TY - JOUR
T1 - Hill-Type Formula and Krein-Type Trace Formula for Hamiltonian Systems
AU - Hu , Xijun
AU - Ou , Yuwei
AU - Wang , Penghui
AU - Zhu , Hao
JO - Analysis in Theory and Applications
VL - 1
SP - 74
EP - 101
PY - 2021
DA - 2021/04
SN - 37
DO - http://doi.org/10.4208/ata.2021.pr80.09
UR - https://global-sci.org/intro/article_detail/ata/18765.html
KW - Hill-type formula, trace formula,  conditional Fredholm determinant, relative Morse index.
AB - <p style="text-align: justify;">In this paper, we give a survey on the Hill-type formula and its applications.&nbsp; 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&nbsp; the relative Morse index.</p>