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Volume 37, Issue 1
Hill-Type Formula and Krein-Type Trace Formula for Hamiltonian Systems

Xijun Hu, Yuwei Ou, Penghui Wang & Hao Zhu

Anal. Theory Appl., 37 (2021), pp. 74-101.

Published online: 2021-04

[An open-access article; the PDF is free to any online user.]

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  • 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.

  • AMS Subject Headings

34B30, 34L15, 34B09, 37C75, 70H14

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-37-74, author = {Hu , XijunOu , YuweiWang , Penghui and Zhu , Hao}, title = {Hill-Type Formula and Krein-Type Trace Formula for Hamiltonian Systems}, journal = {Analysis in Theory and Applications}, year = {2021}, volume = {37}, number = {1}, pages = {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.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2021.pr80.09}, url = {http://global-sci.org/intro/article_detail/ata/18765.html} }
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 -

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.

Hu , XijunOu , YuweiWang , Penghui and Zhu , Hao. (2021). Hill-Type Formula and Krein-Type Trace Formula for Hamiltonian Systems. Analysis in Theory and Applications. 37 (1). 74-101. doi:10.4208/ata.2021.pr80.09
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