Fractional Spectral Degeneracy Operators: Non-Local Phenomena and Anomalous Diffusion

1. Introduction

1.1. Fractional Degeneracy and Non-Local Phenomena

Classical degenerate elliptic operators whose coefficients vanish or blow up along lower-dimensional sets play a central role in the analysis of boundary layers, phase transitions, and PDEs on singular geometries. The development of Spectral Degeneracy Operators (SDOs) [1] has shown that anisotropic degeneracy can be encoded spectrally through prescribed exponents ${\theta }_i\in [1,2)$, but this restriction excludes a broad class of phenomena in which diffusion exhibits fractional or multi-scale vanishing rates. Examples include anomalous diffusion in porous or fractal media, intermittency in plasma turbulence, and geophysical transport in strongly heterogeneous domains.

Fractional calculus provides a mathematically natural extension of scale-dependent diffusion, incorporating non-local interactions and memory effects through fractional derivatives. Allowing ${\theta }_i\in (0,2)$ produces the family of Fractional Spectral Degeneracy Operators (FSDOs), which interpolate continuously between weak and strong degeneracy while retaining intrinsic anisotropy. This generalization captures situations where diffusivity obeys fractional power laws and where the effective geometry near the degeneracy point is governed by a weighted non-Euclidean metric. These features are indispensable in modeling multiscale systems such as turbulent plasma sheets, anomalous transport near singular interfaces, and heterogeneous geological formations.

1.2. Mathematical Challenges and Main Contributions

The analysis of FSDOs introduces difficulties not present in either classical degenerate elliptic theory or standard fractional diffusion. The operator ${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }$ couples two sources of singularity: a degeneracy encoded by the weight

$$W\left(\mathbf{x}\right)=\prod _{i=1}^d{|x_i-a_i|}^{{\theta }_i/2},$$

and the genuinely non-local structure of the fractional operator. This interaction requires a functional setting where anisotropy, fractional regularity, and spatially dependent weights are treated in a unified manner. Weighted fractional Sobolev spaces provide the correct framework and allow sharp control of both the $L^2$-mass and the fractional seminorm near the degeneracy point.

Building on this structure, the main contributions of the present work are:

• We introduce a rigorous definition of FSDOs on bounded domains $\Omega \subset {\mathbb{R}}^d$, allowing for fractional degeneracy exponents ${\theta }_i\in (0,2)$. This construction leads naturally to a corresponding family of fractional weighted Sobolev spaces ${\mathcal{H}}_{\theta }^s(\Omega )$, which encode both non-local effects and anisotropic degeneracy.
• We develop a complete spectral theory for FSDOs, including essential self-adjointness, compactness of the resolvent, and a tensor-product eigenstructure arising from fractional Sturm–Liouville problems. We derive Weyl-type asymptotics with scaling exponent $d/(2-{\theta }_{max})$ and obtain fractional Bessel-type asymptotics for the eigenfunctions near the degeneracy set.
• We establish a family of fractional Landau inequalities that quantify an uncertainty principle between spatial localization near the degeneracy manifold and spectral concentration. The optimal constants are expressed through a fractional Rayleigh quotient and are governed by the ground state of a fractional harmonic oscillator.
• We propose Fractional SDO-Nets (FSDO-Nets), a class of neural architectures incorporating non-locality and adaptive degeneracy at the operator level. Using the fractional spectral gap, we prove well-posedness, stability, and generalization guarantees.
• We extend the theory to Riemannian and Lorentzian manifolds, showing that fractional degeneracy interacts nontrivially with curvature. This yields geometric correction factors in both the Weyl law and the Landau inequalities.

1.3. Relation to Existing Literature

Our work connects to several established research directions. It builds upon the foundational theory of degenerate elliptic equations developed by DiBenedetto [3], the study of inverse problems for degenerate PDEs by Cannarsa et al. [1], and the harmonic-analytic framework for weighted Sobolev spaces introduced by Davies [5]. While related to the extensive literature on fractional Laplacians and non-local operators, the present setting differs fundamentally: here, non-locality emerges from the degeneracy of the diffusion tensor itself, rather than from a convolution-type kernel.

Finally, the neural network aspects extend Physics-Informed Neural Networks (PINNs) [2] and neural operators [4] by embedding fractional degeneracy directly into the architecture. This ensures physical consistency, adaptive resolution, and improved generalization in problems governed by non-local or singular diffusion.

2. Fractional Spectral Degeneracy Operators: Mathematical Foundations

2.1. Definition and Functional Framework

Let $\Omega \subset {\mathbb{R}}^d$ be a bounded Lipschitz domain, and let $\mathbf{a}=(a_1,\dots ,a_d)\in \Omega$ denote the degeneracy center. For a vector of fractional degeneracy exponents

$$\theta =({\theta }_1,\dots ,{\theta }_d)\in {(0,2)}^d,$$

and a fractional order $\alpha \in (0,1]$, we introduce the fractional degeneracy tensor

$${\mathbb{D}}_{\mathbf{a},\theta }^{\alpha }\left(\mathbf{x}\right):=diag\left(|x_1-a_1{|}^{{\theta }_1}{\mathcal{I}}_{{\theta }_1}^{1-\alpha },\dots ,{|x_d-a_d|}^{{\theta }_d}{\mathcal{I}}_{{\theta }_d}^{1-\alpha }\right),$$

(1)

where each ${\mathcal{I}}_{{\theta }_i}^{1-\alpha }$ denotes a one–dimensional weighted fractional integral of order $1-\alpha$, chosen so as to preserve the intrinsic scaling associated with the degeneracy exponent ${\theta }_i$. When $\alpha =1$, the operator ${\mathcal{I}}_{{\theta }_i}^{1-\alpha }$ reduces to the identity, and thus (1) collapses to the classical (local) spectral degeneracy tensor.

Definition 2.1 

(Fractional Spectral Degeneracy Operator). Let $u:\Omega \to \mathbb{R}$ be sufficiently smooth. TheFractional Spectral Degeneracy Operator(FSDO) associated with the parameters $(\mathbf{a},\theta ,\alpha )$ is defined by

$${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }u\left(\mathbf{x}\right):={\nabla }_{\mathbf{x}}·\left({\mathbb{D}}_{\mathbf{a},\theta }^{\alpha }\left(\mathbf{x}\right){\nabla }_{\mathbf{x}}u\left(\mathbf{x}\right)\right),$$

(2)

i.e., ${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }$ is a divergence–form operator with anisotropic, coordinate–wise fractional degeneracy weights.

The operator (2) couples two sources of singular behavior: (i) spectral degeneracy, encoded by the vanishing weights $|x_i-a_i{|}^{{\theta }_i}$, and (ii) nonlocality, encoded through the fractional integral ${\mathcal{I}}_{{\theta }_i}^{1-\alpha }$. To capture this joint structure, we introduce the weighted fractional Sobolev space below.

2.2. Weighted Fractional Sobolev Space

For a fractional order $s\in (0,1]$, define

$${\mathcal{H}}_{\theta }^s(\Omega ):=\left\{u\in L^2(\Omega ):\underset{\Omega \times \Omega }{\int \int }{\displaystyle \frac{{|u\left(\mathbf{x}\right)-u\left(\mathbf{y}\right)|}^2}{\left({\prod }_{i=1}^d{|x_i-a_i|}^{{\theta }_i/2}\right)\left({\prod }_{i=1}^d{|y_i-a_i|}^{{\theta }_i/2}\right){|\mathbf{x}-\mathbf{y}|}^{d+2s}}}d\mathbf{x}d\mathbf{y}<\infty \right\},$$

(3)

and equip the space with the natural Hilbert norm

$${\parallel u\parallel }_{{\mathcal{H}}_{\theta }^s(\Omega )}^2:={\parallel u\parallel }_{L^2(\Omega )}^2+\underset{\Omega \times \Omega }{\int \int }{\displaystyle \frac{{|u\left(\mathbf{x}\right)-u\left(\mathbf{y}\right)|}^2}{\left({\prod }_{i=1}^d{|x_i-a_i|}^{{\theta }_i/2}\right)\left({\prod }_{i=1}^d{|y_i-a_i|}^{{\theta }_i/2}\right){|\mathbf{x}-\mathbf{y}|}^{d+2s}}}d\mathbf{x}d\mathbf{y}.$$

The weights in (3) reflect precisely the degeneracy pattern of the tensor ${\mathbb{D}}_{\mathbf{a},\theta }^{\alpha }\left(\mathbf{x}\right)$: the integrability near the degeneracy point $\mathbf{a}$ depends directly on the exponents ${\theta }_i$. When $s=1$ and ${\theta }_i\in [1,2)$, the seminorm in (3) recovers, up to equivalence, the standard weighted Sobolev seminorm associated with classical (local) spectral degeneracy operators.

2.3. Basic Properties and Coercivity

A central difficulty in the analysis of the operator ${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }$ is the presence of anisotropic fractional degeneracy near the point $\mathbf{a}$. The following result provides the fundamental coercivity estimate ensuring that, despite this degeneracy, the weighted fractional seminorm continues to dominate the $L^2$-norm of functions vanishing on the boundary.

Theorem 2.2 

(Fractional Poincaré Inequality with Spectral Degeneracy). Let $\theta \in {(0,2)}^d$ and $s\in (0,1]$. There exists a constant $C=C(\Omega ,\theta ,s)>0$ such that

$${\parallel u\parallel }_{L^2(\Omega )}\le C{\left(\underset{\Omega \times \Omega }{\int \int }{\displaystyle \frac{{|u\left(\mathbf{x}\right)-u\left(\mathbf{y}\right)|}^2}{\left({\prod }_{i=1}^d{|x_i-a_i|}^{{\theta }_i/2}\right)\left({\prod }_{i=1}^d{|y_i-a_i|}^{{\theta }_i/2}\right){|\mathbf{x}-\mathbf{y}|}^{d+2s}}}d\mathbf{x}d\mathbf{y}\right)}^{1/2},$$

(4)

for all $u\in {\mathcal{H}}_{\theta }^s(\Omega )$ satisfying ${u|}_{\partial \Omega }=0$.

Proof. 

The argument proceeds by combining weighted fractional Hardy inequalities with the classical nondegenerate Poincaré inequality, using a localization strategy adapted to the degeneracy profile.

1. Decomposition of the domain. For $\epsilon >0$ small, set

$${\Omega }_{\epsilon }^{-}:=B_{\epsilon }\left(\mathbf{a}\right)\cap \Omega ,{\Omega }_{\epsilon }^{+}:=\Omega \setminus B_{\epsilon }\left(\mathbf{a}\right).$$

We estimate the $L^2$-norm of u separately on these two regions.

2. Nondegenerate region ${\Omega }_{\epsilon }^{+}$. On ${\Omega }_{\epsilon }^{+}$ the weights

$$|x_i-a_i{|}^{{\theta }_i/2}\ge {\epsilon }^{{\theta }_i/2}$$

are uniformly bounded away from zero. Thus the integrand in (25) dominates the standard fractional Gagliardo seminorm. Since ${u|}_{\partial \Omega }=0$, the classical fractional Poincaré inequality yields

$${\parallel u\parallel }_{L^2\left({\Omega }_{\epsilon }^{+}\right)}^2\le C\underset{\Omega \times \Omega }{\int \int }{\displaystyle \frac{{|u\left(\mathbf{x}\right)-u\left(\mathbf{y}\right)|}^2}{{|\mathbf{x}-\mathbf{y}|}^{d+2s}}}d\mathbf{x}d\mathbf{y}\le C\underset{\Omega \times \Omega }{\int \int }{\displaystyle \frac{{|u\left(\mathbf{x}\right)-u\left(\mathbf{y}\right)|}^2}{{W\left(\mathbf{x}\right)W\left(\mathbf{y}\right)|\mathbf{x}-\mathbf{y}|}^{d+2s}}}d\mathbf{x}d\mathbf{y},$$

(5)

where

$$W\left(\mathbf{x}\right):=\prod _{i=1}^d{|x_i-a_i|}^{{\theta }_i/2}.$$

3. Degenerate region ${\Omega }_{\epsilon }^{-}$. Since $\mathbf{a}$ is the only point where $W\left(\mathbf{x}\right)$ vanishes, we apply a weighted fractional Hardy-type inequality (valid for exponents ${\theta }_i\in (0,2)$):

$${\int }_{{\Omega }_{\epsilon }^{-}}{\displaystyle \frac{{\left|u\left(\mathbf{x}\right)\right|}^2}{W{\left(\mathbf{x}\right)}^2}}d\mathbf{x}\le C\underset{\Omega \times \Omega }{\int \int }{\displaystyle \frac{{|u\left(\mathbf{x}\right)-u\left(\mathbf{y}\right)|}^2}{{W\left(\mathbf{x}\right)W\left(\mathbf{y}\right)|\mathbf{x}-\mathbf{y}|}^{d+2s}}}d\mathbf{x}d\mathbf{y}.$$

(6)

Multiplying both sides by ${\epsilon }^{{\theta }_{max}}$ (where ${\theta }_{max}:={max}_i{\theta }_i$) and using that

$$W\left(\mathbf{x}\right)\asymp \prod _{i=1}^d{|x_i-a_i|}^{{\theta }_i/2}\mathrm{in}{\Omega }_{\epsilon }^{-},$$

we obtain a uniform bound

$${\parallel u\parallel }_{L^2\left({\Omega }_{\epsilon }^{-}\right)}^2\le C\underset{\Omega \times \Omega }{\int \int }{\displaystyle \frac{{|u\left(\mathbf{x}\right)-u\left(\mathbf{y}\right)|}^2}{{W\left(\mathbf{x}\right)W\left(\mathbf{y}\right)|\mathbf{x}-\mathbf{y}|}^{d+2s}}}d\mathbf{x}d\mathbf{y}.$$

(7)

4. Limit as $\epsilon \to 0$. The point $\left\{\mathbf{a}\right\}$ has zero fractional capacity relative to the weight W, so ${lim}_{\epsilon \to 0}{\parallel ·\parallel }_{L^2\left({\Omega }_{\epsilon }^{-}\right)}$ contributes no singular term. Combining (5) and (7) and sending $\epsilon \to 0$ yields

$${\parallel u\parallel }_{L^2(\Omega )}\le C{\left(\underset{\Omega \times \Omega }{\int \int }{\displaystyle \frac{{|u\left(\mathbf{x}\right)-u\left(\mathbf{y}\right)|}^2}{{W\left(\mathbf{x}\right)W\left(\mathbf{y}\right)|\mathbf{x}-\mathbf{y}|}^{d+2s}}}d\mathbf{x}d\mathbf{y}\right)}^{1/2},$$

which is precisely (25). □

3. Spectral Theory of Fractional SDOs

3.1. Self-Adjointness and Compact Resolvent

We now establish the spectral structure of the Fractional Spectral Degeneracy Operator

$${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }u:=\nabla ·\left({\mathbb{D}}_{\mathbf{a},\theta }^{\alpha }\left(\mathbf{x}\right)\nabla u\right),$$

acting on the weighted fractional Sobolev space ${\mathcal{H}}_{\theta }^s(\Omega )$. The degeneracy at $\mathbf{a}$ manifests only in the coefficients of the bilinear form, and—remarkably—does not destroy the spectral discreteness of the operator.

To express the result cleanly, let

$$\mathfrak{a}[u,v]:=\underset{\Omega \times \Omega }{\int \int }{\displaystyle \frac{(u\left(\mathbf{x}\right)-u\left(\mathbf{y}\right))\left(\overline{v\left(\mathbf{x}\right)-v\left(\mathbf{y}\right)}\right)}{{W\left(\mathbf{x}\right)W\left(\mathbf{y}\right)|\mathbf{x}-\mathbf{y}|}^{d+2s}}}d\mathbf{x}d\mathbf{y},W\left(\mathbf{x}\right):=\prod _{i=1}^d{|x_i-a_i|}^{{\theta }_i/2},$$

(8)

denote the closed, symmetric, and positive bilinear form associated to ${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }$ on the domain

$$\mathcal{D}\left(\mathfrak{a}\right)={\mathcal{H}}_{\theta }^s(\Omega ).$$

Theorem 3.1 

(Spectral Decomposition). Let $\mathbf{a}\in int(\Omega )$ and $\theta \in {(0,2)}^d$. Assume $s\in (0,1]$ and $\alpha \in (0,1]$. The operator ${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }$ defined via the form (8) is essentially self-adjoint, strictly positive, and possesses a compact resolvent. As a consequence:

• The spectrum consists solely of a countable set of eigenvalues

  $$0<{\lambda }_1\le {\lambda }_2\le \dots \to \infty .$$
• There exists an orthonormal basis of eigenfunctions

  $${\left\{{\varphi }_k\right\}}_{k=1}^{\infty }\subset {\mathcal{H}}_{\theta }^s(\Omega ),{\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }{\varphi }_k={\lambda }_k{\varphi }_k,$$

  which is complete in $L^2(\Omega )$.
• The operator admits the spectral expansion

  $$u\left(\mathbf{x}\right)=\sum _{k=1}^{\infty }{(u,{\varphi }_k)}_{L^2(\Omega )}{\varphi }_k\left(\mathbf{x}\right),u\in L^2(\Omega ),$$

  with convergence in $L^2(\Omega )$.

Proof. 

The proof proceeds through several steps.

1. Closedness, coercivity, and symmetry of the form. The bilinear form $\mathfrak{a}$ in (8) is symmetric and positive. The weighted fractional Poincaré inequality (Theorem 2.2) ensures the coercivity estimate

$${\parallel u\parallel }_{L^2(\Omega )}^2\le C\mathfrak{a}[u,u],u\in {\mathcal{H}}_{\theta }^s(\Omega ),$$

establishing that $\mathfrak{a}$ defines a closed, densely defined form on $L^2(\Omega )$.

2. Self-adjointness via the KLMN theorem. Since the coefficients of the diffusion tensor ${\mathbb{D}}_{\mathbf{a},\theta }^{\alpha }$ are real and the form is closed and symmetric, the KLMN (Kato–Lions–Milgram–Nelson) theorem implies the existence of a unique self-adjoint operator A such that

$$\mathfrak{a}[u,v]={(Au,v)}_{L^2(\Omega )},u\in \mathcal{D}\left(A\right),v\in \mathcal{D}\left(\mathfrak{a}\right).$$

The operator A coincides with ${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }$ on smooth functions supported away from $\mathbf{a}$. Essential self-adjointness follows because the degeneracy is mild enough (the weights belong to the Muckenhoupt class $A_2$), and the energy form uniquely determines the operator closure.

3. Compactness of the weighted fractional embedding. To show that ${(A+\lambda I)}^{-1}$ is compact, it suffices to prove that the embedding

$${\mathcal{H}}_{\theta }^s(\Omega )\hookrightarrow L^2(\Omega )$$

is compact. This follows from a weighted variant of the fractional Rellich–Kondrachov theorem:

• Away from the point $\mathbf{a}$, the embedding is compact by classical fractional Sobolev theory.
• Near $\mathbf{a}$, the degeneracy enters only through the vanishing weight $W\left(\mathbf{x}\right)$, but the weighted Hardy inequality

  $${\int }_{\Omega }{\displaystyle \frac{{\left|u\left(\mathbf{x}\right)\right|}^2}{W{\left(\mathbf{x}\right)}^2}}\le C\mathfrak{a}[u,u]$$

  prevents concentration of mass at the degeneracy point. Since $\left\{\mathbf{a}\right\}$ has zero fractional capacity with respect to the weight structure, no loss of compactness occurs.

Thus the embedding is compact, implying that $A^{-1}$ is a compact operator on $L^2(\Omega )$.

Step 4: Spectral theorem for compact self-adjoint operators. Since A is positive, self-adjoint, and has compact resolvent, the Hilbert–Schmidt spectral theorem applies. We obtain:

$$\sigma \left(A\right)={\left\{{\lambda }_k\right\}}_{k=1}^{\infty },0<{\lambda }_1\le {\lambda }_2\le \dots \to \infty ,$$

with an orthonormal basis $\left\{{\varphi }_k\right\}$ of $L^2(\Omega )$ satisfying

$$A{\varphi }_k={\lambda }_k{\varphi }_k.$$

This yields the complete spectral decomposition stated in the theorem. □

3.2. Weyl Asymptotics and Fractional Scaling

We analyze the eigenvalue counting function

$$N(\Lambda )=\#\{k:{\lambda }_k\le \Lambda \},$$

where ${\lambda }_k$ are the eigenvalues of ${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }$. Because the degeneracy is anisotropic, the effective geometry is governed by the weighted metric

$$g_{ij}\left(\mathbf{x}\right)={|x_i-a_i|}^{-{\theta }_i}{\delta }_{ij},$$

whose determinant encodes the local spectral density. This metric arises naturally from the principal symbol of the operator and provides a coherent geometric interpretation of the fractional anisotropy. In particular, the largest degeneracy exponent ${\theta }_{max}$ determines the dominant scaling near $\mathbf{a}$, and thus fully controls the Weyl exponent.

Theorem 3.2 

(Fractional Weyl Law). Let ${\left\{{\lambda }_k\right\}}_{k=1}^{\infty }$ denote the eigenvalues of ${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }$, arranged in nondecreasing order and repeated with multiplicity. Then the counting function satisfies the high-energy asymptotic

$$N(\Lambda )\sim C_{\theta ,\alpha }{\Lambda }^{{\displaystyle \frac{d}{2-{\theta }_{max}}}},\Lambda \to \infty ,$$

(9)

where ${\theta }_{max}={max}_{1\le i\le d}{\theta }_i$ and the constant $C_{\theta ,\alpha }$ is given by

$$C_{\theta ,\alpha }={\displaystyle \frac{1}{{\left(2\pi \right)}^d}}{\int }_{\Omega }{\left[det{\left({\mathbb{D}}_{\mathbf{a},\theta }^{\alpha }\left(\mathbf{x}\right)\right)}^{-1}\right]}^{1/2}d\mathbf{x},$$

(10)

with ${\mathbb{D}}_{\mathbf{a},\theta }^{\alpha }\left(\mathbf{x}\right)$ denoting the principal symbol matrix of ${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }$.

Proof. 

The proof relies on a refined heat-kernel asymptotic analysis adapted to the anisotropic degeneracy near $\mathbf{a}$. Let

$$\Theta \left(t\right)=\sum _{k=1}^{\infty }{e}^{-t{\lambda }_k}$$

(11)

denote the heat trace associated with the semigroup $e^{-t{\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }}$. Classical Tauberian theorems imply that (9) follows once the small-time asymptotic behavior of $\Theta \left(t\right)$ is established.

1. Construction of a fractional parametrix. The fractional heat equation

$${\partial }_{t}u+{\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }u=0$$

is governed, at leading order, by the principal symbol

$${\sigma }_{\mathcal{L}}(\mathbf{x},\xi )={\left(\sum _{i=1}^d|x_i-a_i{|}^{{\theta }_i}{\left|{\xi }_i\right|}^2\right)}^{\alpha /2},$$

(12)

whose degeneracy encodes the geometry induced by the weighted metric (). A parametrix for the heat kernel $K(t,\mathbf{x},\mathbf{y})$ is obtained by freezing coefficients at scale $t^{1/(2-{\theta }_{max})}$ and applying a localized pseudo-differential expansion respecting the anisotropic scaling.

2. Small-time asymptotics of the heat trace. Integrating the parametrix diagonal yields the leading-order expansion

$$\Theta \left(t\right)\sim {\left(4\pi t\right)}^{-d/(2-{\theta }_{max})}{\int }_{\Omega }{\left[det{\left({\mathbb{D}}_{\mathbf{a},\theta }^{\alpha }\left(\mathbf{x}\right)\right)}^{-1}\right]}^{1/2}d\mathbf{x},t\to 0^{+},$$

(13)

where the determinant reflects the anisotropic scaling density of the operator.

3. Application of the Tauberian theorem. Karamata’s Tauberian theorem applied to (11) and (13) yields the Weyl asymptotic formula (9) with constant (10). This completes the proof. □

3.3. Fractional Bessel Asymptotics for Eigenfunctions

Near the degeneracy point $\mathbf{a}$, the operator decouples at leading order into coordinate-wise fractional Sturm–Liouville problems. The corresponding change of variables

$${\rho }_i={|x_i-a_i|}^{1-{\theta }_i/2},$$

combined with a Liouville transform, normalizes the principal part of the operator and reveals fractional Bessel equations whose regular solutions are precisely the functions $J_{{\nu }_i}^{\left(\alpha \right)}$. This yields a coherent and coordinate-wise description of boundary-layer behavior of eigenfunctions and explains the scaling structure observed in simulations and applications.

Theorem 3.3 

(Fractional Bessel-Type Asymptotics). Let ${\left\{{\varphi }_k\right\}}_{k=1}^{\infty }$ denote the normalized eigenfunctions of the operator ${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }$. Then, in a sufficiently small neighborhood of the degeneracy point $\mathbf{a}$, the eigenfunctions admit the asymptotic representation

$${\varphi }_k\left(\mathbf{x}\right)\sim \prod _{i=1}^d{|x_i-a_i|}^{{\displaystyle \frac{1-{\theta }_i}{2}}}{J}_{{\nu }_i}^{\left(\alpha \right)}\left(\sqrt{{\lambda }_k}{|x_i-a_i|}^{1-{\theta }_i/2}\right),{\nu }_i={\displaystyle \frac{{\theta }_i-1}{2-{\theta }_i}},$$

(14)

where $J_{{\nu }_i}^{\left(\alpha \right)}$ denotes the fractional Bessel function of order ${\nu }_i$, given as the unique solution (regular at the origin) to the singular fractional Sturm–Liouville equation on a representative one-dimensional interval.

Proof. 1. Localization near the degeneracy point. Near $\mathbf{a}$ the leading-order behaviour of the operator is captured by the coordinate-wise principal part. For $\mathbf{x}$ sufficiently close to $\mathbf{a}$ one has, at leading asymptotic order,

$${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }u\left(\mathbf{x}\right)\approx \sum _{i=1}^d{\mathcal{L}}_i^{\left(\alpha \right)}u\left(\mathbf{x}\right),$$

(15)

where each one-dimensional contribution is the anisotropic fractional radial operator

$${\mathcal{L}}_i^{\left(\alpha \right)}w\left(x_i\right):=-{\displaystyle \frac{d^{\alpha }}{d{x}_i^{\alpha }}}\left(|x_i-a_i{|}^{{\theta }_i}{\displaystyle \frac{d^{\alpha }w}{d{x}_i^{\alpha }}}\right).$$

The approximation (15) neglects lower-order mixed terms which are of smaller order under the anisotropic scaling considered here.

2. Liouville-type change of variables. For each coordinate i introduce the stretched radial variable

$${\rho }_i:={|x_i-a_i|}^{1-{\theta }_i/2}.$$

(16)

Define a Liouville-type transform by

$$u\left(\mathbf{x}\right)=\prod _{i=1}^d{|x_i-a_i|}^{{\displaystyle \frac{1-{\theta }_i}{2}}}v\left(\rho \right),$$

(17)

where $\rho =({\rho }_1,\dots ,{\rho }_d)$. Substituting (17) into the principal part yields, for each coordinate, a one-dimensional fractional radial equation for the profile v with spectral parameter $\lambda$.

3. Identification of the fractional Bessel equation. After the transform (17) and isolating the leading-order terms, the one-dimensional reduced equation may be written in the canonical form (up to lower-order error terms)

$$-{\mathcal{B}}_{{\nu }_i}^{\left(\alpha \right)}\psi \left({\rho }_i\right)=\lambda \psi \left({\rho }_i\right),{\rho }_i\in (0,{\rho }_0),$$

(18)

where ${\mathcal{B}}_{{\nu }_i}^{\left(\alpha \right)}$ is the fractional Bessel operator of order ${\nu }_i$ and $\psi$ denotes the one-dimensional radial part of v. The order ${\nu }_i$ is given by

$${\nu }_i={\displaystyle \frac{{\theta }_i-1}{2-{\theta }_i}},$$

as in (14). The regular solution of (18) at the origin is the fractional Bessel function $J_{{\nu }_i}^{\left(\alpha \right)}$.

4. Reconstructing the multi-dimensional profile. Restoring the Liouville weight yields the local asymptotic representation

$${\varphi }_k\left(\mathbf{x}\right)\approx \prod _{i=1}^d{|x_i-a_i|}^{{\displaystyle \frac{1-{\theta }_i}{2}}}·J_{{\nu }_i}^{\left(\alpha \right)}\left(\sqrt{{\lambda }_k}{\rho }_i\right),$$

which is precisely (14) when ${\rho }_i$ is replaced by the definition (16). The boundary conditions (interpreted in the appropriate fractional trace sense) quantize ${\lambda }_k$ but do not alter the local profile near $\mathbf{a}$.

5. Error control. The lower-order mixed and non-principal terms omitted above may be estimated by standard perturbative arguments. Concretely, for $\mathbf{x}$ in a sufficiently small neighborhood of $\mathbf{a}$ the residual term $R_k\left(\mathbf{x}\right)$, defined by

$${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }{\varphi }_k\left(\mathbf{x}\right)+{\lambda }_k{\varphi }_k\left(\mathbf{x}\right)=R_k\left(\mathbf{x}\right),$$

satisfies $\parallel R_k{\parallel }_{L^2\left(B_r\left(\mathbf{a}\right)\right)}=o\left(1\right)$ as $k\to \infty$ uniformly for small fixed $r>0$. This justifies the asymptotic expansion in (14). □

4. Fractional Landau Inequalities

4.1. Uncertainty Principle for FSDOs

For a function $u\in L^2(\Omega )$ we define two complementary measures of concentration. The spatial spread relative to the degeneracy point $\mathbf{a}$ is

$${\Delta }_x{\left(u\right)}^2:={\int }_{\Omega }{\parallel \mathbf{x}-\mathbf{a}\parallel }^2{\left|u\left(\mathbf{x}\right)\right|}^{2}d\mathbf{x},$$

(19)

and the fractional spectral spread with respect to the fractional singular differential operator (FSDO) ${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }$ is given by

$${\Delta }_{\lambda }^{\alpha }{\left(u\right)}^2:=\sum _{k=1}^{\infty }{\lambda }_k^{2\alpha }{\left|\langle u,{\varphi }_k\rangle \right|}^2=\parallel {\left({\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }\right)}^{\alpha }u{\parallel }_{L^2(\Omega )}^2,$$

(20)

where $\{{\lambda }_k,{\varphi }_k\}$ are the eigenpairs of the FSDO (ordered increasingly and repeated by multiplicity). The quantity ${\Delta }_{\lambda }^{\alpha }\left(u\right)$ measures how much u occupies the high-frequency eigenmodes of the fractional operator and can be interpreted as a generalized momentum spread.

Theorem 4.1 

(Fractional Landau Inequality). Let $u\in {\mathcal{H}}_{\theta }^s(\Omega )$ with $s\ge \alpha$. Then the uncertainty product associated with the spatial concentration around $\mathbf{a}$ and the spectral concentration with respect to the FSDO satisfies

$${\Delta }_x\left(u\right){\Delta }_{\lambda }^{\alpha }\left(u\right)\ge C_{\alpha }(\Omega ,\theta ){\parallel u\parallel }_{L^2(\Omega )}^2,$$

(21)

where the optimal constant admits the variational characterization

$$C_{\alpha }(\Omega ,\theta )={\displaystyle \frac{1}{2}}\underset{k\ge 1}{inf}\left({\displaystyle \frac{{\parallel \parallel \mathbf{x}-\mathbf{a}\parallel }_{\theta }{\varphi }_k{\parallel }_{L^2}}{{\lambda }_k^{\alpha /2}}}\right)\prod _{i=1}^d{\displaystyle \frac{\Gamma (1+\alpha )}{\Gamma (1+{\theta }_i/2)}}.$$

(22)

Proof. 

The proof extends the classical Landau and Heisenberg uncertainty principles to the fractional and anisotropic setting induced by the degeneracies of ${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }$.

1. Fractional commutator framework. Let

$$Xu:=\parallel \mathbf{x}-\mathbf{a}\parallel u,P_{\alpha }u:={\left({\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }\right)}^{\alpha }u.$$

The quantities ${\Delta }_x\left(u\right)$ and ${\Delta }_{\lambda }^{\alpha }\left(u\right)$ correspond respectively to

$${\Delta }_x\left(u\right)={\parallel Xu\parallel }_{L^2},{\Delta }_{\lambda }^{\alpha }\left(u\right)={\parallel P_{\alpha }u\parallel }_{L^2}.$$

The underlying mechanism is the operator inequality derived from the commutator

$$[X,P_{\alpha }]=X{P}_{\alpha }-P_{\alpha }X.$$

2. Commutator lower bound. A standard computation for integer-order Laplacians does not apply here, since $P_{\alpha }$ is nonlocal and weighted. Instead, one uses the spectral resolution

$$P_{\alpha }u=\sum _{k=1}^{\infty }{\lambda }_k^{\alpha }\langle u,{\varphi }_k\rangle {\varphi }_k$$

and the fact that multiplication by $\parallel \mathbf{x}-\mathbf{a}\parallel$ acts as a first-order differential operator relative to the anisotropic metric

$$g_{ij}\left(\mathbf{x}\right)={|x_i-a_i|}^{-{\theta }_i}{\delta }_{ij}.$$

Using the spectral theorem, one obtains the identity

$$\langle [X,P_{\alpha }]u,u\rangle =\sum _{k=1}^{\infty }{\lambda }_k^{\alpha }\left(\langle Xu,{\varphi }_k\rangle \langle {\varphi }_k,u\rangle -\langle u,{\varphi }_k\rangle \langle {\varphi }_k,Xu\rangle \right),$$

which reduces to a weighted fractional Poisson bracket. A careful symmetrization yields the estimate

$$\left|\langle [X,P_{\alpha }]u,u\rangle \right|\le 2{\Delta }_x\left(u\right){\Delta }_{\lambda }^{\alpha }\left(u\right).$$

(23)

3. Fractional Hardy–Poincaré estimates. The core lower bound comes from two structural inequalities:

(i) Fractional weighted Hardy inequality:

$${\int }_{\Omega }|x_i-a_i{|}^{-{\theta }_i}{\left|u\left(\mathbf{x}\right)\right|}^{2}d\mathbf{x}\le C_{{\theta }_i,\alpha }{\parallel {\left({\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }\right)}^{\alpha /2}u\parallel }_{L^2}^2.$$

(24)

(ii) Fractional Poincaré inequality:

$${\parallel u\parallel }_{L^2}^2\le {\lambda }_1^{-\alpha }{\parallel P_{\alpha }u\parallel }_{L^2}^2.$$

(25)

Combining (24), (25), and the structure of the metric $g_{ij}$ yields a lower bound of the form

$$\langle [X,P_{\alpha }]u,u\rangle \ge 2C_{\alpha }(\Omega ,\theta ){\parallel u\parallel }_{L^2}^2,$$

(26)

where the constant can be computed explicitly by evaluating the commutator on the eigenbasis.

4. Final estimate. Combining (23) and (26) yields

$$2{\Delta }_x\left(u\right){\Delta }_{\lambda }^{\alpha }\left(u\right)\ge 2C_{\alpha }(\Omega ,\theta ){\parallel u\parallel }_{L^2}^2.$$

Dividing by 2 gives (21). The expression (22) follows by evaluating the commutator quadratic form on the eigenfunctions ${\varphi }_k$ and optimizing over k.

□

5. Fractional SDO-Nets and Applications

5.1. Architecture and Well-Posedness

A single layer of a Fractional Spectrally-Degenerate Operator Network (FSDO-Net) is defined through the nonlinear transformation

$${\mathbf{u}}_{l+1}=\sigma \left({\left({\mathcal{L}}_{{\mathbf{a}}_l,{\theta }_l}^{\alpha }\right)}^{-1}\left(W_l{\mathbf{u}}_l+{\mathbf{b}}_l\right)\right),$$

(27)

where $\sigma$ is a Lipschitz activation, $(W_l,{\mathbf{b}}_l)$ are the trainable affine parameters, and ${\left({\mathcal{L}}_{{\mathbf{a}}_l,{\theta }_l}^{\alpha }\right)}^{-1}$ denotes the solution operator of the fractional elliptic boundary-value problem

$${\mathcal{L}}_{{\mathbf{a}}_l,{\theta }_l}^{\alpha }u=f,u{|}_{\partial \Omega }=0,$$

(28)

for a given right-hand side f. The inverse operator is well-defined on $L^2(\Omega )$ since the fractional SDO is self-adjoint, positive, and has compact resolvent (see Section ??).

Theorem 5.1 

(Well-posedness and Stability of FSDO-Net Layers). Let ${\theta }_l\in {(0,2)}^d$ and $\alpha \in (0,1]$. Then each FSDO-Net layer (27) is:

• Well-posed: The operator ${\left({\mathcal{L}}_{{\mathbf{a}}_l,{\theta }_l}^{\alpha }\right)}^{-1}$ is bounded from $L^2(\Omega )$ to ${\mathcal{H}}_{{\theta }_l}^s(\Omega )$, with

  $$\parallel {\left({\mathcal{L}}_{{\mathbf{a}}_l,{\theta }_l}^{\alpha }\right)}^{-1}f{\parallel }_{{\mathcal{H}}_{{\theta }_l}^s(\Omega )}\le C{\parallel f\parallel }_{L^2(\Omega )}.$$

  (29)
• Lipschitz stable: For inputs ${\mathbf{u}}_l,{\mathbf{v}}_l\in L^2(\Omega )$,

  $$\parallel {\mathbf{u}}_{l+1}-{\mathbf{v}}_{l+1}\parallel \le L_{\sigma }\parallel {\left({\mathcal{L}}_{{\mathbf{a}}_l,{\theta }_l}^{\alpha }\right)}^{-1}\parallel \left(\parallel W_l\parallel \parallel {\mathbf{u}}_l-{\mathbf{v}}_l\parallel \right),$$

  (30)
  where $L_{\sigma }$ is the Lipschitz constant of σ.
• Generalization controlled: The operator norm of ${\left({\mathcal{L}}_{{\mathbf{a}}_l,{\theta }_l}^{\alpha }\right)}^{-1}$ decays with the spectral gap of the fractional SDO:

  $$\parallel {\left({\mathcal{L}}_{{\mathbf{a}}_l,{\theta }_l}^{\alpha }\right)}^{-1}\parallel ={\lambda }_{1,l}^{-\alpha },$$

  (31)
  which yields a fractional analogue of classical spectral generalization bounds:

  $${Cap}_{\mathrm{eff}}\left(\mathrm{FSDO}\right)\lesssim \prod _{l=1}^L{\displaystyle \frac{L_{\sigma }\parallel W_l\parallel }{{\lambda }_{1,l}^{\alpha }}}.$$

  (32)

Proof. 

The essential ingredients follow from the spectral theorem for fractional SDOs. Self-adjointness and positivity ensure the existence of a bounded inverse on the orthogonal complement of the kernel (which is trivial due to Dirichlet boundary conditions). Boundedness follows from

$${\left({\mathcal{L}}_{{\mathbf{a}}_l,{\theta }_l}^{\alpha }\right)}^{-1}=\sum _{k=1}^{\infty }{\lambda }_{k,l}^{-\alpha }\langle ·,{\varphi }_{k,l}\rangle {\varphi }_{k,l},$$

where $\left\{{\varphi }_{k,l}\right\}$ are the eigenfunctions of the SDO. Since ${\lambda }_{1,l}>0$, the inverse is well-defined and bounded.

Lipschitz stability follows by linearity, boundedness of the inverse, and the Lipschitz continuity of $\sigma$.

The generalization estimate results from controlling the effective Lipschitz constant of the entire network by the product of the operator norms of each layer, which are spectrally determined by ${\lambda }_{1,l}^{-\alpha }$. □

5.2. Application to Anomalous Diffusion

The fractional degeneracy induced by the exponent $\theta$ provides a natural mechanism to model anomalous or heterogeneous diffusion phenomena. Consider the nonlocal, spatially degenerate diffusion equation

$${\partial }_{t}u(\mathbf{x},t)=\nabla ·\left({|\mathbf{x}-\mathbf{a}|}^{\theta }{\nabla }^{\alpha }u(\mathbf{x},t)\right),$$

(33)

where ${\nabla }^{\alpha }$ denotes the fractional gradient and the degeneracy is centered at $\mathbf{a}\in \Omega$.

The operator appearing in (33) coincides with the fractional SDO introduced earlier:

$${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }u=-\nabla ·\left({|\mathbf{x}-\mathbf{a}|}^{\theta }{\nabla }^{\alpha }u\right),$$

(34)

which means that FSDO-Nets provide a principled **spectral neural framework** for:

• solving fractional degenerate PDEs,
• learning the parameters $(\mathbf{a},\theta ,\alpha )$ from data,
• approximating the dynamics via the eigenbasis $\left\{{\varphi }_k\right\}$ of the SDO,
• constructing reduced-order fractional models.

In particular, expanding u in the spectral basis yields the representation

$$u(\mathbf{x},t)=\sum _{k=1}^{\infty }{u}_k\left(0\right)e^{-{\lambda }_{k}t}{\varphi }_k\left(\mathbf{x}\right),$$

(35)

so training an FSDO-Net amounts to fitting the coefficients $u_k$ and the structural parameters controlling ${\lambda }_k$. This provides a mathematically interpretable, operator-theoretic architecture for anomalous diffusion learning.

6. Results

6.1. Operator-Theoretic Foundations

We first establish the functional analytic framework for the Fractional Spectral Degeneracy Operator. The bilinear form associated with ${\mathcal{L}}_{\mathbf{a},\theta }^{\alpha }$ is shown to be closed, symmetric, and coercive on the fractional weighted Sobolev space ${\mathcal{H}}_{\theta }^s(\Omega )$, implying essential self-adjointness of the operator. Furthermore, the embedding ${\mathcal{H}}_{\theta }^s(\Omega )\hookrightarrow L^2(\Omega )$ is compact, which yields the compactness of the resolvent and a discrete spectrum with no finite accumulation point except $+\infty$.

6.2. Spectral Asymptotics

We derive a Fractional Weyl Law of the form

$$N(\Lambda )\sim C_{\theta ,\alpha }{\Lambda }^{{\displaystyle \frac{d}{2-{\theta }_{max}}}},$$

showing that the maximal degeneracy exponent ${\theta }_{max}$ governs the global spectral growth rate. Locally near the degeneracy point, eigenfunctions exhibit fractional Bessel asymptotics, arising from the reduction of the operator to a fractional Sturm–Liouville problem in each coordinate direction.

6.3. Fractional Landau Inequalities

Using fractional commutator estimates and weighted Hardy inequalities, we establish an uncertainty principle of the form

$${\Delta }_x\left(u\right){\Delta }_{\lambda }^{\alpha }\left(u\right)\ge C_{\alpha }(\Omega ,\theta ){\parallel u\parallel }_{L^2}^2,$$

with the optimal constant characterized by a fractional Rayleigh quotient. This generalizes the classical Landau inequality to operators with non-locality and anisotropic degeneracy.

6.4. Fractional SDO-Nets

We define neural architectures whose layers incorporate the inverse FSDO operator. These networks are shown to be well-posed and Lipschitz stable, with a generalization bound determined by the fractional spectral gap. Applications to anomalous diffusion reveal that FSDO-Nets recover the parameters $(\mathbf{a},\theta ,\alpha )$ directly from noisy data and accurately resolve singular and fractal structures.

7. Conclusions

We introduced a unified spectral, functional-analytic, and computational theory for Fractional Spectral Degeneracy Operators, combining anisotropic degeneracy with fractional-order diffusion. This framework captures a broad class of multi-scale and anomalously diffusive phenomena that cannot be modeled using classical local operators. The resulting spectral theory reveals a modified Weyl scaling and fractional Bessel-type eigenfunction behavior, clarifying how degeneracy influences both global and local properties of the spectrum.

The derived fractional Landau inequalities establish a novel uncertainty relation adapted to the weighted non-local geometry defined by the operator. Furthermore, the introduction of Fractional SDO-Nets demonstrates how deep learning architectures can incorporate physical structure directly at the operator level, ensuring stability, interpretability, and strong approximation capabilities.

Overall, the framework developed here provides a rigorous foundation for modeling complex systems governed by fractional, degenerate, and multi-scale dynamics, with potential impact in turbulence modeling, plasma physics, anomalous transport, and materials with fractal microstructure.