. Most of the coordinates of the points xn are in sequential order except in

some cases where xm < xn despite m > n. In this case we have to introduce a

reordering. If, and only if, there are very exceptional cases such that xm = xn

one will encounter a problem in constructing a single-valued potential for all

x ≥ x1. We believe that there are no cases such that xm = xn.

Having extracted the single-valued saw potential in this fashion, with an

infinite potential barrier at x1, we proceed to solve the Schroedinger equation

in each interval region subject to the conditions that the values of Ψ and (dΨ/dx)

must match at the boundaries of all of these infinite number of intervals. This is

a consequence of the conservation of probability for quantum stationary states.

This is how one may find the spectrum of energy levels (bound states) associated

with the single-valued saw potential. It is then when one can ascertain whether

or not one recovers the positive imaginary parts of the non-trivial zeta zeros

in the critical line for the physical energy spectrum. Since the saw potential

can be seen as a hierarchy of deformed potential wells, this procedure (however

daunting) is no different than finding the bound states of a periodic potential

well [21] leading to energy bands and gaps.

Approximate Potentials and Solutions in terms of Whittaker Functions

Let us take the potential in the first region to be given by the exact expression

V1 =

1

x2 + λ1 (E1 = λ1) while the potential in the other intervals are given by

the approximate version

V approx

k

(x) =

1

(Mk x+Nk)2

+ λk, Ek = λk, k = 2, 3, · · ·

(3.5)

It is well known to the experts that the Quantum Mechanics of the 1/x2 potential

on 0 < x <∞ is vey subtle with all sorts of paradoxes whose solutions requires a

sophisticated theoretical machiney [33]. For this reason, we just limit ourselves

to solving the Schroedinger equation.

Setting aside the reordering of xk for the moment, the numerical coefficients

Mk, Nk are obtained by imposing the following conditions

V approx

k

(x = xk) = Ek+1 = λk+1

V approx

k

(x = xk+1) = Vk(x = xk+1) > Ek = λk

(3.6)

In Table-1 we display the first six values of Mk, Nk. The solutions to the

Schroedinger equation in the first region are given in terms of Bessel functions

10

Ψ1(x;E) = a1

√

x J√5/2[

√

E − E1 x] + b1

√

x Y√5/2[

√

E − E1 x]

(3.7)

while the solutions in the other regions are given in terms of the Whittaker

functions after performing the change of variables

z =

Mkx+Nk

M2k

(2Mk

√

Ek − E), k = 2, 3, . . . ,

(3.8)

which convert the Schroedinger equation into the Whittaker differential equation

d2F

dz2

+ (−1

4

+

1

4 − µ

2

z2

) F = 0

(3.9)

for the particular value of the parameter κ = 0. Thus, the general solutions to

the Schroedinger equation in the regions beyond the first interval are given by

linear combinations of two functions related to the Kummer functions [15]

φk(x,E) = M

0,−

√

M2

k

+4

2Mk

(

2

√

Ek − E (xMk +Nk)

Mk

)

(3.10)

χk(x,E) = W

0,−

√

M2

k

+4

2Mk

(

2

√

Ek − E (xMk +Nk)

Mk

)

(3.11)

of the form

Ψk(x,E) = ak φk(x,E) + bk χk(x,E), k = 2, 3, . . . ,

(3.12)

In the most general case κ

6= 0 the differential equation is [15], [16]

d2F

dz2

+ (−1

4

+

κ

z

+

1

4 − µ

2

z2

) F = 0

(3.13)

the two independent solutions are

Mκ,µ(z) = e

− z2 zµ+

1

2 1F1

(

−κ+ µ+ 1

2

; 2µ+ 1; z

)

(3.14)

where 1F1

(

−κ+ µ+ 12 ; 2µ+ 1; z

)

is a confluent hypergeometric function,

1F1(a; b; z) =

∞∑

k=0

(a)k

(b)k

zk

k!

(3.15a)

(a)k = a (a+ 1) (a+ 2) . . . (a+ k − 1); (a)0 = 1

(b)k = b (b+ 1) (b+ 2) . . . (b+ k − 1); (b)0 = 1

(3.15b)

and

Wκ,µ(z) = e

− z2 zµ+

1

2 U

(

−κ+ µ+ 1

2

; 2µ+ 1; z

)

(3.16)

11

with

U(a, b, z) =

1

Γ(a)

∫ ∞

0

dt ta−1 (t+ 1)b−a−1 e−zt

(3.17)

The Whittaker function Wκ,µ(z) is specified by the asymptotic property of

having rapid decrease as z = x→∞ along the positive real axis [15]. In terms

of the confluent hypergeometric function one has [15], [16]

Wκ,µ(z) =

Γ(−2µ)

Γ( 12 − κ− µ)

Mκ,µ(z) +

Γ(2µ)

Γ( 12 − κ+ µ)

Mκ,−µ(z)

(3.18)

from which one can infer that Wκ,µ(z) = Wκ,−µ(z). All other linearly indepen-

dent solutions to Whittaker’s differential equation increase rapidly (in absolute

value) along the positive real axis as x → ∞. In the following subsections we

will have κ = 0, and µk =

M2k+4

2Mk

, k = 2, 3, · · · so there will not be any confusion

between κ and k = 2, 3, · · ·.

To finalize this subsection we may add that one could have also expressed

the wave function solutions in terms of the modified Bessel functions Iν ,Kν or

the Bessel functions Jν , Yν if one wishes. For instance, the Whittaker functions

M0,µ(2z),W0,µ(2z) can be rewritten respectively [16] as

M0,µ(2z) = 2

2µ+ 12 Γ(1 + µ)

√

z Iµ(z); W0,µ(2z) =

√

2

π

√

z Kµ(z) (3.19a)

when µ

6= integer, Iµ(z) and I−µ(z) are linearly independent so that Kµ(z) can

be re-expressed as [16]

Kµ(z) =

π

2sin(µπ)

(I−µ(z) − Iµ(z))

(3.19b)

Kµ(z) tends to zero as |z| → ∞ in the sector |arg z| < π2 for all values of µ.

They also can be rewritten with the Hankel or Bessel functions [17][18][19] as

M0,µ(2z) = (−4i)µ Γ(µ+ 1)

√

z Jµ(iz)

(3.19c)

W0,µ(2z) = i

µ+1

√

π

2

√

z H(1)

µ (iz) = i

µ+1

√

π

2

√

z (Jµ(iz) + i Yµ(iz)) (3.19d)

The Energy spectrum

Before finding the energy spectrum it is required to focus on the wave func-

tion in the first interval. From the condition Ψ1(x1, E) = 0,

Ψ1(x1, E) = a1,E

√

x1 J√5/2[

√

E − E1 x1] +b1,E

√

x1 Y√5/2[

√

E − E1 x1] = 0

(3.20a)

12

one learns that the ratio a1,E/b1,E is given by

a1,E

b1,E

= −

Y√5/2[

√

E − E1 x1]

J√5/2[

√

E − E1 x1]

(3.20b)

and it is an explicit function of the energy E. It is interesting that the order

√

5/2 of the Bessel function is closely related to the Golden mean (1 +

√

5)/2.

When E = E1 = λ1, the Bessel function Y√5/2[

√

E − E1 x1]→ Y√5/2(0) blows

up, so one must have b1,λ1 = 0. Since the Bessel function J

√

5/2(0) is finite and

nonzero, then a1,λ1 is finite and nonzero.

A similar analysis for the wave functions in the other intervals allows to

deduce that when the energy spectrum En = λn, n = 2, 3, · · · , coincides exactly

with the positive imaginary parts of the nontrivial zeta zeroes in the critical

line (except for the first one λ1), it leads to an,λn = ∞; bn,λn = 0, which does

not mean that the wave functions collapse to zero or blow up. This fact simply

follows from the behavior of the Whittaker functions at z = 0

M

0,−

√

M2

k

+4

2Mk

(

2

√

Ek − Ek (xMk +Nk)

Mk

)

= 0, Ek = λk

(3.21a)

W

0,−

√

M2

k

+4

2Mk

(

2

√

Ek − Ek (xMk +Nk)

Mk

)

= ∞, Ek = λk

(3.21b)

so that ak,λk M

0,−

√

M2

k

+4

2Mk

(0) =∞×0; and bk,λk W

0,−

√

M2

k

+4

2Mk

(0) = 0×∞. Since

the latter products are undetermined, the corresponding wave functions are

not well defined (in the k-th interval) when the energy Ek coincides with the

zeta zero λk. Consequently one must find another spectrum associated to well

defined wave functions, even if an,λm

6=∞; bn,λm

6= 0 when m

6= n.

If one matches the values of the wave functions, and their derivatives at

the boundaries of the intervals (resulting from conservation of the quantum

probability), one can deduce the relationships among the numerical amplitude

coefficients. In this case the general recursion relation among the (an, bn) and

(a1, b1) coefficients is of form

Cn = Ln,n−1 Ln−1,n−2 . . .L2,1 C1

(3.22)

where C1 , Cn are are two column matrices whose two entries are comprised

of (a1, b1), (an, bn), respectively, and L’s are the chain of 2 × 2 matrices that

relate the coefficients ai, bi in terms of the previous ones ai−1, bi−1. The entries

of the chain of matrices L’s are comprised of products Whittaker functions and

their derivatives evaluated at different arguments Mixi + Ni associated to the

intervals involved. The four entries of the final 2 × 2 matrix S resulting from

the product of the ladder matrices

S ≡ Ln,n−1 Ln−1,n−2 . . .L2,1

(3.23)

13

are given by S11, S12, S21, S22, respectively. Therefore, the relationships among

the numerical amplitude coefficients is

an,E = a1,E

(

S11(E) − S12(E)

J√5/2(

√

E − E1 x1)

Y√5/2(

√

E − E1 x1)

)

(3.24a)

bn,E = a1,E

(

S21(E) − S22(E)

J√5/2(

√

E − E1 x1)

Y√5/2(

√

E − E1 x1)

)

(3.24b)

The above relations (3.24) determine an,E , bn,E in terms of a1,E , and which

in turn, is determined by imposing the normalization condition on the wave

functions

∫∞

x1

dx Ψ∗(x,E) Ψ(x,E) = 1 as explained below.

Another discrete energy spectrum can be found leading to well defined wave

functions. It is obtained by imposing certain conditions on the values of the

numerical coefficients an,E , bn,E [21]. Because the wave functions Ψ(x,E) must

decrease fast enough towards 0 as x → ∞, as required in order to have a

self-adjoint Hamiltonian operator on the half-line [x1,∞), this will guide us

in choosing the right conditions. As mentioned earlier in eq-(3.18), the Whit-

taker function Wκ,µ(z) is specified by the asymptotic property of having rapid

decrease as z = x→∞ along the positive real axis [15].

The simplest procedure would be to modify the saw potential branches in

the region [xN ,∞), for a very large value of N , by choosing

VN (x) =

1

(MNx+NN )2

+ λN , x ∈ [xN ,∞)

(3.25)

and truncating the coefficient aN,E = 0 so that the wave function in the region

defined by [xN ,∞) becomes

ΨN (x,E) = bN,E χN (x,E) =

bN,E W

0,−

√

M2

N

+4

2MN

(

2

√

EN − E (xMN +NN )

MN

)

, x ∈ [xN ,∞)

(3.26)

so that ΨN (x,E) has a rapid decrease to 0 as x → ∞. Similar results would

follow if one rewrote the solutions in terms of the modified Bessel functions by

simple inspection of eqs-(3.18, 3.19a, 3.19b). After truncating aN,E = 0 one still

has to obey the matching conditions involving the previous interval

ΨN−1(xN , E) = ΨN (xN , E), (

dΨN−1(x,E)

dx

)(xN ) = (

dΨN (x,E)

dx

)(xN )

(3.27)

Hence, after factoring a1,E

6= 0 in (3.24a), one arrives finally at the tran-

scendental equation (in the energy) given by

aN,E = 0 ⇒ S11(E) − S12(E)

J√5/2(

√

E − E1 x1)

Y√5/2(

√

E − E1 x1)

= 0

(3.28)

14

the above equation aN,E = 0 is called the characteristic equation and whose

solutions E = E′1, E

′

2, · · · , E′k(N) yield the discrete energy spectrum. Given N ,

the number of roots of eq-(3.28) is itself dependent on N and for this reason

we label them as 1, 2, · · · , k = k(N). We display in some of the figures (figures

7,11,12 at the end of this work) the graphs of the characteristic equation (3.28),

and its roots, for some values of N .

Having determined the energy spectrum from eq-(3.28) we insert those en-

ergy values into

bN,E′

k

= a1,E′

k

(

S21(E

′

k) − S22(E′k)

J√5/2(

√

E′k − E1 x1)

Y√5/2(

√

E′k − E1 x1)

)

(3.29)

and determine the non-vanishing values for bN,E′1 , bN,E′2 ,

· · · , bN,E′

k(N)

, which

in turn, yield the expression for the wave function ΨN (x,E

′

k) given by eq-(3.26)

and corresponding to the energy values E = E′k, k = 1, 2, 3, · · · , k(N). Eqs-

(3.24) yield the other values of an,E′

k

, bn,E′

k

for n < N , and eq-(3.12) will then

determine the wave functions in all the previous intervals to the N -th one.

We should note that the (global) wave function Ψ(x,E′k) still has an explicit

dependence on the arbitrary a1,E′

k

coefficient. It is then when one can fix its

value from the normalization condition

∫∞

x1

dx Ψ∗(x,E′k) Ψ(x,E

′

k) = 1.

Because the wave functions Ψ(x,E) are fixed to decrease fast enough to-

wards 0 as x → ∞, they are normalizable leading to a finite nonzero value for

the a1,E′

k

coefficients. The wave function Ψ(x,E′k) is comprised (stitched) of

many different pieces from the wave functions in all of the intervals

Ψ1(x,E

′

k),Ψ2(x,E

′

k),Ψ3(x,E

′

k), · · · ,ΨN (x,E′k)

(3.30)

This stitching procedure of the saw potential into many different branches,

and of Ψ(x,E′k) into many different pieces, can roughly be thought of a “frac-

talization” or fragmentation. The discrete energy spectrum will approach the

zeta zeroes only in the asymptotic region (very large values of N) and which

is consistent with the validity of the Bohr-Sommerfeld semi-classical quantiza-

tion condition. The normalization of the wave functions and the study of their

asymptotic properties will be the subject of future investigations.

Infrared Fine-Tuning

To finalize this section we should add that many deformations of the saw

potential are possible such that the small eigenvalues of the deformed Hamilto-

nian coincide with the first zeroes of zeta and without changing the asymptotic

energy spectrum. One method described here is based in a judicious scaling

of the set of zeroes (which allowed the construction of the saw potential) by a

logistic function,

This scaling allows to deform the potential and fit the first zeroes as shown

in figure-(8). The infrared part of the spectrum is comprised of the low energy

solutions of the characteristic equation eq-(3.28) and leads to values which don’t

15

coincide to the low values of the zeta zeroes. We can still maintain the same func-

tional form (shape) of the potential while modifying the values of E1, E2, E3, · · ·

such thatÊk =λ̂k = Ek f(k), where f(k) is an interval-dependent scaling

factor.

Hence, the modified potential in each interval is now given by

V̂k(x) =

1

(M̂k x +N̂k )2

+λ̂k,

λ̂k = Ek f(k), k = 1, 2, 3, · · ·

(3.31)

Any function f(k) converging to 1 as k → ∞ can be used as the scaling

function. For example, the following logistic function (where γ is negative)

f(x) =

αeγx + 1

βeγx + 1

(3.32)

To facilitate (speed up) the computations, it is convenient to choose a new

basis for the wavefunctions. Hence, eqs-(3.7), (3.10), (3.11) and (3.12) are now

replaced by

Ψk(x;E) = ck

√

x+

Nk

Mk

J√

M2

k

+4

2Mk

[(x+

Nk

Mk

)

√

E − Ek]

+dk

√

x+

Nk

Mk

Y√

M2

k

+4

2Mk

[(x+

Nk

Mk

)

√

E − Ek]

(3.33)

For k = 1 one recovers eq-(3.7) with M1 = 1 and N1 = 0.

The wave function solutions associated to the the deformed potential (3.31)

are now given by

Ψ̂k(x,E) = ck

√

x+

N̂k

M̂k

J√

M̂k

2+4

2M̂k

[(x+

N̂k

M̂k

)

√

E −λ̂k]

+dk

√

x+

N̂k

M̂k

Y√

M̂k

2+4

2M̂k

[(x+

N̂k

M̂k

)

√

E −λ̂k]

(3.34)

From (3.19c, 3.19d), we get cn,E + idn,E = (−4i)µΓ(1 + µ)an,E with µ =

√

M2n+4

2Mn

. Due to the change of basis (3.33, 3.34), the characteristic equation

(3.28) an,E = 0 is now replaced by cn,E + idn,E = 0. To fine tune its solutions,

in order to match the low values of the first zeta zeros, we may choose the

interval number n = 11 and fine tune the values of α, β, γ in eq-(3.32), starting

at α = 500, β = 1000, γ = −1, by solving the equation numerically so that the

first two zeroes of (cn + idn)(E) (for n = 11) converge to E = E1 = λ1 and

E = E2 = λ2. The fine tuning is achieved by alternatively changing the value of

α, γ, β, γ, α ... until convergence. See figure-(8). In figures-(9,10) we display the

graphs of the wave functions for n = 11 intervals and corresponding to E1, E2,

respectively.

16

Now it remains to fine tune the parameters of the modified potential, for

larger and larger values of the interval number n, in order for the solutions cn,E+

idn,E = 0 to match not only the first two zeroes but the other consecutive zeroes.

Consequently we have to repeat the same procedure for the next zeroes. As one

increases the values of the interval-number n (towards infinity) the parameters

of the scaling function (3.32) α, β, γ can be fine tuned such that the spectrum

converges to the zeta zeroes.

4 Concluding Remarks

The Bohr-Sommerfeld formula, in conjunction with the full-fledged Riemann-

von Mangoldt counting formula (without any truncations) for the number N(E)

of zeroes in the critical strip, with an imaginary part greater than 0 and less

than or equal to E, was used to construct a saw potential comprised of an

infinite number of branches. And this potential, in turn, generates explicitly

the spectrum of energy levels associated to the solutions of the Schroedinger

equation in each interval. The discrete energy spectrum will approach the zeta

zeroes only in the asymptotic region (very large values of N). In this respect

our construction is self-referential since the potential itself was constructed using

the zeta zeroes.

Our potential is fragmented, “fractal”, in the sense that is comprised of

infinite fragments obtained from an infinite number of tree-like branches. One

finds also that there are cases when xm < xn despite m > n, see figures (4-

5). The location of these interval points, for example x8 < x7, appears to be

“random”. Absence of evidence of order is not evidence of its absence. Random

looking patterns may have hidden order. Ideally one would like to have other

potentials. For example, related to the prime number distribution. However, the

exact prime number counting function found by Riemann requires a knowledge

of the zeta zeroes in the critical line, and once again one would end up with a

self-referential construction. A potential based on the number counting function

using the Riemann-Siegel Θ is another possibility worth exploring.

One could contemplate the possibility that the location of the nontrivial

zeta zeroes in the critical line could behave as a quasicrystal array of atoms as

postulated by [22],[34]. Instead of studying the bound states associated with

our aperiodic saw potential, we may study the diffraction process from these

atomic sites and see whether or not the patterns have sharp Bragg diffraction

peaks as required in quasicrystals.

As described by Freeman Dyson and others [34], [35], a quasicrystal is a

distribution of discrete point masses whose Fourier transform is a distribution

of discrete point frequencies. Or to say it more briefly, a quasicrystal is a pure

point distribution that has a pure point spectrum. Namely, it is an aperiodic but

ordered (quasiperiodic) crystal, hence its name quasicrystal. This definition also

includes as a special case the ordinary crystals, which are periodic distributions

17

with periodic spectra. Odlyzko [36] has published a computer calculation of the

Fourier transform of the zeta-function zeroes. The calculation shows precisely

the expected structure of the Fourier transform, with a sharp discontinuity at

every logarithm of a prime or prime-power number and nowhere else. In other

words, he showed that the distribution of the zeta zeros only appears random

but is actually highly correlated to the distribution of primes by performing a

Fourier transform.

There are some problems and questions raised in [35] that need to be ad-

dressed. For instance, if somehow one manages to get the classification of one-

dimensional quasicrystals, we still have to face two huge issues : The classifi-

cation will be an uncountable list of classes, each with uncountable elements.

And given the many cases of models with the same diffraction, the classification

is highly likely to be not nice. If we have a list of all quasicrystals, how do

we check if the zeroes of the Riemann zeta function are or are not in the list,

without knowing already all the zeroes?

Another issue raised in [35] is that the zeroes of zeta are not a Delone set,

and this provides difficulty navigating the following issue: Let Λ be the set of

zeroes. Let Λ′ be the set obtained by moving all the zeroes, such that the n-th

zero is moved by at most 1/n. Then a diffraction pattern cannot differentiate

between Λ and Λ′. Even if we were able to construct the Riemann quasicrystal

associated with the nontrivial zeta zeros in the critical line this does not exclude

the possibility of having zeroes off the critical line in case the RH fails.

The Fibonacci chain is the quintessential example of a one-dim quasicrys-

tal. It is a one-dimensional aperiodic sequence which is a subset of the ring of

Dirichlet integers Z[τ ] = Z + Zτ , where τ is the Golden mean τ = (1 +

√

5)/2.

The Fibonacci numbers are generated recursively by summing the previous two

numbers. The tribonacci numbers are like the Fibonacci numbers, but instead

of starting with two predetermined terms, the sequence starts with three pre-

determined terms and each term afterwards is the sum of the preceding three

terms. The tetranacci numbers start with four predetermined terms, each term

afterwards being the sum of the preceding four terms, and so forth. It is war-

ranted to explore this possibility for the construction of a quasicrystal using

chains based on generalized Fibonacci numbers [37].

Finally, we ought to explore the possibility that the one-dimensional qua-

sicrystal might be obtained from the projection of a lattice in infinite dimensions,

like the infinite simplex A∞, via the cut and projection mechanism involving ir-

rational angles (irrational fraction of 2π). The irrational angles that generate the

Dirichlet integers corresponding to the Fibonacci chains are the most physically

interesting. One of us (KI) suggests that the special Dirichlet-integer-generating

angles are a key to vastly narrowing the infinite universe of 1D quasicrystals that

mathematicians are seeking for. The infinite simplex A∞ is very relevant be-

cause the distribution of prime numbers correlates exactly with the distribution

of prime simplexes, and prime A-lattices, within any bound as shown in [38],

and where he introduced simplex-integers, as a form of geometric symbolism

for numbers, such that the An lattice series (made of n-simplexes) corresponds

logically to the integers. We shall leave this project for future work.

18

Table 1: Table of values of E′1(k) and Mk, Nk.

k Mk

Nk

E′1(k)

1

0.

1.

2

-0.189471

0.858503

3

-0.306782

0.733875

4

-0.530412

0.770909

28.8823

5

-0.431255

0.569456

29.1688

6

-0.778562

0.706055

29.2093

Acknowledgements

One of us (CCP) thanks M. Bowers for assistance.

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22

Figure 1: Number counting function

Figure 2: Number counting function using Riemann-Siegel Θ

23

Figure 3: x(V)

Figure 4: Approximate Potential V(x) versus exact potential

24

Figure 5: Symmetric approximate potential

Figure 6: Fine-tuned potential

25

Figure 7: Characteristic function a11(E) for 11 intervals, when α = β

Figure 8: Characteristic function (c11 + i d11)(E) for 11 intervals, when α =

665.142, β = 1407.11, γ = −0.628615, inset of the fine-tuning logistic function

f(k)

26

Figure 9: Wave function for 11 intervals for the eigenvalue E1(11) = 14.1347

where a1,E has been scaled to 100 for convenience

Figure 10: Wave function for 11 intervals for the eigenvalue E2(11) = 21.0222

where a1,E has been scaled to 20 for convenience

27

Figure 11: Characteristic function a13(E) for 13 intervals, when α = β

Figure 12: Characteristic function a15(E) for 15 intervals, when α = β

28