Hamiltonian of the One-Dimensional Torsion Schrödinger Equation in a Complex-Valued Basis of Mathieu Functions

An analytical method for calculating the matrix elements of the Hamiltonian of the torsion Schrödinger equation in a basis of Mathieu functions is developed. The matrix elements are represented by integrals of the product of three Mathieu functions, and also the derivatives of these functions. Analytical expressions for the matrix elements are obtained by approximating the Mathieu functions by Fourier series and are products of the corresponding Fourier expansion coefficients. It is shown that replacing high-order Mathieu functions by one harmonic leads to insignificant errors in the calculation.