[Paper Review] Transmutation operators: construction and applications

Vladislav V. Kravchenko, Sergii M. Torba+1 more|arXiv (Cornell University)|Aug 1, 2017

Numerical methods for differential equations8 references3 citations

TL;DR

This paper presents new exact representations for solutions of the one-dimensional Schrödinger equation using transmutation operators, leveraging Fourier-Legendre and Hermite series expansions of the transmutation kernel. The key contribution is ω-independent approximation accuracy, enabling high-precision computation of large sets of eigendata for variable-coefficient equations.

ABSTRACT

Recent results on the construction and applications of the transmutation (transformation) operators are discussed. Three new representations for solutions of the one-dimensional Schrödinger equation are considered. Due to the fact that they are obtained with the aid of the transmutation operator all the representations possess an important for practice feature. The accuracy of the approximate solution is independent of the real part of the spectral parameter. This makes the representations especially useful in problems requiring computation of large sets of eigendata with a nondeteriorating accuracy. Applications of the exact representations for the transmutation operators to partial differential equations are discussed as well. In particular, it is shown how the methods based on complete families of solutions can be extended onto equations with variable coefficients.

Motivation & Objective

• To develop exact, numerically tractable representations for solutions of the one-dimensional Schrödinger equation with variable potentials.
• To construct the transmutation kernel K(x,t) using orthogonal polynomial expansions (Legendre and Hermite) for improved computational efficiency.
• To ensure that the approximation error in solution representations remains independent of the spectral parameter ω, especially for large or complex ω.
• To extend methods based on complete families of solutions—such as the method of fundamental solutions—to partial differential equations with variable coefficients.
• To provide a systematic framework for generating complete systems of solutions for equations like (Δ − q(x))u = 0 using transmutation operators.

Proposed method

• Represent the transmutation kernel K(x,t) as a Fourier-Legendre series: K(x,t) = Σₙ₌₀^∞ [βₙ(x)/x] Pₙ(t/x), with coefficients βₙ(x) derived from formal powers φₖ(x) and Legendre polynomial coefficients.
• Construct formal powers φₖ(x) recursively using solutions f(x) of f'' − q(x)f = 0 with f(0)=1, f'(0)=0, and iterated integrals involving f²(s).
• Derive a solution representation u(ω,x) = e^{iωx} + Σₙ₌₀^∞ iⁿβₙ(x)jₙ(ωx), where jₙ are spherical Bessel functions, ensuring uniform convergence and ω-independent error bounds.
• Apply a second representation using Hermite polynomials: K(x,t) expanded in Hₙ(t)e^{-t²}, leading to u(ω,x) = e^{iωx} + √π e^{-ω²/4} Σₙ₌₀^N cₙ(x)(iω)^n with coefficients cₙ(x) computed from integrals of K(x,t)Hₙ(t).
• Establish error bounds independent of ω: |u(ω,x) − u_N(ω,x)| ≤ ε_N(x) × C, where C depends only on the imaginary part of ω, ensuring stable accuracy across large spectral ranges.
• Apply the transmutation operator T to harmonic functions (e.g., real and imaginary parts of z^m) and fundamental solutions (e.g., log|x−Z|) to generate complete systems of solutions for (Δ − q(x))u = 0.

Experimental results

Research questions

• RQ1 Can the transmutation kernel K(x,t) be represented in a closed-form series expansion that enables efficient and accurate numerical computation?
• RQ2 Does the proposed series representation for solutions of the Schrödinger equation maintain uniform accuracy across varying spectral parameters ω, especially for large or complex ω?
• RQ3 To what extent can transmutation-based methods extend the method of fundamental solutions to partial differential equations with variable coefficients?
• RQ4 How can complete families of solutions for (Δ − q(x))u = 0 be systematically generated using transmutation operators?
• RQ5 Can the error in truncated series approximations of solutions be bounded independently of ω, ensuring robustness in large-scale eigendata computations?

Key findings

• The transmutation kernel K(x,t) admits an exact representation as a Fourier-Legendre series: K(x,t) = Σₙ₌₀^∞ [βₙ(x)/x] Pₙ(t/x), with βₙ(x) computable from formal powers φₖ(x) and Legendre polynomial coefficients.
• The solution u(ω,x) of the Schrödinger equation is represented as u(ω,x) = e^{iωx} + Σₙ₌₀^∞ iⁿβₙ(x)jₙ(ωx), where jₙ are spherical Bessel functions, and this series converges uniformly in x.
• The approximation error |u(ω,x) − u_N(ω,x)| is bounded by ε_N(x) × √(2x) for real ω and by ε_N(x) × sinh(Cx)/C for complex ω with |Im ω| ≤ C, proving ω-independent accuracy.
• A second representation using Hermite polynomials yields u(ω,x) = e^{iωx} + √π e^{-ω²/4} Σₙ₌₀^N cₙ(x)(iω)^n, with error bounded by π^{1/4} e^{(Im ω)^2/2} ε_N(x), again independent of ω.
• Applying the transmutation operator T to harmonic polynomials generates a complete system of solutions for (Δ − q(x))u = 0, explicitly given in terms of φₖ(x) and monomials in y.
• The method extends the method of fundamental solutions to variable-coefficient PDEs: T[log|x−Z|] is expressed as a series involving Legendre functions Qₙ(Z/x), enabling construction of fundamental solutions for (Δ − q(x))u = 0.