A Fractal Dirac Eigenvalue Problem: Spectral Properties and Numerical Examples

a Xi :2502.10529 2 [ma h.SP] 18 Ma 2025
A FRACTAL DIRAC EIGENVALUE PROBLEM:
SPECTRAL PROPERTIES AND NUMERICAL EXAMPLES
F. AYC¸A C¸ETINKAYA, GAGE PLOTT
Abs ac . In his pape , we s udy a Di ac bounda y alue p oblem whe e he ope a o is
conside ed wi h a de i a i e o o de α∈(0,1], known as he Fα-de i a i e. We p o e some
spec al p ope ies o eigen alues and eigen unc ions and we p esen nume ical examples o
demons a e he p ac ical implica ions o ou app oach.
1. In oduc ion
The wo d ac al de i es om he La in wo d ac us which means c acked. F ac als exhibi
unique geome ic p ope ies, showcasing a ac al dimension ha su passes hei opological
dimension. These in ica e s uc u es display sel -simila i y a a ying scales, combining epe -
i i e and andom p ocesses. The in ica e na u e o ac als poses challenges o con en ional
calculus me hods in calcula ing de i a i es and in eg als. The connec ion be ween ac al ge-
ome y and na u al phenomena, as seen in clouds, moun ains, and ligh ning, unde sco es a
complexi y ha de ies con en ional ma hema ical amewo ks, as highligh ed in [16]. Mo e-
o e , in con as o Euclidean geome y, de e mining he size o ac als in ol es non- i ial
conside a ions o measu emen s like leng h, su ace a ea, and olume, as discussed in [3,13].
F ac als, cha ac e ized by in ica e pa e ns, showcase sel -simila i y and o en display di-
mensions ha a e non-in ege and complex [15,17]. The middle- hi d se o Can o is one
o he mos well-known examples o ac als [2]. F ac als a e o en oo i egula o ha e any
smoo h di e en iable s uc u e de ined on hem, and his esul s in deli e ing he me hods and
echniques o o dina y calculus as powe less and inapplicable. The echniques and me hods o
c ea e calculus on he ac al se s and cu es a e s udied in [3,14,21].
In ecen wo k, Pa a e and Gangal [18,19] in oduced Fα-calculus, a o m o Riemannian-
like calculus g ounded in he ac al subse s o he eal line. This calculus is dis inguished
by i s algo i hmic simplici y compa ed o o he me hods. Fα-calculus is a gene aliza ion o
he o dina y calculus ha add esses he cases whe e s anda d calculus is inapplicable. In his
calculus, an in eg al o o de α∈(0,1] called he Fα-in eg al is de ined which makes i possible
o in eg a e unc ions wi h ac al suppo Fo dimension α.
Fu he mo e, a de i a i e o o de α∈(0,1], known as he Fα-de i a i e, acili a es he
di e en ia ion o unc ions such as he Can o s ai case unc ion and he Weie s ass unc ion.
In con as o he classical ac ional de i a i e, he Fα-de i a i e aligns he geome ical o de
o he de i a i e wi h he domain o he unc ion’s suppo , lending i a dis inc physical
in e p e a ion [6,20,22]. No ably, he Fα-de i a i e is local, which is in con as o he non-
local na u e o he classical ac ional de i a i e. This locali y is c ucial in physics, whe e all
Da e: Ma ch 19, 2025.
Key wo ds and ph ases. F ac al calculus, F ac al de i a i e, Di ac eigen alue p oblem, Eigen alues, Eigen unc-
ions.
1
2 F. A. C¸ETINKAYA, G. PLOTT
measu emen s a e inhe en ly local. Mo eo e , Fα-calculus e ains much o he simplici y ound
in o dina y calculus [4].
Di e en ial equa ions o e ac al domains a e o en e e ed o as ac al di e en ial equa-
ions. S udies conce ning ac al di e en ial equa ions ha e been an impo an a ea o esea ch
in ecen yea s. Fo ins ance, in [9] Golmankhaneh and Tun¸c p o e he exis ence and unique-
ness heo ems o he linea and non-linea ac al di e en ial equa ions and hey gi e he
ac al Lipschi z condi ion on he Fα-calculus. In [5] Golmankhaneh and Ca ini gi e di -
e ence equa ions on ac al se s and hei co esponding ac al di e en ial equa ions. They
de ine an analogue o he classical Eule me hod in ac al calculus and hey sol e ac al di -
e en ial equa ions by using his ac al Eule me hod. In [10] Golmankhaneh and Tun¸c gi e
he analogues o Laplace and Sumudu ans o ms, which ha e an impo an ole in con ol
enginee ing p oblems, and hey sol e linea di e en ial equa ions on Can o -like se s by u i-
lizing he ac al Sumudu ans o ms. F ac al di e en ial equa ions we e sol ed by de ining
ac al Mellin, Laplace, and Fou ie ans o ms in [11]. Re a ded, neu al, and enewal delay
di e en ial equa ions wi h cons an coe icien s in he ac al domain a e sol ed h ough he
me hod o s eps and employing Laplace ans o m in [8].
In ligh o he abo e-gi en li e a u e, we s udy he Di ac eigen alue p oblem gene a ed by
he ac al di e en ial equa ion
(1.1) ℓα := Dα
F 2−p(x) 1=λ 1
−Dα
F 1+ (x) 2=λ 2, x ∈[0, π]
and he bounda y condi ions
(1.2) U1( ) := 1(0) = 0,
(1.3) U2( ) := 1(π) = 0,
whe e Dα
Findica es he Fα-de i a i e in oduced in [18], = 1
2,λis a eal spec al
pa ame e , p(x), (x)∈ Lα
2(0, π) a e eal alued unc ions whe e Lα
2(0, π) is he space o squa e
Fα-in eg able unc ions on [0, π], i.e.
Zπ
0
| (x)|2dα
Fx < ∞
holds o : [0, π]→Ras Sch( ) is an α-pe ec se .
Al hough nume ous s udies add ess a ious di e en ial equa ions p oblems wi hin he ac al
calculus amewo k, he e a e ela i ely ew ha ocus on eigen alue p oblems in he con ex
o ac al calculus. Fo ins ance, in [23] C¸e inkaya and Golmakhaneh explo e a ac al S u m–
Liou ille p oblem, while in [1] Allah e die and Tuna p o e he exis ence and uniqueness
heo em o he solu ions o such p oblems. In his wo k, we ex end he esul s in [23] o he
Di ac se ing and p o ide nume ical examples o demons a e he p ac ical implica ions o ou
app oach. We belie e ha his wo k will con ibu e o u he s udies ela ed o he eigen alue
p oblems gene a ed wi h Fα-de i a i e and hei applica ions.
The s uc u e o he pape is as ollows. In Sec ion 2 we in oduce a sel -adjoin ope a o
and we gi e some o he i ues o eigen alues and ec o - alued eigen unc ions. In Sec ion 3
we p esen nume ical examples o illus a e he applicabili y o ou conclusions. In Sec ion 4
we close he pape wi h some concluding ema ks.
A FRACTAL DIRAC EIGENVALUE PROBLEM 3
2. Spec al p ope ies
In he con ex o ac al analysis desc ibed in [4], an inne p oduc in he Hilbe space
Lα
2(0, π) can be de ined by
h , gi=Zπ
0 1(x)g1(x) + 2(x)g2(x)dα
Fx
whe e = ( 1, 2)T∈ Lα
2(0, π) and g= (g1, g2)T∈ Lα
2(0, π).
Theo em 1. The ope a o ℓαis sel -adjoin in Lα
2(0, π).
P oo . Le and gbe he solu ions o he bounda y alue p oblem (1.1)–(1.3). Using he
de ini ion o he inne p oduc we ha e
hℓα , gi − h , ℓαgi=Zπ
0Dα
F 2−p(x) 1g1dα
Fx+Zπ
0−Dα
F 1+ (x) 2g2dα
Fx
−Zπ
0
1Dα
Fg2−p(x)g1dα
Fx−Zπ
0
2−Dα
Fg1+ (x)g2dα
Fx
=Zπ
0Dα
F 2g1+ 2Dα
Fg1−Dα
F 1g2− 1Dα
Fg2dα
Fx
=Zπ
0
Dα
F 2g1− 1g2dα
Fx.
Hence
(2.1) hℓα , gi − h , ℓαgi= 2g1− 1g2χF(x)π
x=0.
Using he bounda y condi ions (1.2) and (1.3), we see ha his e m anishes, so
hℓα , gi − h , ℓαgi= 0,
comple ing he p oo . 
Assume ha he bounda y alue p oblem (1.1)–(1.3) has a non i ial solu ion
(x, λ0) =  1(x, λ0)
2(x, λ0)
o a ce ain λ0, called an eigen alue, and he co esponding solu ion (x, λ0) is called a ec o -
alued eigen unc ion.
Lemma 2. The ec o - alued eigen unc ions (x, λ1)and g(x, λ2)co esponding o di e en
eigen alues λ16=λ2a e o hogonal.
P oo . Since (x, λ1) and g(x, λ2) a e solu ions o (1.1), we ha e
Dα
F 2(x, λ1)−p(x) 1(x, λ1) = λ1 1(x, λ1),
−Dα
F 1(x, λ1) + (x) 2(x, λ1) = λ1 2(x, λ1),
Dα
Fg2(x, λ2)−p(x)g1(x, λ2) = λ2g1(x, λ2),
−Dα
Fg1(x, λ2) + (x)g2(x, λ2) = λ2g2(x, λ2).
4 F. A. C¸ETINKAYA, G. PLOTT
Mul iplying hese equa ions by g1(x, λ2), g2(x, λ2), − 1(x, λ1), and − 2(x, λ1), espec i ely,
and summing, we ge
Dα
F 2(x, λ1)g1(x, λ2) + 2(x, λ1)Dα
Fg1(x, λ2)
−Dα
F 1(x, λ1)g2(x, λ2)− 1(x, λ1)Dα
Fg2(x, λ2)
= (λ1−λ2) 1(x, λ1)g1(x, λ2) + 2(x, λ1)g2(x, λ2).
Tha le -hand side is
Zπ
0
Dα
F 2(x, λ1)g1(x, λ2)− 1(x, λ1)g2(x, λ2)dα
Fx,
which becomes a bounda y e m ha anishes by condi ions (1.2), (1.3). Hence
(λ1−λ2)Zπ
0 1(x, λ1)g1(x, λ2) + 2(x, λ1)g2(x, λ2)dα
Fx= 0.
Since λ16=λ2, he inne p oduc mus be ze o, so he eigen unc ions a e o hogonal. 
Co olla y 3. The eigen alues o he bounda y alue p oblem (1.1)–(1.3) a e eal.
Le ϕ(·, λ) = ϕ1(·, λ)
ϕ2(·, λ)and ψ(·, λ) = ψ1(·, λ)
ψ2(·, λ)be he solu ions o (1.1) unde he ini ial
condi ions
(2.2) ϕ1(0, λ) = 0, ϕ2(0, λ) = 1, ψ1(π, λ) = 0, ψ2(π, λ) = 1.
Then
(2.3) U1(ϕ) = U2(ψ) = 0.
Deno e
(2.4) ∆(λ) = ϕ2(·, λ)ψ1(·, λ)−ϕ1(·, λ)ψ2(·, λ).
This unc ion ∆(λ) is called he cha ac e is ic unc ion o (1.1)–(1.3). One checks ha ∆(λ)
does no depend on xand is en i e in λ. I has an a mos coun able se o ze os {λn}. I can
be easily seen ha he cha ac e is ic unc ion does no depend on x. Indeed,
Dα
F∆(λ) = Dα
Fϕ2(·, λ)ψ1(·, λ) + ϕ2(·, λ)Dα
Fψ(·, λ)−Dα
Fϕ1(·, λ)
−ϕ(·, λ)Dα
Fψ2(·, λ)(1.1)
=p(·)−λϕ1(·, λ)ψ1(·, λ)
+ϕ2(·, λ) (·)−λψ2(·, λ) + ψ2(·, λ)λ− (·)ϕ2(·, λ)
−ϕ1(·, λ)λ+p(·)ψ1(·, λ) = 0.
Subs i u ing x= 0 and x=πin (2.4) and aking (2.2) in o conside a ion we ha e
(2.5) ∆(λ) = U1(ψ) = −U2(ϕ).
Theo em 4. The ze os {λn}o he cha ac e is ic unc ion coincide wi h he eigen alues o he
bounda y alue p oblem (1.1)–(1.3). The unc ions ϕ(x, λn)and ψ(x, λn)a e eigen unc ions,
and he e exis s a sequence {βn}such ha
(2.6) ψ(x, λn) = βnϕ(x, λn), βn6= 0.
A FRACTAL DIRAC EIGENVALUE PROBLEM 5
P oo . I λ0is a ze o o ∆(λ), i.e. ∆(λ0) = 0, hen by cons uc ion ψ(·, λ0) is a mul iple
o ϕ(·, λ0). Bo h sa is y he bounda y condi ions, so λ0is indeed an eigen alue, and ψ(·, λ0),
ϕ(·, λ0) a e eigen unc ions.
Con e sely, i λ0is an eigen alue and 0is a co esponding (nonze o) eigen unc ion sa is ying
(1.2)–(1.3), we can ma ch 0wi h one o ϕ(·, λ0) o ψ(·, λ0) (depending on ini ial condi ions),
showing ∆(λ0) = 0. One also sees each eigen alue is simple om he geome ic poin o iew.

We de ine he weigh numbe s {αn}o (1.1)–(1.3) by
(2.7) αn:= Zπ
0ϕ2
1(x, λn) + ϕ2
2(x, λn)dα
Fx.
Lemma 5. The ollowing ela ion holds:
(2.8) βnαn=Dα
F(λn),
whe e βna e de ined by (2.6) and Dα
F(λ) = Dα
F,λ∆(λ).
P oo . Since ϕ(x, λn) and ψ(x, λ) sol e (1.1), mul iply hem in a s anda d way and in eg a e
o e [0, π] o ge
Sα
F(λn)−Sα
F(λ)Zπ
0ϕ1(x, λn)ψ1(x, λ) + ϕ2(x, λn)ψ2(x, λ)dα
Fx= ∆(λn)−∆(λ).
As λ→λn, he di e ence quo ien leads o
Dα
F(λn) = Zπ
0ϕ1(x, λn)ψ1(x, λn) + ϕ2(x, λn)ψ2(x, λn)dα
Fx.
Subs i u ing (2.6) in o he exp ession abo e, we ob ain
Dα
F(λn) = βnZπ
0ϕ2
1(x, λn) + ϕ2
2(x, λn)dα
Fx.
Now, using (2.7), we simpli y his o
Dα
F(λn) = βnαn.
Thus, we a i e a (2.8). 
Co olla y 6. The eigen alues o (1.1)–(1.3) a e simple om he algeb aic poin o iew, i.e.
Dα
F∆(λn)6= 0.
P oo . Since αn6= 0, βn6= 0, we ge by i ue o (2.8) ha Dα
F∆(λn)6= 0. 
3. Nume ical Examples
In his sec ion, we p esen nume ical examples o illus a e he applicabili y o ou conclu-
sions. We used he ou h-o de classical Runge-Ku a me hod and he ou h-o de ac al
Runge-Ku a me hod as desc ibed in [12], whose equa ions a e gi en in ac al o m below. Fo
s i p oblems, o he nume ical echniques may be mo e sui able. We ha e plo ed solu ions o
hese equa ions using he Ma plo lib Py hon package.
yn+1 =yn+hα
Fkn1+ 2kn2+ 2kn3+kn4
6,

6 F. A. C¸ETINKAYA, G. PLOTT
whe e
kn1= (Sα
F(x), yn),
kn2= Sα
F(x) + 1
2hα
F, yn+1
2hα
Fkn1,
kn3= Sα
F(x) + 1
2hα
F, yn+1
2hα
Fkn2,
kn4= (Sα
F(x) + hα
F, yn+hα
Fkn3),
hα
F=Sα
F(xn+1)−Sα
F(xn).
We app oxima ed he in eg al s ai case unc ion Sα
F(x) by a powe law xα(an app oxima ion
alid o ce ain se s) desc ibed in [18]. In a p oblem whe e he exac s uc u e o he ac al
se Fis mo e c ucial han he scaling beha io o α, one can e e o he implemen a ion in
[12] o he coa se-g ained mass unc ion, γα
δ(F, a, b).
Below a e h ee examples conside ing (1.1)–(1.3) on he eal line (wi h sui able bounda y
app oxima ions).
Example 7. We de ine p(x) = 1
1+Sα
F(x),q(x) = 1
1+(Sα
F(x))2, and scaling indices α= [0.8,0.9,1.0].
The nume ical eigen alues (deno ed ˜
λn) appea in Table 1. E o s o he case α= 1 a e shown
in Table 2.
Table 1. Nume ically compu ed eigen alues ˜
λn o a ious me hods and α.
Me hod α˜
λ1˜
λ2˜
λ3˜
λ4
Classical N/A 0.347524 1.176747 2.055970 3.020643
F ac al 0.8 0.413400 1.438434 2.566015 -
F ac al 0.9 0.378385 1.301643 2.296227 -
F ac al 1.0 0.347685 1.176925 2.056040 3.020692
Table 2. Magni ude o e o be ween classical and ac al me hods o α= 1.
˜
λnClassical |∆˜
λn|
˜
λ10.347524 1.61 ×10−4
˜
λ21.176747 1.78 ×10−4
˜
λ32.055970 7.00 ×10−5
˜
λ43.020643 4.90 ×10−5
The plo s o he i s eigen unc ions 1=y1(x)and 2=y2(x) o he classical calculus and
ac al calculus me hods ac oss he h ee scaling indices a e shown below in Figu e 1.
A FRACTAL DIRAC EIGENVALUE PROBLEM 7
Figu e 1
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
y1(x)
classical
alpha=0.8
alpha=0.9
alpha=1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
0.

0
0.

5
0.50
0.55
0.

0
0.

5
y2(x)
classical
alpha=0.8
alpha=0.9
alpha=1.0
Fo easie isual e i ica ion ha he classical and ac al me hods ag ee when α= 1, zoomed-
in plo s a e also p o ided in Figu e 2.
Figu e 2
1.

1.5 1.

1.7 1.8
x
0.000
0.025
0.050
0.075
0.100
0.125
0.150
0.175
y1(x)
classical
alpha=1.0
1.

1.5 1.

1.7 1.8
x
0.

0
0.

5
0.50
0.55
0.

0
0.

5
y2(x)
classical
alpha=1.0
Example 8. Le p(x) = Sα
F(x)+1,q(x) = (Sα
F(x))2+1, and α= [0.8,0.9,1.0]. The eigen alues
and e o s a e shown in Tables 3 and 4.
Table 3. Nume ically compu ed eigen alues ˜
λn o a ious me hods and α.
Me hod α˜
λ1
Classical N/A 1.544759
F ac al 0.8 1.516625
F ac al 0.9 1.530339
F ac al 1.0 1.544186
8 F. A. C¸ETINKAYA, G. PLOTT
Table 4. Magni ude o e o be ween classical and ac al me hods o α= 1.
˜
λnClassical |∆˜
λn|
˜
λ11.544759 5.73 ×10−4
The plo s o he i s eigen unc ions 1=y1(x)and 2=y2(x) o he classical calculus and
ac al calculus me hods ac oss he h ee scaling indices a e shown below in Figu e 3.
Figu e 3
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x

2.5

2.0

1.5

1.0

0.5
0.0
y1(x)
classical
alpha=0.8
alpha=0.9
alpha=1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x

2.0

1.5

1.0

0.5
0.0
0.5
1.0
y2(x)
classical
alpha=0.8
alpha=0.9
alpha=1.0
Fo easie isual e i ica ion ha he classical and ac al me hods nea ly ag ee when α= 1,
zoomed-in plo s a e also p o ided in Figu e 4.
Figu e 4
1.

1.5 1.

1.7 1.8
x
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
y1(x)
classical
alpha=1.0
1.

1.5 1.

1.7 1.8
x
−2.0
−1.5
−1.0
−0.5
0.0
0.5
1.0
y2(x)
classical
alpha=1.0
Example 9. Le p(x) = eSα
F(x),q(x) = e−Sα
F(x), and α= [0.8,0.9,1.0]. Tables 5 and 6 show
he eigen alues and e o s.
A FRACTAL DIRAC EIGENVALUE PROBLEM 9
Table 5. Nume ically compu ed eigen alues ˜
λn o a ious me hods and α.
Me hod α˜
λ1˜
λ2˜
λ3˜
λ4˜
λ5˜
λ6
Classical N/A 0.148677 0.458639 0.865004 1.452401 2.170184 2.965759
F ac al 0.8 0.210897 0.644622 1.301299 2.201887 - -
F ac al 0.9 0.175896 0.542309 1.057334 1.790641 2.656072 -
F ac al 1.0 0.148792 0.458986 0.865601 1.453235 2.171232 2.966991
Table 6. Magni ude o e o be ween classical and ac al me hods o α= 1.
˜
λnClassical |∆˜
λn|
˜
λ10.148677 1.15 ×10−4
˜
λ20.458639 3.47 ×10−4
˜
λ30.865004 5.97 ×10−4
˜
λ41.452401 8.34 ×10−4
˜
λ52.170184 1.05 ×10−3
˜
λ62.965759 1.23 ×10−3
The plo s o he i s eigen unc ions 1=y1(x)and 2=y2(x) o he classical calculus and
ac al calculus me hods ac oss he h ee scaling indices a e shown below in Figu e 5.
Figu e 5
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
0.00
0.02
0.0

0.0

0.08
y1(x)
classical
alpha=0.8
alpha=0.9
alpha=1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
0.2
0.

0.

0.8
1.0
y2(x)
classical
alpha=0.8
alpha=0.9
alpha=1.0
Fo easie isual e i ica ion ha he classical and ac al me hods ag ee when α= 1, zoomed-
in plo s a e also p o ided in Figu e 6.