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

We study a family of Laguerre–Sobolev orthogonal polynomials associated with a Sobolev inner product arising from second–order boundary value problems on the semi–infinite interval $(0,+\infty)$. These polynomials generate an orthogonal basis of test functions vanishing at the endpoints and are especially well suited for the spectral approximation of Schrödinger–type problems with singular potentials. Explicit connection formulas with classical Laguerre polynomials are obtained, together with recurrence relations and asymptotic properties of the corresponding coefficients. A generating function involving Bessel functions is also derived. As an application, we develop a fully diagonalized Laguerre–Sobolev spectral method for Dirichlet problems with singular potentials. The method avoids the solution of linear systems and can be implemented recursively. Numerical experiments for a Schrödinger–type equation with inverse–distance potential confirm spectral accuracy and exponential convergence.

💡 Research Summary

The paper introduces a new family of Laguerre‑Sobolev orthogonal polynomials tailored to the half‑line (0,∞) and demonstrates how these polynomials can be used to construct a fully diagonalized spectral method for Dirichlet boundary‑value problems involving singular potentials, in particular the Schrödinger‑type equation with an inverse‑distance term V(x)=1/x.

The authors begin by recalling the classical Laguerre polynomials L_n^{(α)}(x), their orthogonality with respect to the weight t^{α}e^{-t}, norm ‖L_n^{(α)}‖²=Γ(n+α+1)/n!, three‑term recurrence, differentiation formula, and outer asymptotics in the complex plane. They also present the Hardy‑Hille generating function involving Bessel functions, which will later serve as a template for the Sobolev case.

A Sobolev inner product is defined as
⟨u,v⟩_λ = λ∫₀^∞ u(x)v(x)/x dx + ∫₀^∞ u′(x)v′(x) dx,
where λ>0 is a parameter originating from the variational formulation of the Schrödinger operator. The space of test functions is taken as P x e^{-x/2}, i.e., polynomials multiplied by the decaying factor x e^{-x/2} that vanish at both endpoints. Applying Gram–Schmidt orthogonalization to the monomials {x^k e^{-x/2}} with respect to ⟨·,·⟩_λ yields a sequence {S_n(x)} of Sobolev orthogonal polynomials. By construction, S_n shares the leading coefficient of the classical Laguerre polynomial L_n^{(1)}(x).

A central result is the simple connection formula
L_n^{(1)}(x) = S_n(x) + a_{n-1} S_{n-1}(x), n≥1,
with coefficients a_n satisfying the linear recurrence
a_n = (n+2)/(4λ+2(n+1)) – n(n+1)/(4λ+2(n+1)) a_{n-1}, a_0 = 2/(4λ+2).
From this recurrence the authors prove 0<a_n<1 for all n and obtain an explicit representation
a_n = (n+2) n! L_n^{(1)}(−4λ) L_{n+1}^{(1)}(−4λ).
Using refined ratio asymptotics for Laguerre polynomials they derive the first‑order asymptotic expansion
a_n = 1 – √(4λ)/√n + O(1/n).

Further algebraic manipulation leads to a closed form for S_n in terms of Laguerre polynomials evaluated at the negative argument −4λ:
S_n(x) L_n^{(1)}(−4λ)^{n+1} = Σ_{k=0}^n (−1)^{n−k} L_k^{(1)}(x) L_k^{(1)}(−4λ)^{k+1}.
From this identity the authors prove the uniform limit
lim_{n→∞} √{n+1} S_n(x) L_n^{(1)}(x) = 0,
valid on any compact subset of ℂ\ (0,∞).

A generating function is also obtained:
∑_{n=0}^∞ S_n(x) L_n^{(1)}(−4λ)^{n+1} ω^n = (1−ω)^{−1} exp