Organizers of this minisymposium are
The spectral theory has one of its origins in the analysis of wave processes. In particular, finding time-harmonic solutions often leads to eigenvalue problems for suitable Hamilton operators. The respective eigenfunctions correspond to the standing waves of the system. As a prominent textbook example one can mention the wave equation describing the oscillation of an elastic membrane. The corresponding Hamiltonian is the Dirichlet Laplacian, its eigenvalues are (the squares of) the eigenfrequencies of the membrane, and the associated eigenfunctions correspond to the stable membrane profiles. Eigenvalues, eigenfunctions, and their generalizations (e.g., scattering poles) play a central role in the analysis of linear acoustic, electromagnetic, and quantum-mechanical systems and in the studies of nonlinear wave equations by means of the linearization method.
This section is devoted to the spectral analysis of differential operators that appear in the study of linear and nonlinear wave phenomena, in particular, Laplacians on manifolds, generators of semigroups for Dirac, Schrödinger, and elastic wave equations, as well as linearized operators of the bifurcation analysis. A special emphasis is put on the contemporary applications of the spectral theory in mathematical physics, photonics, and material science that involve homogenization methods, PDEs on graphs, dissipative boundary conditions on non-smooth interfaces, and optimization problems for eigenvalues.
A particular attention will be paid to the state of art in the modern branches of the theory of partial differential operators, such as the theory of boundary triples and random spectral theory.