Srdečně vás zveme na poslední letošní MAFIA Seminar na KM FJFI. V příštím týdnu se konají hned dva: 10. 12. a 12. 12., vždy od 13.15 hodin v místnosti T112.
10. 12. 2019: Orif Ibrogimov (ČVUT v Praze) - Sharp spectral inequalities for complex perturbations of the indefinite Laplacian.
Abstract:
We discuss quantitative bounds for eigenvalues of complex perturbations of the one-dimensional indeffinite Laplacian. For integrable potentials, we obtain optimal spectral enclosures which accommodate also embedded eigenvalues. This is a substantial improvement of a recent result due to Behrndt and Trunk. Moreover, we discuss analogous questions for complex-valued Lp-potentials(1 < p < ∞) and establish quantitative estimates which considerably improve existing results even for real-valued potentials. Weak coupling asymptotics and a connection to the well-known conjecture of Behrndt will also be addressed.
12. 12. 2019: Borbala Gerhat (University of Bern / Queen's University Belfast)-Schur complement dominant operator matrices.
Abstract:
In mathematical physics, matrix (di erential) operators arise naturally in applications as coupled systems of partial di erential equations. Up to now, the spectral analysis of such problems has typically been tackled by means of perturbation theory. We propose to view operator matrices in a more general setting, which allows our results to fully abstain from any perturbative argument.
Rather than requiring the matrix to act in a Hilbert space H, we extend its action to a suitable distributional triple D H D0 and restrict it to its maximal domain in H. The crucial point in our approach is the choice of the spaces D and D0 which are essentially determined by the Schur complement of the matrix. We show spectral equivalence between the resulting operator matrix in H and its Schur complement, eventually implying closedness and non-empty resolvent set of the matrix. Finally, we apply our abstract results to the damped wave equation with possibly unbounded and singular damping, as well as to more general second order matrix di erential operators with singular coecients. By means of our methods, the previously used regularity assumptions can be lowered substantially in both cases.