Reconstruction of Sturm–Liouville Equations in Impedance Form

A method for solving a general inverse coefficient problem for a Sturm–Liouville equation in impedance form $$\left(r^{2}(x)u^{\prime}(x)\right)^{\prime}+\lambda r^{2}(x)u(x)=0$$ , $$x\in\left(0,L\right)$$ , is proposed. It is based on Neumann series of Bessel functions (NSBF) representations for solutions of a related Schrödinger equation. The whole procedure reduces to a solution of a couple of systems of linear algebraic equations for the NSBF coefficients. Solving the first one, we recover a pair of characteristic functions of two Sturm–Liouville problems, while the solution of the second one leads to the knowledge of the NSBF coefficients on $$\left[0,L\right]$$ , and in particular of the first NSBF coefficient $$g_{0}(x)$$ . The unknown coefficient $$r(x)$$ in the Sturm–Liouville equation is related with $$g_{0}(x)$$ as $$r(x)=r(0)\left(g_{0}(x)+1\right)$$ , that means that $$r(x)$$ is recovered directly from the first component of the solution vector of the second system of linear algebraic equations. The approach leads to a simple and accurate numerical algorithm. Numerical efficiency is illustrated by test examples.