Eigenfunction and Green’s function asymptotics for Hill’s equation with symmetric single well potential

A. Kabataş

doi:10.37863/umzh.v74i2.6246

A. Kabataş KaradenizTechn.Univ.,Trabzon,Turkey

DOI: https://doi.org/10.37863/umzh.v74i2.6246

Keywords: Hill's equation, symmetric single well potential, periodic and semi-periodic eigenfunctions, Green's functions, asymptotics

Abstract

UDC 517.9

This paper is devoted to determine the asymptotic formulae for eigenfunctions of the Hill's equation with symmetric single well potential under periodic and semi-periodic boundary conditions. The obtained results for eigenvalues by H. Coşkun and the others (2019) are used. With these estimates on the eigenfunctions, Green's functions related to the Hill's equation are obtained. The method is based on the work of C. T. Fulton (1977) to derive Green's functions in an asymptotical manner. We need the derivatives of the solutions in this method. Therefore, the asymptotic approximations for the derivatives of the eigenfunctions are also calculated with different types of restrictions on the potential.

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