Year: 2021
Author: Xijun Hu, Yuwei Ou, Penghui Wang, Hao Zhu
Analysis in Theory and Applications, Vol. 37 (2021), Iss. 1 : pp. 74–101
Abstract
In this paper, we give a survey on the Hill-type formula and its applications. Moreover, we generalize the Hill-type formula for linear Hamiltonian systems and Sturm-Liouville systems with any self-adjoint boundary conditions, which include the standard Neumann, Dirichlet and periodic boundary conditions. The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions. Further, based on the Hill-type formula, we derive the Krein-type trace formula. As applications, we give nontrivial estimations for the eigenvalue problem and the relative Morse index.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.2021.pr80.09
Analysis in Theory and Applications, Vol. 37 (2021), Iss. 1 : pp. 74–101
Published online: 2021-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 28
Keywords: Hill-type formula trace formula conditional Fredholm determinant relative Morse index.