Transmutation operators: construction and applications
For researchers in spectral theory and PDEs, this provides a practical method to maintain accuracy in large-scale eigenvalue computations, though the results are domain-specific and incremental.
The paper presents new representations for solutions of the 1D Schrödinger equation using transmutation operators, achieving accuracy independent of the spectral parameter's real part, which is crucial for computing large eigendata sets without accuracy loss.
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.