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<p>This paper investigates the eigenvalues and generalized eigenfunctions of a Sturm–Liouville problem in which the eigenparameter appears in the boundary conditions. By employing operator pencil theory in a Hilbert space, combined with an appropriate integral transformation, we prove that the system of generalized eigenfunctions forms a Riesz basis. Furthermore, we demonstrate that the spectrum consists of a countably infinite set of real, discrete eigenvalues accumulating at infinity and provide a lower bound estimation for the eigenvalues.</p>

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Properties of Eigenvalues and Generalized Eigenfunctions for Sturm–Liouville Problem with Eigenparameter-Dependent Boundary Conditions

  • 期刊论文

  • 发表日期：

    2025年11月5日

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  • Zhiyu Li,
  • Zhaowen Zheng,
  • Yinghan Zhang

摘要

This paper investigates the eigenvalues and generalized eigenfunctions of a Sturm–Liouville problem in which the eigenparameter appears in the boundary conditions. By employing operator pencil theory in a Hilbert space, combined with an appropriate integral transformation, we prove that the system of generalized eigenfunctions forms a Riesz basis. Furthermore, we demonstrate that the spectrum consists of a countably infinite set of real, discrete eigenvalues accumulating at infinity and provide a lower bound estimation for the eigenvalues.