Eigenfunction and Green’s Function Asymptotics for Hill’s Equation with Symmetric Single-Well Potential

We determine the asymptotic formulas for eigenfunctions of Hill’s equation with symmetric single-well potential under periodic and semiperiodic boundary conditions. We apply the results for eigenvalues obtained by H. Coşkun, et al. in (2019). By using these estimates for eigenfunctions, we obtain Green’s functions related to Hill’s equation. The method based on the work by C. T. Fulton (1977) is used to derive Green’s functions in the asymptotic manner. We need the derivatives of the solutions in this method. Therefore, the asymptotic approximations for the derivatives of the eigenfunctions are also computed with different types of restrictions imposed on the potential.