A Fractal Dirac Eigenvalue Problem: Spectral Properties and Numerical Examples

arXiv:2502.10529v2 [math.SP] 18 Mar 2025 A FRACTAL DIRAC EIGENVALUE PROBLEM: SPECTRAL PROPERTIES AND NUMERICAL EXAMPLES F. AYC¸A C¸ETINKAYA, GAGE PLOTT Abstract. In this paper, we study a Dirac boundary value problem where the operator is considered with a derivative of order α∈(0,1], known as the Fα-derivative. We prove some spectral properties of eigenvalues and eigenfunctions and we present numerical examples to demonstrate the practical implications of our approach. 1. Introduction The word fractal derives from the Latin word fractus which means cracked. Fractals exhibit unique geometric properties, showcasing a fractal dimension that surpasses their topological dimension. These intricate structures display self-similarity at varying scales, combining repetitive and random processes. The intricate nature of fractals poses challenges for conventional calculus methods in calculating derivatives and integrals. The connection between fractal geometry and natural phenomena, as seen in clouds, mountains, and lightning, underscores a complexity that defies conventional mathematical frameworks, as highlighted in [16]. Moreover, in contrast to Euclidean geometry, determining the size of fractals involves non-trivial considerations for measurements like length, surface area, and volume, as discussed in [3,13]. Fractals, characterized by intricate patterns, showcase self-similarity and often display dimensions that are non-integer and complex [15,17]. The middle-third set of Cantor is one of the most well-known examples of fractals [2]. Fractals are often too irregular to have any smooth differentiable structure defined on them, and this results in delivering the methods and techniques of ordinary calculus as powerless and inapplicable. The techniques and methods to create calculus on the fractal sets and curves are studied in [3,14,21]. In recent work, Parvate and Gangal [18,19] introduced Fα-calculus, a form of Riemannianlike calculus grounded in the fractal subsets of the real line. This calculus is distinguished by its algorithmic simplicity compared to other methods. Fα-calculus is a generalization of the ordinary calculus that addresses the cases where standard calculus is inapplicable. In this calculus, an integral of order α∈(0,1] called the Fα-integral is defined which makes it possible to integrate functions with fractal support Fof dimension α. Furthermore, a derivative of order α∈(0,1], known as the Fα-derivative, facilitates the differentiation of functions such as the Cantor staircase function and the Weierstrass function. In contrast to the classical fractional derivative, the Fα-derivative aligns the geometrical order of the derivative with the domain of the function’s support, lending it a distinct physical interpretation [6,20,22]. Notably, the Fα-derivative is local, which is in contrast to the nonlocal nature of the classical fractional derivative. This locality is crucial in physics, where all Date: March 19, 2025. Key words and phrases. Fractal calculus, Fractal derivative, Dirac eigenvalue problem, Eigenvalues, Eigenfunctions. 1 2 F. A. C¸ETINKAYA, G. PLOTT measurements are inherently local. Moreover, Fα-calculus retains much of the simplicity found in ordinary calculus [4]. Differential equations over fractal domains are often referred to as fractal differential equations. Studies concerning fractal differential equations have been an important area of research in recent years. For instance, in [9] Golmankhaneh and Tun¸c prove the existence and uniqueness theorems for the linear and non-linear fractal differential equations and they give the fractal Lipschitz condition on the Fα-calculus. In [5] Golmankhaneh and Cattini give difference equations on fractal sets and their corresponding fractal differential equations. They define an analogue of the classical Euler method in fractal calculus and they solve fractal differential equations by using this fractal Euler method. In [10] Golmankhaneh and Tun¸c give the analogues of Laplace and Sumudu transforms, which have an important role in control engineering problems, and they solve linear differential equations on Cantor-like sets by utilizing the fractal Sumudu transforms. Fractal differential equations were solved by defining fractal Mellin, Laplace, and Fourier transforms in [11]. Retarded, neutral, and renewal delay differential equations with constant coefficients in the fractal domain are solved through the method of steps and employing Laplace transform in [8]. In light of the above-given literature, we study the Dirac eigenvalue problem generated by the fractal differential equation (1.1) ℓαf:= Dα Ff2−p(x)f1=λf1 −Dα Ff1+r(x)f2=λf2, x ∈[0, π] and the boundary conditions (1.2) U1(f) := f1(0) = 0, (1.3) U2(f) := f1(π) = 0, where Dα Findicates the Fα-derivative introduced in [18], f=f1 f2,λis a real spectral parameter, p(x), r(x)∈ Lα 2(0, π) are real valued functions where Lα 2(0, π) is the space of square Fα-integrable functions on [0, π], i.e. Zπ 0 |f(x)|2dα Fx < ∞ holds for f: [0, π]→Ras Sch(f) is an α-perfect set. Although numerous studies address various differential equations problems within the fractal calculus framework, there are relatively few that focus on eigenvalue problems in the context of fractal calculus. For instance, in [23] C¸etinkaya and Golmakhaneh explore a fractal Sturm– Liouville problem, while in [1] Allahverdiev and Tuna prove the existence and uniqueness theorem for the solutions of such problems. In this work, we extend the results in [23] to the Dirac setting and provide numerical examples to demonstrate the practical implications of our approach. We believe that this work will contribute to further studies related to the eigenvalue problems generated with Fα-derivative and their applications. The structure of the paper is as follows. In Section 2 we introduce a self-adjoint operator and we give some of the virtues of eigenvalues and vector-valued eigenfunctions. In Section 3 we present numerical examples to illustrate the applicability of our conclusions. In Section 4 we close the paper with some concluding remarks. A FRACTAL DIRAC EIGENVALUE PROBLEM 3 2. Spectral properties In the context of fractal analysis described in [4], an inner product in the Hilbert space Lα 2(0, π) can be defined by hf, gi=Zπ 0f1(x)g1(x) + f2(x)g2(x)dα Fx where f= (f1, f2)T∈ Lα 2(0, π) and g= (g1, g2)T∈ Lα 2(0, π). Theorem 1. The operator ℓαis self-adjoint in Lα 2(0, π). Proof. Let fand gbe the solutions of the boundary value problem (1.1)–(1.3). Using the definition of the inner product we have hℓαf, gi − hf, ℓαgi=Zπ 0Dα Ff2−p(x)f1g1dα Fx+Zπ 0−Dα Ff1+r(x)f2g2dα Fx −Zπ 0 f1Dα Fg2−p(x)g1dα Fx−Zπ 0 f2−Dα Fg1+r(x)g2dα Fx =Zπ 0Dα Ff2g1+f2Dα Fg1−Dα Ff1g2−f1Dα Fg2dα Fx =Zπ 0 Dα Ff2g1−f1g2dα Fx. Hence (2.1) hℓαf, gi − hf, ℓαgi=f2g1−f1g2χF(x)π x=0. Using the boundary conditions (1.2) and (1.3), we see that this term vanishes, so hℓαf, gi − hf, ℓαgi= 0, completing the proof.  Assume that the boundary value problem (1.1)–(1.3) has a nontrivial solution f(x, λ0) = f1(x, λ0) f2(x, λ0) for a certain λ0, called an eigenvalue, and the corresponding solution f(x, λ0) is called a vectorvalued eigenfunction. Lemma 2. The vector-valued eigenfunctions f(x, λ1)and g(x, λ2)corresponding to different eigenvalues λ16=λ2are orthogonal. Proof. Since f(x, λ1) and g(x, λ2) are solutions of (1.1), we have Dα Ff2(x, λ1)−p(x)f1(x, λ1) = λ1f1(x, λ1), −Dα Ff1(x, λ1) + r(x)f2(x, λ1) = λ1f2(x, λ1), Dα Fg2(x, λ2)−p(x)g1(x, λ2) = λ2g1(x, λ2), −Dα Fg1(x, λ2) + r(x)g2(x, λ2) = λ2g2(x, λ2). 4 F. A. C¸ETINKAYA, G. PLOTT Multiplying these equations by g1(x, λ2), g2(x, λ2), −f1(x, λ1), and −f2(x, λ1), respectively, and summing, we get Dα Ff2(x, λ1)g1(x, λ2) + f2(x, λ1)Dα Fg1(x, λ2) −Dα Ff1(x, λ1)g2(x, λ2)−f1(x, λ1)Dα Fg2(x, λ2) = (λ1−λ2)f1(x, λ1)g1(x, λ2) + f2(x, λ1)g2(x, λ2). That left-hand side is Zπ 0 Dα Ff2(x, λ1)g1(x, λ2)−f1(x, λ1)g2(x, λ2)dα Fx, which becomes a boundary term that vanishes by conditions (1.2), (1.3). Hence (λ1−λ2)Zπ 0f1(x, λ1)g1(x, λ2) + f2(x, λ1)g2(x, λ2)dα Fx= 0. Since λ16=λ2, the inner product must be zero, so the eigenfunctions are orthogonal.  Corollary 3. The eigenvalues of the boundary value problem (1.1)–(1.3) are real. Let ϕ(·, λ) = ϕ1(·, λ) ϕ2(·, λ)and ψ(·, λ) = ψ1(·, λ) ψ2(·, λ)be the solutions of (1.1) under the initial conditions (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. Denote (2.4) ∆(λ) = ϕ2(·, λ)ψ1(·, λ)−ϕ1(·, λ)ψ2(·, λ). This function ∆(λ) is called the characteristic function of (1.1)–(1.3). One checks that ∆(λ) does not depend on xand is entire in λ. It has an at most countable set of zeros {λn}. It can be easily seen that the characteristic function does not 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)r(·)−λψ2(·, λ) + ψ2(·, λ)λ−r(·)ϕ2(·, λ) −ϕ1(·, λ)λ+p(·)ψ1(·, λ) = 0. Substituting x= 0 and x=πin (2.4) and taking (2.2) into consideration we have (2.5) ∆(λ) = U1(ψ) = −U2(ϕ). Theorem 4. The zeros {λn}of the characteristic function coincide with the eigenvalues of the boundary value problem (1.1)–(1.3). The functions ϕ(x, λn)and ψ(x, λn)are eigenfunctions, and there exists a sequence {βn}such that (2.6) ψ(x, λn) = βnϕ(x, λn), βn6= 0. A FRACTAL DIRAC EIGENVALUE PROBLEM 5 Proof. If λ0is a zero of ∆(λ), i.e. ∆(λ0) = 0, then by construction ψ(·, λ0) is a multiple of ϕ(·, λ0). Both satisfy the boundary conditions, so λ0is indeed an eigenvalue, and ψ(·, λ0), ϕ(·, λ0) are eigenfunctions. Conversely, if λ0is an eigenvalue and f0is a corresponding (nonzero) eigenfunction satisfying (1.2)–(1.3), we can match f0with one of ϕ(·, λ0) or ψ(·, λ0) (depending on initial conditions), showing ∆(λ0) = 0. One also sees each eigenvalue is simple from the geometric point of view.  We define the weight numbers {αn}of (1.1)–(1.3) by (2.7) αn:= Zπ 0ϕ2 1(x, λn) + ϕ2 2(x, λn)dα Fx. Lemma 5. The following relation holds: (2.8) βnαn=Dα F(λn), where βnare defined by (2.6) and Dα F(λ) = Dα F,λ∆(λ). Proof. Since ϕ(x, λn) and ψ(x, λ) solve (1.1), multiply them in a standard way and integrate over [0, π] to get 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, the difference quotient leads to Dα F(λn) = Zπ 0ϕ1(x, λn)ψ1(x, λn) + ϕ2(x, λn)ψ2(x, λn)dα Fx. Substituting (2.6) into the expression above, we obtain Dα F(λn) = βnZπ 0ϕ2 1(x, λn) + ϕ2 2(x, λn)dα Fx. Now, using (2.7), we simplify this to Dα F(λn) = βnαn. Thus, we arrive at (2.8).  Corollary 6. The eigenvalues of (1.1)–(1.3) are simple from the algebraic point of view, i.e. Dα F∆(λn)6= 0. Proof. Since αn6= 0, βn6= 0, we get by virtue of (2.8) that Dα F∆(λn)6= 0.  3. Numerical Examples In this section, we present numerical examples to illustrate the applicability of our conclusions. We used the fourth-order classical Runge-Kutta method and the fourth-order fractal Runge-Kutta method as described in [12], whose equations are given in fractal form below. For stiff problems, other numerical techniques may be more suitable. We have plotted solutions to these equations using the Matplotlib Python package. yn+1 =yn+hα Fkn1+ 2kn2+ 2kn3+kn4 6, 6 F. A. C¸ETINKAYA, G. PLOTT where kn1=f(Sα F(x), yn), kn2=fSα F(x) + 1 2hα F, yn+1 2hα Fkn1, kn3=fSα F(x) + 1 2hα F, yn+1 2hα Fkn2, kn4=f(Sα F(x) + hα F, yn+hα Fkn3), hα F=Sα F(xn+1)−Sα F(xn). We approximated the integral staircase function Sα F(x) by a power law xα(an approximation valid for certain sets) described in [18]. In a problem where the exact structure of the fractal set Fis more crucial than the scaling behavior of α, one can refer to the implementation in [12] of the coarse-grained mass function, γα δ(F, a, b). Below are three examples considering (1.1)–(1.3) on the real line (with suitable boundary approximations). Example 7. We define 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 numerical eigenvalues (denoted ˜ λn) appear in Table 1. Errors for the case α= 1 are shown in Table 2. Table 1. Numerically computed eigenvalues ˜ λnfor various methods and α. Method α˜ λ1˜ λ2˜ λ3˜ λ4 Classical N/A 0.347524 1.176747 2.055970 3.020643 Fractal 0.8 0.413400 1.438434 2.566015 - Fractal 0.9 0.378385 1.301643 2.296227 - Fractal 1.0 0.347685 1.176925 2.056040 3.020692 Table 2. Magnitude of error between classical and fractal methods for 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 plots of the first eigenfunctions f1=y1(x)and f2=y2(x)for the classical calculus and fractal calculus methods across the three scaling indices are shown below in Figure 1. A FRACTAL DIRAC EIGENVALUE PROBLEM 7 Figure 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 For easier visual verification that the classical and fractal methods agree when α= 1, zoomedin plots are also provided in Figure 2. Figure 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. Let p(x) = Sα F(x)+1,q(x) = (Sα F(x))2+1, and α= [0.8,0.9,1.0]. The eigenvalues and errors are shown in Tables 3 and 4. Table 3. Numerically computed eigenvalues ˜ λnfor various methods and α. Method α˜ λ1 Classical N/A 1.544759 Fractal 0.8 1.516625 Fractal 0.9 1.530339 Fractal 1.0 1.544186 8 F. A. C¸ETINKAYA, G. PLOTT Table 4. Magnitude of error between classical and fractal methods for α= 1. ˜ λnClassical |∆˜ λn| ˜ λ11.544759 5.73 ×10−4 The plots of the first eigenfunctions f1=y1(x)and f2=y2(x)for the classical calculus and fractal calculus methods across the three scaling indices are shown below in Figure 3. Figure 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 For easier visual verification that the classical and fractal methods nearly agree when α= 1, zoomed-in plots are also provided in Figure 4. Figure 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. Let p(x) = eSα F(x),q(x) = e−Sα F(x), and α= [0.8,0.9,1.0]. Tables 5 and 6 show the eigenvalues and errors. A FRACTAL DIRAC EIGENVALUE PROBLEM 9 Table 5. Numerically computed eigenvalues ˜ λnfor various methods and α. Method α˜ λ1˜ λ2˜ λ3˜ λ4˜ λ5˜ λ6 Classical N/A 0.148677 0.458639 0.865004 1.452401 2.170184 2.965759 Fractal 0.8 0.210897 0.644622 1.301299 2.201887 - - Fractal 0.9 0.175896 0.542309 1.057334 1.790641 2.656072 - Fractal 1.0 0.148792 0.458986 0.865601 1.453235 2.171232 2.966991 Table 6. Magnitude of error between classical and fractal methods for 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 plots of the first eigenfunctions f1=y1(x)and f2=y2(x)for the classical calculus and fractal calculus methods across the three scaling indices are shown below in Figure 5. Figure 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 For easier visual verification that the classical and fractal methods agree when α= 1, zoomedin plots are also provided in Figure 6.