Perturbation of eigenvalues of the Klein–Gordon operators

Abstract

We prove inclusion theorems for both spectra and essential spectra as well as two-sided bounds for isolated eigenvalues for Klein-Gordon type Hamiltonian operators. We first study the operators of the form JG, where J, G are selfadjoint operators on a Hilbert space, J = J * = J −1 and G is positive definite and then we apply these results to obtain the bounds of the Klein-Gordon eigenvalues under the change of the electrostatic potential. The developed general theory allows applications to some other instances, as e.g. the Sturm-Liouville problems with indefinite weight.

Limits of stabilization of a networked hyperbolic system with a circle

“…Gugat and Gerster (2019) discuss the limits of stabilizability for the starshaped networks of strings, inspired by Coron. In Gugat and Gerster (2019), they show that the system is stabilizable if the lengths of the arcs are sufficiently small and that it is not stabilizable in some other cases. Nakić and Veselić (2020) consider the perturbation of eigenvalues of our discussed operator, although their analysis primarily approached the topic here considered from an abstract operator perspective.…”

From Lieb–Thirring Inequalities to Spectral Enclosures for the Damped Wave Equation

“…Basic eigenvalue bounds for weakly damped wave systems can be deduced from [18]. The perturbation of eigenvalues of abstract damped wave operators has been recently studied in the framework of Krein spaces in [27]. Resolvent estimates for an abstract dissipative operator are derived in [4,29].…”