Approximative spectral methods

• Applicant: Professor Dr. Carsten Trunk
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 574006433

Eigenvalues and spectra play a key role in applications such as fluid mechanics, magnetohydrodynamics, electromagnetics or quantum mechanics. Typically, the underlying physical problems are described by partial differential equations (PDEs) or coupled systems of PDEs, often subject to damping or viscosity which destroy symmetry. It is a major challenge to obtain reliable information on eigenvalues and spectra of such infinite dimensional problems, both analytically and numerically. First, the lack of symmetry makes eigenvalues and spectra very sensitive to perturbations and, secondly, numerical approximations of spectra such as finite dimensional approximations or domain truncation are unreliable. There are two undesirable effects: • spectral pollution, where approximating eigenvalues converge to limits that are no true spectral points (so-called spurious eigenvalues), • spectral invisibility, where true spectral points are not obtained as limits of approximating eigenvalues. The absence of these two undesirable effects is called spectral exactness. The suggested project will build upon recent research on the powerful concepts of essential numerical ranges and spectral approximations for unbounded linear operators. This topic was pioneered by the proposed Mercator fellow and it allows a unique combination with the novel results on differential operators exhibiting only indefinite symmetries by the host and his research group

• DFG Programme: Research Grants