Complete systems of recursive integrals and Taylor series for solutions of Sturm-Liouville equations

Abstract

Consider an arbitrary complex-valued, twice continuously differentiable, nonvanishing function $\varphi$ defined on a finite segment $[a,b]\subset \mathbb{R}$. Let us introduce an infinite system of functions constructed in the following way. Each subsequent function is a primitive of the preceding one multiplied or divided by $\varphi$ alternately. The obtained system of functions is a generalization of the system of powers ${(x-x_{0}%)^{k}}_{k=0}^{\infty}$. We study its completeness as well as the completeness of its subsets in different functional spaces. This system of recursive integrals results to be closely related to so-called $L$-bases arising in the theory of transmutation operators for linear ordinary differential equations. Besides the results on the completeness of the system of recursive integrals we show a deep analogy between the expansions in terms of the recursive integrals and Taylor expansions. We prove a generalization of the Taylor theorem with the Lagrange form of the remainder term and find an explicit formula for transforming a generalized Taylor expansion of a function in terms of the recursive integrals into a usual Taylor expansion. As a direct corollary of the formula we obtain the following new result concerning solutions of the Sturm-Liouville equation. Given a regular nonvanishing complex valued solution $y_0$ of the equation $y^{\prime \prime }+q(x)y=0$, $x\in (a,b)$, assume that it is $n$ times differentiable at a point $x_0$. We present explicit formulas for calculating the first $n$ derivatives at $x_0$ for any solution of the equation $u^{\prime \prime }+q(x)u=\lambda u$. That is, an explicit map transforming the Taylor expansion of $y_0$ into the Taylor expansion of $u$ is constructed.

arXiv comment17 pages, 0 figure

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Cited by in corpus (9)

• Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions
• Transmutations, L-bases and complete families of solutions of the stationary Schrödinger equation in the plane
• Transmutations and spectral parameter power series in eigenvalue problems
• Wave polynomials, transmutations and Cauchy's problem for the Klein-Gordon equation
• Analytic approximation of transmutation operators and related systems of functions
• Construction of a transmutation for the one-dimensional Schrödinger operator and a representation for solutions
• On Sturm-Liouville Equations with Several Spectral Parameters
• Supersymmetric generalized power functions
• A method for computation of scattering amplitudes and Green functions of whole axis problems