Premium
Comparison theorems for self‐adjoint linear Hamiltonian eigenvalue problems
Author(s) -
Hilscher Roman Šimon
Publication year - 2014
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200314
Subject(s) - mathematics , eigenvalues and eigenvectors , hamiltonian (control theory) , mathematical analysis , boundary value problem , controllability , dirichlet boundary condition , dirichlet eigenvalue , hamiltonian system , nonlinear system , spectral theory , pure mathematics , dirichlet's principle , hilbert space , mathematical optimization , quantum mechanics , physics
In this work we derive new comparison results for (finite) eigenvalues of two self‐adjoint linear Hamiltonian eigenvalue problems. The coefficient matrices depend on the spectral parameter nonlinearly and the spectral parameter is present also in the boundary conditions. We do not impose any controllability or strict normality assumptions. Our method is based on a generalization of the Sturmian comparison theorem for such systems. The results are new even for the Dirichlet boundary conditions, for linear Hamiltonian systems depending linearly on the spectral parameter, and for Sturm–Liouville eigenvalue problems with nonlinear dependence on the spectral parameter.