The search for a Hamiltonian whose
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
( ∑
Enδ(E − En)
1
(V − E)3/2
)
dE +
∑
Enδ(V−En)
1
(V − V )1/2
=
− 1
2
∑
En1
(V − En)3/2
+
∑
Enδ(V − 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 =
∑
En1
(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) =
∑
En1
(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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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