The image illustrates a resolvent estimate of an operator function, where red indicates large values and blue small values

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 information

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
The Swedish Research Council, 2013–2018
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:

  1. To derive two-sided estimates of all eigenvalues of self-adjoint operator functions and apply the results to photonic crystals.
  2. To establish proofs of accumulation of eigenvalues to the essential spectrum for a class of rational operator functions.
  3. To derive enclosures of the spectrum for operator functions.
  4. To derive resolvent estimates for operator functions.


Engström, C., Torshage, A. (2018). Accumulation of complex eigenvalues of a class of analytic operator functions. Journal of Functional Analysis. 275. 442-477.

Engström, C., Torshage, A. (2018). Spectral properties of conservative, dispersive, and absorptive photonic crystals. GAMM - Mitteilungen. 41. 1-16.

Engström, C., Langer, H., Tretter, C. (2017). Rational eigenvalue problems and applications to photonic crystals. Journal of Mathematical Analysis and Applications. 445. 240-279.

Engström, C., Torshage, A. (2017). On equivalence and linearization of operator matrix functions with unbounded entries. Integral equations and operator theory. 89. 465-492.

Engström, C., Torshage, A. (2017). Enclosure of the numerical range of a class of non-selfadjoint rational operator functions. Integral equations and operator theory. 88. 151-184.