Transmutation operators and a new representation for solutions of perturbed Bessel equations. (English) Zbl 1478.34095
Summary: New representations for an integral kernel of the transmutation operator and for a regular solution of the perturbed Bessel equation of the form
\[
-u''+\left(\frac{\ell(\ell+1)}{x^2}+q(x)\right)u=\omega^2u
\]
are obtained. The integral kernel is represented as a Fourier-Jacobi series. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to \(\omega\). For the coefficients of the series convenient for numerical computation recurrent integration formulas are obtained. The new representation improves the ones previously obtained by the authors for large values of \(\omega\) and \(\ell\) and for noninteger values of \(\ell\). The results are based on application of several ideas from the classical transmutation (transformation) operator theory, asymptotic formulas for the solution, results connecting the decay rate of the Fourier transform with the smoothness of a function, the Paley-Wiener theorem, and some results from constructive approximation theory. We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy.
MSC:
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
| 34B24 | Sturm-Liouville theory |
| 41A10 | Approximation by polynomials |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 65L15 | Numerical solution of eigenvalue problems involving ordinary differential equations |
Keywords:
Jacobi polynomials; Neumann series of Bessel functions; Paley-Wiener theorem; perturbed Bessel equation; Sturm-Liouville theory; transmutation operatorReferences:
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