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

Kabataş A.

UKRAINIAN MATHEMATICAL JOURNAL, vol.74, no.2, pp.218-231, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 74 Issue: 2
  • Publication Date: 2022
  • Doi Number: 10.1007/s11253-022-02059-5
  • Journal Name: UKRAINIAN MATHEMATICAL JOURNAL
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Page Numbers: pp.218-231
  • Karadeniz Technical University Affiliated: Yes

Abstract

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. Coskun, 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.