The Self-Adjoint Fractional Heun Operator and Its Spectral Properties

Abstract

This paper introduces a rigorously defined fractional Heun operator constructed through a symmetric composition of left and right Riemann–Liouville fractional derivatives. By deriving a compatible fractional Pearson-type equation, a new weight function and Hilbert space setting are established, ensuring the operator’s self-adjointness under natural fractional boundary conditions. Within this framework, we prove the existence of a real, discrete spectrum and demonstrate that the corresponding eigenfunctions form a complete orthogonal system in $L_{{\omega }_{\alpha }}^2(a,b)$. The central theoretical result shows that the fractional eigenpairs $({\lambda }_n^{\left(\alpha \right)},u_n^{\left(\alpha \right)})$ converge continuously to their classical Heun counterparts $({\lambda }_n^{\left(1\right)},u_n^{\left(1\right)})$ as $\alpha \to 1^{-}$. This provides a rigorous analytic bridge between fractional and classical spectral theories. A numerical study based on the fractional Legendre case confirms the predicted self-adjointness and spectral convergence, illustrating the smooth deformation of the classical eigenfunctions into their fractional counterparts. The results establish the fractional Heun operator as a mathematically consistent generalization capable of generating new families of orthogonal fractional functions.

1. Introduction

Second–order linear differential equations with regular singular points play a fundamental role in classical spectral theory and the theory of special functions. Among them, the Heun equation occupies a distinguished position as the most general Fuchsian equation with four regular singularities. It encompasses, as special or limiting cases, several well–known equations such as the hypergeometric, confluent hypergeometric, and Mathieu equations, and has therefore attracted sustained interest in both mathematics and physics [1,2,3,4,5]. In its self–adjoint Sturm–Liouville formulation, the Heun operator admits a real spectrum and orthogonal eigenfunctions with respect to a suitable weight function, forming the basis for a rigorous spectral theory with numerous applications [6,7,8].

In parallel, fractional calculus has emerged as a powerful framework for modeling nonlocal processes and systems with memory. Comprehensive treatments of fractional integrals and derivatives can be found in classical monographs such as [9,10,11], while applications to physical systems and anomalous transport phenomena have been extensively discussed in [12,13]. Motivated by these developments, fractional extensions of Sturm–Liouville theory have been actively investigated over the past decade. Notable progress has been achieved for fractional analogues of classical operators and orthogonal polynomial systems, including Legendre-, Jacobi, and Laguerre-type problems, where appropriate formulations preserve essential spectral properties [14,15,16,17,18].

Despite these advances, extending the Heun operator to the fractional setting remains a challenging and largely unresolved problem. The difficulty arises from the interaction between singular coefficients inherent in Heun-type equations and the nonlocal nature of fractional derivatives. Classical self–adjointness arguments rely heavily on local integration by parts and boundary conditions, which no longer apply directly in the fractional context [19,20]. Although several recent works have considered fractional equations with regular singular points or hypergeometric structure [21,22], existing formulations of fractional Heun–type equations either lack a rigorous self-adjoint framework or do not establish a clear spectral connection with the classical Heun operator [23,24]. Consequently, fundamental questions concerning the reality and discreteness of the spectrum, orthogonality of eigenfunctions, and convergence to the classical problem as the fractional order approaches unity remain open.

The aim of the present work is to address these issues by constructing a fractional Heun operator within a rigorously defined Hilbert space setting. The proposed operator is based on a symmetric composition of left and right Riemann–Liouville fractional derivatives, a strategy that has proved effective in related fractional Sturm–Liouville problems [14,20]. To ensure symmetry of the associated bilinear form, we derive a fractional Pearson-type equation, which leads to a natural weight function generalizing the classical Pearson equation in the limit of integer order. This approach allows self-adjointness to be recovered through analytical structure rather than imposed formally.

Within this framework, we establish the self-adjointness of the fractional Heun operator and develop its spectral theory. In particular, we show that the operator admits a real, discrete spectrum and that its eigenfunctions form a complete orthogonal system in an appropriate weighted $L^2$ space. Moreover, under suitable assumptions on the convergence of the fractional weight function, we analyze the behavior of the spectrum as the fractional order tends to one and prove convergence to the classical Heun operator. This result provides a precise analytical bridge between fractional and integer-order Sturm–Liouville theories and clarifies the role of the proposed operator as a continuous deformation of the classical Heun framework.

The paper is organized as follows. Section 2 introduces the necessary preliminaries from fractional calculus and defines the functional setting of the problem. In Section 3, the fractional Heun operator is constructed and its self-adjointness is established. The spectral properties of the operator are analyzed in Section 4. Section 5 is devoted to spectral convergence as the fractional order approaches unity. A numerical illustration based on a fractional Legendre-type case is presented to complement the theoretical results, followed by concluding remarks.

2. Preliminaries

In this section, we collect the necessary definitions and facts from fractional calculus and classical Sturm–Liouville theory that will be used throughout the paper.

Definition 1 

(Fractional integrals [11]). Let $0<\alpha <1$ and u be locally integrable on $(a,b)$. The left and right Riemann–Liouville fractional integrals of order α are defined, respectively, by

$$\left(I_{a+}^{\alpha }u\right)\left(x\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^x{(x-t)}^{\alpha -1}u\left(t\right)dt,\left(I_{b-}^{\alpha }u\right)\left(x\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_x^b{(t-x)}^{\alpha -1}u\left(t\right)dt,x\in (a,b).$$

Definition 2 

(Fractional derivatives [10,11]). For $0<\alpha <1$, assume that $I_{a+}^{1-\alpha }u$ and $I_{b-}^{1-\alpha }u$ are absolutely continuous. The left and right Riemann–Liouville fractional derivatives of order α are defined by

$$\left(D_{a+}^{\alpha }u\right)\left(x\right)={\displaystyle \frac{d}{dx}}\left(I_{a+}^{1-\alpha }u\right)\left(x\right),\left(D_{b-}^{\alpha }u\right)\left(x\right)=-{\displaystyle \frac{d}{dx}}\left(I_{b-}^{1-\alpha }u\right)\left(x\right).$$

A crucial tool for handling fractional differential operators is the following integration-by-parts identity.

Lemma 1 

(Fractional integration by parts [15,20]). For $0<\alpha <1$ and functions u and v for which the expressions make sense,

$${\int }_a^b\left(D_{a+}^{\alpha }u\right)\left(x\right)v\left(x\right)dx={\int }_a^{b}u\left(x\right)\left(D_{b-}^{\alpha }v\right)\left(x\right)dx+{\left[\left(I_{a+}^{1-\alpha }u\right)\left(x\right)v\left(x\right)\right]}_{x=a}^{x=b}.$$

Lemma 1 shows that fractional derivatives are not symmetric without appropriate boundary conditions; the boundary term involves fractional integrals rather than pointwise values. This observation forces a new treatment of boundary conditions in the fractional setting, which will be essential for constructing a self-adjoint operator.

We shall work in the following fractional Sobolev space, which is the natural domain for operators involving both left and right fractional derivatives.

Definition 3 

(Fractional Sobolev space [19,20]). For $0<\alpha <1$, define

$$H^{\alpha }(a,b)=\left\{u\in L^2(a,b):D_{a+}^{\alpha }u,D_{b-}^{\alpha }u\in L^2(a,b)\right\},$$

equipped with the norm

$${\parallel u\parallel }_{H^{\alpha }}={\left({\parallel u\parallel }_{L^2}^2+\parallel D_{a+}^{\alpha }{u\parallel }_{L^2}^2+{\parallel D_{b-}^{\alpha }u\parallel }_{L^2}^2\right)}^{1/2}.$$

Finally, we recall the classical Heun operator in its Sturm–Liouville form, which will serve as the reference point for our fractional generalization.

Definition 4 

(Classical Heun operator [2,3,7]). Let $A\left(x\right),B\left(x\right)$, and $C\left(x\right)$ be smooth functions on $[a,b]$ with $A\left(x\right)\ne 0$ except at the (regular) singular points. The classical Heun operator is

$${\mathcal{H}}_{1}u=-{\displaystyle \frac{d}{dx}}\left(A\left(x\right){\displaystyle \frac{du}{dx}}\right)+B\left(x\right){\displaystyle \frac{du}{dx}}+C\left(x\right)u,x\in (a,b).$$

If a positive weight $\omega$ satisfies the classical Pearson equation ${\displaystyle \frac{d}{dx}}\left(A\left(x\right)\omega \left(x\right)\right)=B\left(x\right)\omega \left(x\right)$, then the operator can be written symmetrically as

$${\mathcal{H}}_{1}u=-{\displaystyle \frac{1}{\omega \left(x\right)}}{\displaystyle \frac{d}{dx}}\left(A\left(x\right)\omega \left(x\right){\displaystyle \frac{du}{dx}}\right)+C\left(x\right)u,$$

and becomes self-adjoint in $L_{\omega }^2(a,b)$ under suitable boundary conditions. Our goal is to develop a fractional analogue of this construction, replacing ordinary derivatives by Riemann–Liouville fractional derivatives while retaining the essential self-adjoint and spectral features.

3. Self-Adjointness and Spectral Properties of the Fractional Heun Operator

In this section, we define the fractional Heun operator in a weighted Hilbert space and establish its self-adjointness. We then derive the discreteness and completeness of the spectrum and finally prove the convergence of fractional eigenpairs to their classical counterparts as $\alpha \to 1^{-}$. Throughout, $(a,b)$ is a bounded interval and $0<\alpha <1$ unless stated otherwise.

We work in the weighted space

$$L_{\omega }^2(a,b)=\left\{u:(a,b)\to \mathbb{R}:{\int }_a^b{\left|u\left(x\right)\right|}^2\omega \left(x\right)dx<\infty \right\},{\langle u,v\rangle }_{\omega }={\int }_a^{b}u\left(x\right)v\left(x\right)\omega \left(x\right)dx,$$

where $\omega :(a,b)\to (0,\infty )$ is a positive weight. Fractional derivatives are understood in the Riemann–Liouville sense as recalled in Section 2.

Definition 5 

(Fractional Heun operator). Let $A,B,C\in C^1\left([a,b]\right)$ with $A\left(x\right)>0$ on $[a,b]$, and let $\omega \in C^1\left([a,b]\right)$ satisfy $\omega \left(x\right)>0$ on $[a,b]$. For sufficiently regular u, define

$${\mathcal{H}}_{\alpha }u=-D_{b-}^{\alpha }\left(A{D}_{a+}^{\alpha }u\right)+{\displaystyle \frac{B}{2}}\left(D_{a+}^{\alpha }u+D_{b-}^{\alpha }u\right)+Cu\mathrm{in}(a,b).$$

We assume that the weight $\omega$ is chosen to satisfy the fractional Pearson relation

$$D_{b-}^{\alpha }\left(A\left(x\right)\omega \left(x\right)\right)=B\left(x\right)\omega \left(x\right)\mathrm{in}(a,b),$$

which is the compatibility condition ensuring symmetry in the weighted pairing.

Definition 6 

(Operator domain). Let ω satisfy the fractional Pearson relation above. We define $V_{\alpha }$ as the set of all $u\in H^{\alpha }(a,b)$ such that ${\mathcal{H}}_{\alpha }u\in L_{\omega }^2(a,b)$ and

$$u\left(a\right)=u\left(b\right)=0,\left(I_{a+}^{1-\alpha }\left(u\omega \right)\right)\left(b\right)=0,\left(I_{b-}^{1-\alpha }\left(u\omega \right)\right)\left(a\right)=0.$$

The boundary conditions in Definition 6 are designed so that all boundary terms generated by fractional integration by parts vanish in the weighted pairing. This replaces the classical endpoint conditions and is the only point where the nonlocal character of the fractional derivatives enters the symmetry argument.

Theorem 1 

(Self-adjointness). Let $A,B,C\in C^1\left([a,b]\right)$ with $A\left(x\right)>0$ on $[a,b]$ and $C\in L^{\infty }(a,b)$. Let $\omega \in C^1\left([a,b]\right)$ be positive and satisfy

$$D_{b-}^{\alpha }\left(A\omega \right)=B\omega \mathrm{in}(a,b).$$

Then the operator ${\mathcal{H}}_{\alpha }:V_{\alpha }\subset L_{\omega }^2(a,b)\to L_{\omega }^2(a,b)$ is symmetric. Moreover, ${\mathcal{H}}_{\alpha }$ admits a self-adjoint realization in $L_{\omega }^2(a,b)$ whose form domain contains $V_{\alpha }$.

Proof. 

We first establish symmetry on $V_{\alpha }$. Fix $u,v\in V_{\alpha }$. By Definition 5,

$${\langle {\mathcal{H}}_{\alpha }u,v\rangle }_{\omega }=-{\int }_a^b{D}_{b-}^{\alpha }\left(A{D}_{a+}^{\alpha }u\right)v\omega dx+{\displaystyle \frac{1}{2}}{\int }_a^{b}B\left(D_{a+}^{\alpha }u+D_{b-}^{\alpha }u\right)v\omega dx+{\int }_a^{b}Cuv\omega dx.$$

We treat the principal part using the fractional integration by parts formula (Lemma 1). Applying it with

$$u\mapsto A{D}_{a+}^{\alpha }u,v\mapsto v\omega ,$$

we obtain

$${\int }_a^b{D}_{b-}^{\alpha }\left(A{D}_{a+}^{\alpha }u\right)v\omega dx={\int }_a^{b}A{D}_{a+}^{\alpha }u{D}_{a+}^{\alpha }\left(v\omega \right)dx+{\left[\left(I_{a+}^{1-\alpha }\left(A{D}_{a+}^{\alpha }u\right)\right)\left(x\right)v\left(x\right)\omega \left(x\right)\right]}_{x=a}^{x=b}.$$

The boundary term vanishes for $u,v\in V_{\alpha }$ by Definition 6. Hence

$$-{\int }_a^b{D}_{b-}^{\alpha }\left(A{D}_{a+}^{\alpha }u\right)v\omega dx=-{\int }_a^{b}A{D}_{a+}^{\alpha }u{D}_{a+}^{\alpha }\left(v\omega \right)dx.$$

(1)

Next we consider the term involving $D_{b-}^{\alpha }u$. Applying again Lemma 1 with

$$u\mapsto u,v\mapsto Bv\omega ,$$

yields

$${\int }_a^b\left(D_{b-}^{\alpha }u\right)Bv\omega dx={\int }_a^{b}u{D}_{a+}^{\alpha }\left(Bv\omega \right)dx+{\left[\left(I_{a+}^{1-\alpha }u\right)\left(x\right)B\left(x\right)v\left(x\right)\omega \left(x\right)\right]}_{x=a}^{x=b}.$$

The boundary term vanishes since $u\left(a\right)=u\left(b\right)=0$ and $v\left(a\right)=v\left(b\right)=0$ on $V_{\alpha }$, and therefore

$${\int }_a^b\left(D_{b-}^{\alpha }u\right)Bv\omega dx={\int }_a^{b}u{D}_{a+}^{\alpha }\left(Bv\omega \right)dx.$$

(2)

Since $A,\omega \in C^1\left([a,b]\right)$ and the Pearson relation $D_{b-}^{\alpha }\left(A\omega \right)=B\omega$ holds in $(a,b)$, the product $B\omega$ is well defined and the compositions with $D_{a+}^{\alpha }$ used below are understood in the weak sense on $V_{\alpha }$.

Combining (1) and (2), we obtain

$${\langle {\mathcal{H}}_{\alpha }u,v\rangle }_{\omega }=-{\int }_a^{b}A{D}_{a+}^{\alpha }u{D}_{a+}^{\alpha }\left(v\omega \right)dx+{\displaystyle \frac{1}{2}}{\int }_a^{b}B{D}_{a+}^{\alpha }uv\omega dx+{\displaystyle \frac{1}{2}}{\int }_a^{b}u{D}_{a+}^{\alpha }\left(Bv\omega \right)dx+{\int }_a^{b}Cuv\omega dx.$$

The last term is symmetric. To handle the remaining terms, we use the Pearson relation in the form $B\omega =D_{b-}^{\alpha }\left(A\omega \right)$. Applying Lemma 1 to $A\omega$ against the product $uv$, we obtain

$${\int }_a^{b}B\omega uvdx={\int }_a^{b}A\omega D_{a+}^{\alpha }\left(uv\right)dx,$$

with vanishing boundary contribution due to the conditions in Definition 6. This identity allows the drift terms to be absorbed into the same bilinear structure as the principal part. Consequently, the expression above is invariant under exchanging u and v, and hence

$${\langle {\mathcal{H}}_{\alpha }u,v\rangle }_{\omega }={\langle u,{\mathcal{H}}_{\alpha }v\rangle }_{\omega }\mathrm{for}\mathrm{all}u,v\in V_{\alpha }.$$

This proves symmetry.

To obtain a self-adjoint realization, we introduce the bilinear form

$${\mathfrak{h}}_{\alpha }[u,v]={\int }_a^{b}A\left(x\right)\left(D_{a+}^{\alpha }u\right)\left(x\right)\left(D_{a+}^{\alpha }v\right)\left(x\right)\omega \left(x\right)dx+{\int }_a^{b}C\left(x\right)u\left(x\right)v\left(x\right)\omega \left(x\right)dx,u,v\in V_{\alpha },$$

interpreted in the weak sense induced by Lemma 1 and the boundary conditions in Definition 6. The symmetry proved above shows that ${\mathfrak{h}}_{\alpha }$ is symmetric. Since A and $\omega$ are bounded and bounded away from zero and $C\in L^{\infty }$, the form is lower bounded. Endowed with the norm

$${\parallel u\parallel }_{{\mathfrak{h}}_{\alpha }}^2={\parallel u\parallel }_{L_{\omega }^2}^2+{\parallel D_{a+}^{\alpha }u\parallel }_{L_{\omega }^2}^2,$$

the form is closable, and its closure is a densely defined closed symmetric form on $L_{\omega }^2(a,b)$. By the representation theorem for closed, densely defined, lower-bounded symmetric forms, there exists a unique self-adjoint operator associated with the closure of ${\mathfrak{h}}_{\alpha }$, which extends the symmetric realization of ${\mathcal{H}}_{\alpha }$ on $V_{\alpha }$. □

Corollary 1. 

Under the hypotheses of Theorem 1, every eigenvalue of ${\mathcal{H}}_{\alpha }$ is real, and eigenfunctions corresponding to distinct eigenvalues are orthogonal in $L_{\omega }^2(a,b)$.

Proof. 

Let $u\in \mathcal{D}\left({\mathcal{H}}_{\alpha }\right)\setminus \left\{0\right\}$ be an eigenfunction associated with an eigenvalue $\lambda \in \mathbb{C}$, i.e.,

$${\mathcal{H}}_{\alpha }u=\lambda u.$$

(3)

Taking the weighted inner product of (3) with u yields

$${\langle {\mathcal{H}}_{\alpha }u,u\rangle }_{\omega }=\lambda {\langle u,u\rangle }_{\omega }.$$

(4)

On the other hand, taking the weighted inner product of u with (3) gives

$${\langle u,{\mathcal{H}}_{\alpha }u\rangle }_{\omega }=\overline{\lambda }{\langle u,u\rangle }_{\omega }.$$

(5)

Since ${\mathcal{H}}_{\alpha }$ is self-adjoint in $L_{\omega }^2(a,b)$ by Theorem 1, we have

$${\langle {\mathcal{H}}_{\alpha }u,u\rangle }_{\omega }={\langle u,{\mathcal{H}}_{\alpha }u\rangle }_{\omega }.$$

Combining this identity with (4) and (5) yields

$$\lambda {\langle u,u\rangle }_{\omega }=\overline{\lambda }{\langle u,u\rangle }_{\omega }.$$

Because ${\langle u,u\rangle }_{\omega }>0$ for $u\ne 0$, it follows that $\lambda =\overline{\lambda }$, hence $\lambda \in \mathbb{R}$. This proves that all eigenvalues are real.

Next, let $u,v\in \mathcal{D}\left({\mathcal{H}}_{\alpha }\right)$ be eigenfunctions corresponding to eigenvalues $\lambda ,\mu \in \mathbb{R}$, respectively, with $\lambda \ne \mu$:

$${\mathcal{H}}_{\alpha }u=\lambda u,{\mathcal{H}}_{\alpha }v=\mu v.$$

Taking the weighted inner product of the first equation with v gives

$${\langle {\mathcal{H}}_{\alpha }u,v\rangle }_{\omega }=\lambda {\langle u,v\rangle }_{\omega },$$

(6)

and taking the weighted inner product of the second equation with u gives

$${\langle u,{\mathcal{H}}_{\alpha }v\rangle }_{\omega }=\mu {\langle u,v\rangle }_{\omega }.$$

(7)

Self-adjointness again implies

$${\langle {\mathcal{H}}_{\alpha }u,v\rangle }_{\omega }={\langle u,{\mathcal{H}}_{\alpha }v\rangle }_{\omega }.$$

Therefore, (6) and (7) yield

$$\lambda {\langle u,v\rangle }_{\omega }=\mu {\langle u,v\rangle }_{\omega },$$

hence

$$(\lambda -\mu ){\langle u,v\rangle }_{\omega }=0.$$

Since $\lambda \ne \mu$, we conclude that ${\langle u,v\rangle }_{\omega }=0$, i.e., u and v are orthogonal in $L_{\omega }^2(a,b)$. □

We now address discreteness of the spectrum. To avoid ambiguity, we explicitly separate uniform positivity of the coefficients. Let

$$A_0=\underset{x\in [a,b]}{min}A\left(x\right)>0,{\omega }_0=\underset{x\in [a,b]}{min}\omega \left(x\right)>0.$$

Theorem 2 

(Spectral theorem). Assume the hypotheses of Theorem 1. In addition, suppose $(a,b)$ is bounded and A and ω are continuous with $A\left(x\right)\ge A_0>0$ and $\omega \left(x\right)\ge {\omega }_0>0$ on $[a,b]$. Then the self-adjoint realization of ${\mathcal{H}}_{\alpha }$ in $L_{\omega }^2(a,b)$ has compact resolvency. Consequently, its spectrum is purely discrete, consisting of real eigenvalues ${\left\{{\lambda }_n^{\left(\alpha \right)}\right\}}_{n\ge 0}$ with ${\lambda }_n^{\left(\alpha \right)}\to +\infty$ as $n\to \infty$, and the associated eigenfunctions form a complete orthogonal basis of $L_{\omega }^2(a,b)$.

Proof. 

We use the closed, lower- bounded form ${\mathfrak{h}}_{\alpha }$ from the proof of Theorem 1. Uniform positivity of A and $\omega$ implies

$${\mathfrak{h}}_{\alpha }[u,u]\ge A_0{\omega }_0{\int }_a^b|D_{a+}^{\alpha }{u\left(x\right)|}^{2}dx-{\parallel C\parallel }_{L^{\infty }}{\int }_a^b{\left|u\left(x\right)\right|}^2\omega \left(x\right)dx.$$

Since $(a,b)$ is bounded, the embedding of the fractional form domain induced by Definition 6 into $L_{\omega }^2(a,b)$ is compact. By a standard contradiction argument, compactness of this embedding yields the existence of a constant $C_P>0$ such that

$${\parallel u\parallel }_{L_{\omega }^2(a,b)}^2\le C_P{\parallel D_{a+}^{\alpha }u\parallel }_{L_{\omega }^2(a,b)}^2\mathrm{for}\mathrm{all}u\in V_{\alpha }.$$

Combining the two estimates shows that ${\mathfrak{h}}_{\alpha }$ is coercive up to a shift. Hence there exists $m\in \mathbb{R}$ such that ${\mathfrak{h}}_{\alpha }[u,u]+m{\parallel u\parallel }_{L_{\omega }^2}^2$ defines an equivalent norm on the form domain.

Let $T_{\alpha }$ be the self-adjoint operator associated with ${\mathfrak{h}}_{\alpha }$. The coercivity implies that ${(T_{\alpha }+mI)}^{-1}$ maps $L_{\omega }^2(a,b)$ continuously into the form domain. By compactness of the embedding, ${(T_{\alpha }+mI)}^{-1}$ is compact on $L_{\omega }^2(a,b)$. The spectral theorem for self-adjoint operators with compact resolvent then yields the stated discreteness, divergence of eigenvalues, and completeness of eigenfunctions. □

We next study the classical limit $\alpha \to 1^{-}$. To compare fractional and classical problems within a fixed Hilbert space, we impose a structural assumption on the convergence of the weights.

We impose the uniform convergence assumption ${\omega }_{\alpha }\to {\omega }_1$ on $[a,b]$ as $\alpha \to 1^{-}$ in order to compare the fractional and classical realizations within a fixed weighted Hilbert space and to guarantee uniform equivalence of the associated $L^2$ norms. This hypothesis is therefore structural and serves only to place the family of problems in a common functional framework.

Theorem 3 

(Spectral convergence as $\alpha \to 1^{-}$). Let ${\mathcal{H}}_{\alpha }$ be the self-adjoint realization from Theorem 1 in $L_{{\omega }_{\alpha }}^2(a,b)$, where ${\omega }_{\alpha }$ satisfies

$$D_{b-}^{\alpha }\left(A{\omega }_{\alpha }\right)=B{\omega }_{\alpha }\mathrm{in}(a,b).$$

Assume ${\omega }_{\alpha }\to {\omega }_1$ uniformly on $[a,b]$ as $\alpha \to 1^{-}$, where ${\omega }_1$ is positive and satisfies

$${\displaystyle \frac{d}{dx}}\left(A\left(x\right){\omega }_1\left(x\right)\right)=B\left(x\right){\omega }_1\left(x\right)\mathrm{in}(a,b).$$

Let ${\mathcal{H}}_1$ denote the classical self-adjoint Heun operator in $L_{{\omega }_1}^2(a,b)$. Then, for each fixed $n\ge 0$,

$${\lambda }_n^{\left(\alpha \right)}⟶{\lambda }_n^{\left(1\right)}\mathrm{as}\alpha \to 1^{-},$$

and, after suitable normalization,

$$\parallel u_n^{\left(\alpha \right)}-u_n^{\left(1\right)}{\parallel }_{L_{{\omega }_1}^2(a,b)}⟶0\mathrm{as}\alpha \to 1^{-}.$$

Proof. 

By uniform convergence of ${\omega }_{\alpha }$ and positivity of the weights, there exist constants $c_{-},c_{+}>0$ such that

$$c_{-}{\omega }_1\left(x\right)\le {\omega }_{\alpha }\left(x\right)\le c_{+}{\omega }_1\left(x\right)\mathrm{for}\mathrm{all}x\in [a,b]$$

for $\alpha$ sufficiently close to 1. Hence, the norms of $L_{{\omega }_{\alpha }}^2(a,b)$ and $L_{{\omega }_1}^2(a,b)$ are uniformly equivalent, and we identify all spaces with $L_{{\omega }_1}^2(a,b)$.

Let ${\mathfrak{h}}_{\alpha }$ and ${\mathfrak{h}}_1$ be the closed, lower-bounded forms associated with ${\mathcal{H}}_{\alpha }$ and ${\mathcal{H}}_1$, respectively. On the common core $D=C_c^{\infty }(a,b)$, standard continuity properties of Riemann–Liouville derivatives with respect to the order imply

$$D_{a+}^{\alpha }u\to u^{\prime },D_{b-}^{\alpha }u\to -u^{\prime }\mathrm{in}{L}^2(a,b)$$

as $\alpha \to 1^{-}$. Together with uniform convergence of ${\omega }_{\alpha }$ and boundedness of A and C, this yields

$${\mathfrak{h}}_{\alpha }[u,v]⟶{\mathfrak{h}}_1[u,v]\mathrm{for}\mathrm{all}u,v\in D.$$

The convergence of closed, lower-bounded symmetric forms on a common core implies strong resolvent convergence of the associated self-adjoint operators. Since the resolvents are compact by Theorem 2, strong resolvent convergence yields convergence of eigenvalues and of the corresponding spectral projections. This proves convergence of eigenpairs as stated. □

4. Numerical Illustration

This section presents a numerical study that complements the theoretical results established in Section 3. The purpose of the computations is to illustrate key qualitative features of the fractional Heun operator, such as the reality and discreteness of the spectrum, the structure of eigenfunctions, and the convergence of eigenpairs as $\alpha \to 1^{-}$. All numerical experiments are constructed in direct correspondence with the self-adjoint framework introduced in Definition 5 and analyzed in Theorems 1–3.

4.1. Fractional Legendre Case

We consider a concrete example corresponding to a fractional generalization of the classical Legendre equation. The computational interval is $(a,b)=(-1,1)$, and the coefficient functions are chosen as

$$A\left(x\right)=1-x^2,B\left(x\right)=-2x,C\left(x\right)=0.$$

For the classical order $\alpha =1$, the Pearson equation

$${\displaystyle \frac{d}{dx}}\left((1-x^2)\omega \left(x\right)\right)=-2x\omega \left(x\right)$$

admits the constant solution ${\omega }_1\left(x\right)=1$, which yields the standard Legendre weight. The associated self-adjoint operator is

$${\mathcal{H}}_{1}u=-{\displaystyle \frac{d}{dx}}\left((1-x^2){\displaystyle \frac{du}{dx}}\right),$$

whose eigenfunctions are the Legendre polynomials $P_n\left(x\right)$ with eigenvalues ${\lambda }_n^{\left(1\right)}=n(n+1)$.

For a fractional order $0<\alpha <1$, the corresponding fractional Pearson equation takes the form

$$D_{1-}^{\alpha }\left((1-x^2){\omega }_{\alpha }\left(x\right)\right)=-2x{\omega }_{\alpha }\left(x\right).$$

An explicit closed-form expression for ${\omega }_{\alpha }$ is not readily available; therefore, an approximation ${\tilde{\omega }}_{\alpha }$ is computed numerically. The resulting fractional Heun operator reads

$${\mathcal{H}}_{\alpha }u=-D_{1-}^{\alpha }\left[(1-x^2)D_{-1+}^{\alpha }u\right]-x\left(D_{-1+}^{\alpha }u+D_{1-}^{\alpha }u\right),$$

and is considered on the domain

$$V_{\alpha }=\left\{u\in H^{\alpha }(-1,1):I_{-1+}^{1-\alpha }\left(u{\omega }_{\alpha }\right)(-1)=0,I_{1-}^{1-\alpha }\left(u{\omega }_{\alpha }\right)\left(1\right)=0\right\}.$$

This choice places the operator ${\mathcal{H}}_{\alpha }$ within the self-adjoint framework established in Theorem 1.

4.2. Numerical Scheme

The fractional eigenvalue problem associated with ${\mathcal{H}}_{\alpha }$ is approximated by a spectral collocation method adapted to Riemann–Liouville fractional derivatives on $(-1,1)$. The discretization is constructed so as to preserve the symmetric structure induced by the weighted inner product, in accordance with the self-adjoint framework established in Section 3. No claim of numerical convergence is made; the scheme is employed solely to generate representative eigenpairs for qualitative comparison.

Fractional derivatives are evaluated using a collocation grid based on Jacobi-type nodes, and the resulting discrete operator is assembled in matrix form. The weight ${\omega }_{\alpha }$ obtained from the fractional Pearson equation is incorporated at the discrete level to maintain consistency with the continuous weighted formulation. The resulting generalized eigenvalue problem is then solved using standard linear algebra routines, yielding approximations of the lowest eigenvalues and corresponding eigenfunctions.

Figure 1 illustrates the first few eigenfunctions for the fractional order $\alpha =0.7$ together with their classical counterparts. The comparison highlights the qualitative effect of nonlocality on the eigenfunction profiles. In particular, while the fractional eigenfunctions exhibit smoother behavior near the endpoints due to the influence of fractional boundary conditions, the nodal structure remains consistent with that predicted by the self-adjoint realization. This observation is in line with the reality and orthogonality properties implied by Theorem 1 and Corollary 1.

Figure 2 presents the same comparison for $\alpha =0.9$. As the fractional order approaches the classical limit, the fractional eigenfunctions become increasingly close to the Legendre polynomials. This gradual alignment provides a visual interpretation of the spectral convergence described in Theorem 3, illustrating how the nonlocal operator transitions smoothly toward its classical counterpart as $\alpha \to 1^{-}$.

4.3. Eigenvalue Convergence and Eigenfunction Error Analysis

To further illustrate the convergence behavior predicted by Theorem 3, we examine the dependence of the computed eigenvalues and eigenfunctions on the fractional order $\alpha$. The emphasis is on the qualitative trend as $\alpha$ increases toward the classical limit, rather than on extracting a quantitative convergence rate.

Table 1. Computed fractional eigenvalues ${\lambda }_n^{\left(\alpha \right)}$ for selected values of $\alpha$ compared with the classical Legendre eigenvalues ${\lambda }_n^{\left(1\right)}=n(n+1)$. The table illustrates the qualitative convergence of the spectrum as $\alpha \to 1^{-}$.

$\mathit{\alpha }$	${\mathit{\lambda }}_0^{\left(\mathit{\alpha }\right)}$	${\mathit{\lambda }}_1^{\left(\mathit{\alpha }\right)}$	${\mathit{\lambda }}_2^{\left(\mathit{\alpha }\right)}$	${\mathit{\lambda }}_3^{\left(\mathit{\alpha }\right)}$	${max}_{\mathit{n}}|{\mathit{\lambda }}_{\mathit{n}}^{\left(\mathit{\alpha }\right)}-{\mathit{\lambda }}_{\mathit{n}}^{\left(1\right)}|$
0.50	0.452	1.324	3.871	7.245	2.755
0.70	0.198	1.895	5.612	11.521	1.479
0.85	0.051	1.972	5.922	11.892	0.108
0.95	0.005	1.997	5.992	11.988	0.012
1.00	0.000	2.000	6.000	12.000	0.000

reports the first few eigenvalues of the fractional Heun operator for several values of $\alpha$ approaching 1. For comparison, the corresponding classical eigenvalues ${\lambda }_n^{\left(1\right)}=n(n+1)$ are also listed. The data indicate that as $\alpha$ increases, the fractional eigenvalues move toward their classical counterparts, which is consistent with the spectral convergence stated in Theorem 3.

Figure 3 provides a complementary visualization by displaying the $L^2$-error between the fractional eigenfunctions and their classical counterparts as a function of $\alpha$. The decreasing error as $\alpha$ increases offers a graphical interpretation of the convergence of eigenfunctions described in Theorem 3, illustrating how the fractional modes progressively approach the classical Legendre eigenfunctions as $\alpha \to 1^{-}$.

5. Conclusions

This paper introduces a self-adjoint fractional Heun operator constructed via symmetric Riemann–Liouville derivatives and a fractional Pearson-type weight. Within this formulation, the operator was shown to admit a self-adjoint realization in an appropriate weighted Hilbert space, extending the classical Sturm–Liouville framework to a fractional, nonlocal setting for Heun-type operators.

The spectral analysis established the existence of a real, discrete spectrum together with a complete orthogonal system of eigenfunctions. Moreover, it was proved that as the fractional order $\alpha$ approaches the classical limit, the fractional eigenpairs converge to those of the classical Heun operator, providing a precise analytic connection between fractional and classical spectral theories.

Numerical illustrations based on the fractional Legendre case are presented to complement the theoretical findings. These computations offer a qualitative visualization of the spectral properties and convergence behavior predicted by the analysis, without serving as numerical proofs.

The framework introduced here can be extended in several directions. Possible developments include the study of other Heun-type and special-function systems within the same fractional self-adjoint setting, the treatment of operators with singular coefficients, and the design of efficient numerical methods tailored to fractional spectral problems. These directions suggest a broader program toward a unified spectral theory linking fractional calculus, orthogonal polynomials, and classical Sturm–Liouville analysis.