Construction of transmutation operators and application to direct and inverse spectral problem

A new approach for solving the classical direct and inverse Sturm-Liouville problems on finite and infinite intervals is presented. It is based on the Gel'fand-Levitan-Marchenko integral equations and recent results on the functional series representations for the transmutation (transformation) operator kernels [1-5]. New representations of solutions to Sturm-Liouville equation are obtained enjoying the following feature important for practical applications. Partial sums of the series admit estimates independent of the real part of the square root of the spectral parameter which makes it especially convenient for the approximate solution of spectral problems. Numerical methods based on the proposed approach for solving direct problems allow one to compute large sets of eigendata with a nondeteriorating accuracy. Solution of the inverse problems reduces directly to a system of linear algebraic equations. In the talk some numerical illustrations will be presented.