Project: Spectral properties of unbounded operator functions
Advanced time-dependent materials are used to model viscoelastic properties of wood, wave propagation of ultrasound, and systems with thermal memory. Localized vibrations in those systems are related to the spectral properties of an operator function. In this project, we study classes of operator functions with applications to integro-differential equations.
The image illustrates a resolvent estimate of an operator function, where red indicates large values and blue small values (Torshage, A., 2020).
Project manager at Linnaeus University Christian Engström Other project members Heinz Langer, Vienna University of Technology, Austria Axel Torshage, Umeå University, Sweden Christiane Tretter, University of Bern, Switzerland Participating organizations Linnaeus University and Umeå University, Sweden; Vienna University of Technology, Austria; University of Bern, Switzerland Financier The Swedish Research Council, 2013–2018 Timetable 2013– Subject Mathematics (Department of Mathematics, Faculty of Technology)
More about the project
Operator functions whose values are linear operators occur in various branches of engineering and physics, e.g., in electromagnetics, magnetohydrodynamics, and continuum mechanics. This project concerns the spectral theory of unbounded operator functions, where the non-linear dependence of the spectral parameter is a consequence of modeling advanced materials more realistically.
In particular, we study problems where the materials have memory. This leads for the time-evolution problem to integro-differential equations with a convolution term. Common models of this type are the Drude-Lorentz model in electromagnetics and Boltzmann damping in viscoelastic materials.
The objectives of the project are:
To derive two-sided estimates of all eigenvalues of self-adjoint operator functions and apply the results to photonic crystals.
To establish proofs of accumulation of eigenvalues to the essential spectrum for a class of rational operator functions.
To derive enclosures of the spectrum for operator functions.
To derive resolvent estimates for operator functions.