spectral analysis; non-self-adjoint operators; Riesz basis; damped wave equation; Dirac operator; perturbation theory
Cuenin Jean-Claude, Siegl Petr (2018), Eigenvalues of one-dimensional non-self-adjoint Dirac operators and applications, in Letters in Mathematical Physics
Bögli Sabine, Siegl Petr, Tretter Christiane (2017), Approximations of spectra of Schrödinger operators with complex potentials on ℝ d, in Communications in Partial Differential Equations
, 42(7), 1001-1041.
Mityagin Boris, Siegl Petr, Viola Joseph (2017), Differential operators admitting various rates of spectral projection growth, in Journal of Functional Analysis
, 272, 3129-3175.
Krejčiřík David, Raymond Nicolas, Royer Julien, Siegl Petr (2017), Non-accretive Schrödinger operators and exponential decay of their eigenfunctions, in Israel Journal of Mathematics
, 221, 779-802.
Lotoreichik Vladimir, Siegl Petr (2017), Spectra of definite type in waveguide models, in Proceedings of the American Mathematical Society
, 145, 1231-1246.
Siegl Petr, Štampach František (2017), Spectral analysis of non-self-adjoint Jacobi operator associated with Jacobian elliptic functions, in Operators and Matrices
, 11, 901-928.
Ibrogimov O.O., Siegl P., Tretter C. (2016), Analysis of the essential spectrum of singular matrix differential operators, in Journal of Differential Equations
, 260(4), 3881-3926.
Dohnal Tomáš, Siegl Petr (2016), Bifurcation of eigenvalues in nonlinear problems with antilinear symmetry, in Journal of Mathematical Physics
, 57, 093502.
Siegl Petr, Štampach František (2016), On extremal properties of Jacobian elliptic functions with complex modulus, in Journal of Mathematical Analysis and Applications
, 442(2), 627-641.
Boris Mityagin, Petr Siegl, Local form-subordination condition and Riesz basisness of root systems, in Journal d'Analyse Mathématique
Krejcirik David, Raymond Nicolas, Royer Julien, Siegl Petr, Reduction of dimension as a consequence of norm-resolvent convergence and applications, in Mathematika
The spectral analysis of non-self-adjoint operators is a very active and rapidly developing field. The steadily growing interest in this area originates in the discovery of new phenomena being very different from the self-adjoint case. These new effects have been observed in mathematical studies as well as in more applied investigations in physics and numerical analysis and have many dramatic consequences, e.g. for the (non-)reliability of numerical approximations. While self-adjoint operators are intrinsically connected with quantum mechanics, non-self-adjoint problems appear in classical branches of physics as hydro- and magnetohydrodynamics, damped systems, quantum resonances, but also in much more recently emerging areas such as superconductivity, optics, and graphene or graphene-like structures. In spite of the rapid development of the latter on experimental side e.g. balanced loss/gain materials or active photonic honeycomb lattices simulating the graphene physics, the corresponding mathematical foundations are still in their infancy. One of the main reasons is that, unlike self-adjoint theory, the analysis of non-self-adjoint problems is much less developed and fragmented, mainly comprising a collection of diverse advanced methods; the pseudospectral analysis is a typical example. This difference originates in the lack of the most powerful self-adjoint tools like spectral theorem and variational principles. Thus new and more general non-self-adjoint methods must be developed. The expected outcomes of the project are of importance to applications since the analytic results on spectra, pseudospectra or basis properties will put numerical or non-rigorous results on solid grounds. The project has the following topical structure:A Non-self-adjoint matrix Schrödinger and Dirac operators,B Damped wave equation with singular damping,C Transition between spectrum and pseudospectrum,D Applications (graphene waveguides, active photonic lattices, Bose-Einstein condensates).