🔢 mathematics

Laguerre-Sobolev orthogonal Polynomials and Boundary Value Problems on a semi-infinite domain

This paper introduces a family of Laguerre-Sobolev orthogonal polynomials derived from boundary value problems on a semi-infinite domain, establishing their theoretical properties and utilizing them to develop a fully diagonalized, recursively implementable spectral method that achieves exponential convergence for solving Schrödinger-type equations with singular potentials.

Original authors: Cleonice F. Bracciali, Miguel A. Piñar

Published 2026-02-09

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Cleonice F. Bracciali, Miguel A. Piñar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a complex puzzle: a mathematical equation that describes how a particle moves in a universe where the forces get weird and wild near the starting line (zero) and stretch out forever. This is a "boundary value problem" on a semi-infinite road.

The authors of this paper, Cleonice Bracciali and Miguel Piñar, are like master carpenters who have built a new, specialized set of tools to solve this specific puzzle more efficiently than anyone else.

Here is the story of their work, broken down into everyday concepts:

1. The Old Tools vs. The New Problem

For a long time, mathematicians have used "Laguerre polynomials" as their go-to tools for solving equations on infinite roads. Think of these as standard, sturdy hammers. They work great for most jobs.

However, when the problem involves singular potentials (mathematical "potholes" or infinite spikes in the force field, like an inverse-distance potential where the force gets huge as you get close to zero) or specific rules about how the solution must behave at the very start and the very end, the standard hammers start to slip. They require solving massive, complicated systems of equations, which is slow and computationally heavy.

2. The New Tool: Laguerre-Sobolev Polynomials

The authors invented a new family of tools called Laguerre-Sobolev orthogonal polynomials.

The "Sobolev" Twist: Imagine a standard hammer that only cares about the shape of the wood. A "Sobolev" hammer is special because it also cares about the grain and the smoothness of the wood at the same time. In math terms, these new polynomials are designed to "know" about the derivative (the rate of change) and the boundary conditions (the rules at the start and finish) right from the moment they are created.
The Result: Because these tools are built with the rules baked into them, they fit the puzzle perfectly without needing to force the pieces together.

3. The "Magic Shortcut" (Diagonalization)

This is the most exciting part of the paper. Usually, when you use spectral methods (a fancy way of breaking a problem into waves), you end up with a giant, tangled web of linear equations that a computer has to solve all at once. It's like trying to untangle a knot of 1,000 headphones.

The authors' method is fully diagonalized.

The Analogy: Instead of untangling the whole knot, their method gives you a magic wand that instantly separates every single headphone from the others.
The Benefit: You don't need to solve a giant system of equations. You can calculate the answer recursively. This means you can build the solution step-by-step, like stacking bricks, where each new brick only depends on the one before it. It's fast, simple, and avoids the computational bottleneck.

4. The Connection to the Classics

The authors didn't just invent these tools from thin air; they showed exactly how these new tools relate to the old, trusted Laguerre polynomials.

They derived connection formulas: Think of this as a translation dictionary. If you know how to use the old hammers, this paper tells you exactly how to switch to the new ones.
They found a generating function: This is like a master recipe that can instantly bake any number of these new polynomials, using a special ingredient called a "Bessel function" (a type of wave pattern often found in physics).

5. The Proof: Does it Work?

The authors tested their new method on a specific type of Schrödinger equation (the equation that governs quantum particles) with a "singular" potential (the $1/x$ problem).

The Test: They tried to approximate a solution that fades away smoothly (exponential decay) and another that fades away more slowly (rational decay).
The Result:
- For the smooth, fading solution, the new method was incredibly accurate. The error dropped like a stone falling down a well (exponential convergence). It was like hitting the bullseye with a laser.
- For the slower-fading solution, the method still worked, but the "bullseye" was harder to hit, showing that the method is best suited for problems where the solution behaves nicely at infinity.

Summary

In short, this paper introduces a new, specialized set of mathematical building blocks designed specifically for difficult physics problems on infinite domains. By baking the boundary rules directly into the blocks, the authors created a method that is fast, recursive, and avoids complex calculations, offering a highly accurate way to solve problems that were previously difficult to handle with standard tools.

Technical Summary: Laguerre–Sobolev Orthogonal Polynomials and Boundary Value Problems on a Semi-Infinite Domain

Problem Statement
The paper addresses the numerical solution of second-order boundary value problems (BVPs) on the semi-infinite interval $(0, +\infty)$ , specifically focusing on stationary Schrödinger-type equations with singular potentials, such as the inverse-distance potential $V(x) = 1/x$ . While classical spectral methods based on Laguerre polynomials are well-established for smooth solutions on semi-infinite domains, they can become inefficient when dealing with boundary conditions at infinity or singular coefficients in the differential operator. The authors seek to develop a framework that incorporates derivative information and boundary behavior directly into the approximation space to handle these singularities more effectively.

Methodology
The authors introduce a family of Laguerre–Sobolev orthogonal polynomials, denoted $\{S_n(x)\}_{n \geq 0}$ , associated with a specific Sobolev inner product derived from the variational formulation of the target BVP. The inner product is defined as:
$\langle u, v \rangle_\lambda = \lambda \int_0^{+\infty} \frac{u(x)v(x)}{x} dx + \int_0^{+\infty} u'(x)v'(x) dx$
The test functions are chosen from the space $\mathcal{P} \times x e^{-x/2}$ , ensuring they vanish at both endpoints $0$ and $+\infty$ .

The core of the methodology involves:

Construction: Generating the orthogonal basis $\{S_n(x)\}$ via the Gram-Schmidt process applied to the sequence $\{x^k x e^{-x/2}\}$ with respect to the Sobolev inner product.
Analytical Derivation: Establishing explicit connection formulas between the new Sobolev polynomials $S_n(x)$ and the classical Laguerre polynomials $L_n^{(1)}(x)$ . This includes deriving recurrence relations for the connection coefficients and analyzing their asymptotic behavior using refined ratio asymptotics for Laguerre polynomials.
Generating Function: Deriving a closed-form generating function for the Sobolev polynomials involving Bessel functions.
Spectral Algorithm: Developing a fully diagonalized spectral method. By utilizing the connection formulas, the Fourier–Sobolev coefficients of the solution are computed recursively. This approach avoids solving linear systems, as the resulting algorithm requires only the evaluation of integrals involving classical Laguerre polynomials.

Key Contributions

New Orthogonal Family: The definition and analysis of a specific family of Laguerre–Sobolev orthogonal polynomials tailored for BVPs with singular potentials on $(0, +\infty)$ .
Connection Formulas: The derivation of explicit relations linking $S_n(x)$ to $L_n^{(1)}(x)$ , including a recurrence relation for the connection coefficients $a_n$ and their asymptotic expansion ( $a_n \sim 1 - \sqrt{4\lambda/n}$ ).
Generating Function: The provision of a generating function for the Sobolev polynomials expressed in terms of Bessel functions of the first kind.
Diagonalized Algorithm: The formulation of a recursive spectral method that computes the solution's expansion coefficients without solving linear systems, thereby reducing computational complexity while maintaining stability for singular problems.

Results
The paper presents numerical experiments solving a non-homogeneous Schrödinger equation with an inverse-distance potential ( $V(x)=1/x$ ) and Dirichlet boundary conditions.

Exponential Convergence: For a solution exhibiting exponential decay ( $u(x) = x \cos x e^{-x}$ ), the method demonstrates spectral accuracy and exponential convergence. The error in the Sobolev norm decreases rapidly as the number of terms increases.
Limitations: The authors note that for solutions that do not decay exponentially (e.g., rational decay), the performance of the approximants is worse, as expected for spectral methods on semi-infinite domains.
Efficiency: The recursive nature of the algorithm allows for computational costs comparable to standard Laguerre methods while offering improved stability for the specific class of singular problems addressed.

Significance
The paper claims that the proposed Laguerre–Sobolev spectral method offers a valuable alternative to classical Laguerre methods for problems involving singular coefficients and boundary conditions at infinity. By embedding derivative information and boundary behavior directly into the approximation space, the method achieves a fully diagonalized system. This eliminates the need to solve linear systems, significantly reducing computational complexity. The work provides a rigorous analytical foundation (connection formulas, asymptotics, generating functions) that enables the practical implementation of these methods for Schrödinger-type problems with singular potentials.