THE MOMENTS OF THE EIGENVALUE
DISTRIBUTIONS OF GENERALISED
CANTOR CHAINS
Roland J. Etienne
Physics Department, Lycée Bel-Val
L-4402 Belvaux, Grand-Duchy of Luxemburg
&
Junior Research Group Fractal Geometry and Stochastics,
University of Siegen
D-57068 Siegen, Germany
[email protected]
Presentation outline
• Introduction
 A (not so) short history
 The eigenvalue counting function
• Generalised Cantor strings
 Fractal strings
 The spectrum of a Cantor string
• Cantor chains
 The model and its Laplacian
 The moments of their eigenvalue distributions
• Summary and outlook
A (not so) short history
Vibrating
Can
Modified
Extension
Weyl‘s
Wolfskehl
Weyl‘s
Probably
Lorentz’
one hear
string
conjecture:
conjecture
Weyl-Berry
problem
of
the
lecture:
the
Weyl‘s
first
shape
conjecture
counting
conjecture
proved,
proven
of aoffor
but
proven
drum?
to
eigenvalues
smooth
there
domains
inisthe
the
•Prove
~550
B.C.
Pythagoreans
The
second
that
the
term
number
in
of
•boundaries
1589
V.boundary
Galilei
one-dimensional
with
fractal
more…
of
the
domain
caseis
asymptotic
sufficiently
expansion
high
• 1636
M. Mersenne
under certain
mild
proportional
overtones
which
to
area
lie
of
• 1713
B. Taylor
conditions
between
is
• mid
18th boundary
ct. ν and
J.&D.ν+dν
Bernoulli
independent J.ofD’Alembert
shape but
proportionalL.toEuler
volume
J.-L. Lagrange
Weyl-Berry
Frequency
Riemann-conjecture
String
First
First
Mersenne’s
natural
as
description
the
proportional
conjecture
limit
law
laws:
ever
of
inas
a
finite
to
terms
ansquare
formulated
collection
inverse
root
of
of
inbeads
the
•1.
1891
F.spectral
Pockels
f ~ofl-1differential
• 1910
H.A.
Lorentz
tension
mathematical
joined
equations
problem
by
of the
massless
string:
terms:
2.
f~
√T
•springs
1911
Weyl
as
theH.
number
of
f~
√T
Frequency
inversely
-1 Weyl
•3.
1912
H.
f
~
√
σ
beads
approaches
proportional
to length
of
Derivation
of
•infinity
1966
M.their
Kac total
while
string:
(σ:
linear
mass
density)
fundamental
frequency
• 1979
V.-1Ivrii
mass remains
fixed
f ~M.V.
l Berry
• 1979
Fractal strings
• 1993
Lapidus / Pomerance
The eigenvalue counting function
Let Ω be a bounded domain in n with boundary δΩ
and consider the following EVP
-Δu = λu in Ω and u|δΩ = 0
Then there exists a countable set of eigenvalues
0 < λ1 ≤ λ2 ≤ λ3 ≤… ≤ λk ≤…
each eigenvalue being repeated according to (algebraic)
multiplicity. For a given positive λ, the “eigenvalue
counting function” is then
N(λ) := #{(0 <) λk < λ }
The eigenvalue counting function
Weyl’s estimate:
If Ω is sufficiently „smooth“ with n-dimensional volume
|Ω|, then
N(λ) = (2π)-nBn |Ω| λn/2+R(λ)
as λ→¥, with Bn the volume of the unit ball in n.
For these domains, the remainder term is
R(λ) = O(λ(n-1)/2)
Berry’s conjecture:
Weyl’s estimate valid for domains with fractal boundary,
but the remainder term should read
R(λ) = O(λh/2)
where h is the Hausdorff-dimension of the boundary.
The eigenvalue counting function
Weyl-Berry conjecture
Disproved in 1985 by J. Brossard and R. Carmona, but
modified in 1987 by J. Fleckinger and M.L. Lapidus
(Minkowski- instead of Hausdorff-dimension).
Modified Weyl-Berry conjecture:
Proved in 1993 by M.L. Lapidus and C. Pomerance in
the one-dimensional case (fractal strings).
Fractal strings
Fractal String:
An ordinary fractal string Ω is a one-dimensional drum
with fractal boundary δΩ.
Cantor String:
A fractal string is called Cantor string if its boundary is a
Cantor set.
Triadic Cantor String:
The boundary of the triadic Cantor String is the standard
triadic Cantor set.
The spectrum of a Cantor string
An example for the
eigenfunctions of a triadic Cantor string:
The spectrum of a Cantor string
The spectrum of a Cantor string
In 1993, M.L. Lapidus and C. Pomerance showed that
for a Cantor string the eigenvalue counting function N(λ)
is bounded:
cλ1/2-cs,infλs/2 < N(λ) < cλ1/2-cs,supλs/2
How is it possible to gain more information about these
bounds and their origins?
Cantor chains
• String: continuous mass distribution along a
segment.
• Monoatomic Chain: assembly of N identical
masses coupled by harmonic springs of stiffness K.
Cantor chains
For a monoatomic chain it is easily possible to give a
matrix representation of the Laplacian:
 0 0 0 0 .... 0 0 
 1 2  1 0 .... 0 0 


 0  1 2  1 .... 0 0 
K

0
0

1
....
....
....
....

m
.... .... .... .... ....  1 0 


0
0
0
....

1
2

1


 0 0 0 .... 0 0 0 


The eigenvalues are then given by:
2
   i 
K    i 


   max   sin 
 
i  4   sin 
m   2  N  1  
  2  N  1  
2
Cantor chains
Cantor chain:
•A Cantor chain is obtained by combining monoatomic chains
with increasing basic frequencies in accordance with the
Cantor set construction.
•The number of masses in the different chains of the obtained
pre-fractal is chosen such that chains of higher order than the
iteration level do not contribute to the overall spectrum.
•The part of the spectrum up to the maximal frequency of the
fundamental chain allows a comparison to the Cantor string
spectrum, which is recovered for the iteration level - and
therefore the number of masses - going to infinity.
Cantor chains
The spectrum of the Cantor chain:
Cantor chains
Comparison of the spectra:
Cantor chains
Cantor chain:
The matrix representation of the Laplacian is then:
The dynamical matrix is a
block diagonal matrix with
the block matrices taken
with multiplicity aj from
the set of matrices of type:
0 ....
 2 1 0
 1 2  1 0 ....
K


b 2 j  for
Note:
the
triadic
Cantor
m  0  1 2  1 ....
 and b=3

chain a=2
....
....
....
....
....


The moments of their eigenvalue distributions
RMT:
Expressing the Laplacian in matrix form makes it
possible to use RMT-formalism to compute the moments
of the eigenvalue distribution. Indeed these moments are
normalised traces of powers of the dynamical matrix D.
Traces of powers of D:
As the eigenvalues are known, it is straightforward to
calculate the traces of the different powers of D.
2k
k
maxorder
N
  i 
K
k
j
2j k
trD  
 a b   4 m   sin  2  N  1 
j 0
i 1


The moments of their eigenvalue distributions
Pseudo-traces I:
In order to allow a comparison to the Cantor string
spectrum a cut-off frequency has to be introduced,
leaving us with the following “pseudo-traces”:

ptr Dtrunc
where:
k

 K
 4 
 m
k
maxorder

j 0
a  b
j

2j k
N j
  i 

  sin 
i 1
 2  N  1 
 2  N  1  arcsin b  j 
N  j  




2k
The moments of their eigenvalue distributions
Pseudo-traces II:
The second sum in the above is then calculated by
Euler-Maclaurin summation:
N j
  i 
sin



i 1
 2  N  1 
2k
N j


0
2k
  i 
sin 
 di  Rk
 2  N  1 
The moments of their eigenvalue distributions
Pseudo-traces III:
and after long and tedious algebraic transformations one
obtains:
k

ptr Dtrunc
k

 K
/ 4  
 m
j
1, k  1

maxorder
1
1
a

2 j
2 j 
3 b 

     1  b 2 F1 
2k  1 arcsin b  maxorder  j 0  b 
k

2


1  a maxorder1  1 
 

2 
a 1 
maxorder
 1 k  2k 
1 

i
j
2j k
   1  sin 2  i  c  N  j   cot c  i  
  a  b    2 k   

k

i
2
c

i
j 0
i

1





with:
c

2  N  1
The moments of their eigenvalue distributions
Pseudo-traces IV:
The last term in this quite cumbersome expression
can be bounded:
0
maxorder

j 0
a  b
j

2j k
 1 k  2k 
1  k maxorder

i
   1  sin 2  i  c  N  j   cot c  i  
  2 k   
   a
k

i
2
c

i
i 1 

 5


The moments of their eigenvalue distributions
A few remarks:
The first term is of order bmaxorder resp. N, thus
relating it to the „volume-term“ in Weyl‘s law,
whereas the two other terms being of order amaxorder
resp. Ns describe the influence of the boundary.
It is also interesting to note that the first term results
from the integral in Euler-Maclaurin summation,
whereas the other terms originate from the correction
terms.
Summary and outlook
Back to M.L. Lapidus and C. Pomerance:
Let a standard fractal string with Minkowski dimension D ∈ (0, 1) be
given. If this string has no oscillations of order D in its spectrum, does it
follow that it is Minkowski measurable?
Theorem: For a given D in the critical interval [0, 1], this inverse spectral
problem in dimension D has a positive answer if and only if ζ(s) does not
have any zero on the vertical line Re s = D.
Since ζ(s) has zeros on the critical line Re s = 1/2, the following corollary
is obtained:
Corollary: The inverse spectral problem is not true when D = 1/2. On the
other hand, it is true for every D ∈ [0, 1]\{1/2}, if and only if the Riemann
hypothesis holds. In other words, the spectral operator is invertible for all
fractal strings of dimension D ≠ 1/2 if and only if the Riemann hypothesis
holds.
Summary and outlook
What now?
•As in the case of smooth domains with symmetries, the selfsimilarities of many fractals lead to oscillations in the
spectrum.
•These oscillations are strongly related to the Minkowskimeasurability of the fractals under consideration and to the
zeroes of the Riemann Zeta-function.
•Although Cantor chains are a sub-class of fractals that are
not Minkowski-measurable, the models and methods
developed here can also be applied to Minkowski-measurable
fractals.
Summary and outlook
What now?
•Preliminary results for a class of Minkowski-measurable
chains, i.e. a-chains seem to indicate a completely different
behaviour of the boundary related terms.
•Although this was already to be suspected through the results
of Lapidus and Pomerance, the approach used here is
completely different.
•New insights are probably to be gained from a thorough and
in-depth study of the relation between the Minkowskimeasurability of fractal chains and the behaviour of the
moments of their eigenvalue distribution.
Thank you!
Contact:
[email protected]