Trigonometric Interpolation Method for One-Dimensional Quantum-Mechanical Problems

A rapidly converging difference method, based on harmonic analysis, is described. It can be applied to periodic or nonperiodic bound‐state problems of the general Sturm–Liouville type. Numerical examples for the Mathieu problem and for the harmonic oscillator show considerable accuracy. Advantages and disadvantages of the method are discussed in a comparison with Harris's matrix transformation technique and with direct integration methods. The set of difference equations representing a quantum‐mechanical problem constitutes a symmetric matrix eigenvalue problem which is approximately equivalent to the algebraic problem obtained by using a finite trigonometric basis. Basis functions associated with the difference method are related to the Dirichlet kernel. In an approximation which corresponds to the difference method, these basis functions can be treated in a similar way as Dirac's $\delta$ function.