Spectral theory of differential operators on graphs

Student Number : 9804032J - PhD thesis - School of Mathematics - Faculty of ScienceThe focus of this thesis is the spectral structure of second order self-adjoint differential operators on graphs. Various function spaces on graphs are defined and we define, in terms of both differential systems and the afore noted function spaces, boundary value problems on graphs. A boundary value problem on a graph is shown to be spectrally equivalent to a system with separated boundary conditions. An example is provided to illustrate the fact that, for Sturm-Liouville operators on graphs, self-adjointness does not necessarily imply regularity. We also show that since the differential operators considered are self-adjoint the algebraic and geometric eigenvalue multiplicities are equal. Asymptotic bounds for the eigenvalues are found using matrix Pr¨ufer angle methods. Techniques common in the area of elliptic partial differential equations are used to give a variational formulation for boundary value problems on graphs. This enables us to formulate an analogue of Dirichlet-Neumann bracketing for boundary value problems on graphs as well as to establish a min-max principle. This eigenvalue bracketing gives rise to eigenvalue asymptotics and consequently eigenfunction asymptotics. Asymptotic approximations to the Green’s functions of Sturm-Liouville boundary value problems on graphs are obtained. These approximations are used to study the regularized trace of the differential operators associated with these boundary value problems. Inverse spectral problems for Sturm-Liouville boundary value problems on graphs resembling those considered in Halberg and Kramer, A generalization of the trace concept, Duke Math. J. 27 (1960), 607-617, for Sturm-Liouville problems, and Pielichowski, An inverse spectral problem for linear elliptic differential operators, Universitatis Iagellonicae Acta Mathematica XXVII (1988), 239-246, for elliptic boundary value problems, are solved. Boundary estimates for solutions of non-homogeneous boundary value problems on graphs are given. In particular, bounds for the norms of the boundary values of solutions to the non-homogeneous boundary value problem in terms of the norm of the non-homogeneity are obtained and the eigenparameter dependence of these bounds is studied. Inverse nodal problems on graphs are then considered. Eigenfunction and eigenvalue asymptotic approximations are used to provide an asymptotic expression for the spacing of nodal points on each edge of the graph from which the uniqueness of the potential, for given nodal data, is deduced. An explicit formula for the potential in terms of the nodal points and eigenvalues is given

The Coloured Drawing of the Dulle Griet in the Kunstpalast, Düsseldorf : New Findings on its Status and Dating

editorial reviewedThe Düsseldorf coloured drawing of the Dulle Griet was examined at the Royal Institute for Cultural Heritage (KIK-IRPA) in Brussels in collaboration with the Kunstpalast, Düsseldorf, using infrared reflectography, infrared photography, transmitted light photography, macro-XRF and micro-Raman spectroscopy. The study of the paper revealed the watermark of Niklaus Heusler, a Basel paper manufacturer, giving 1578 as the earliest possible execution date for the drawing, and thus confirming its status as a copy. Examination and analysis of the colours suggest that they are most likely original. Various hypotheses are proposed as to the drawing’s original function, attribution and context, taking into account the early history of Bruegel’s Dulle Griet. Thanks to its early dating, the drawing is a key witness to the appearance of the Dulle Griet painting at an early stage in its history, and as such it provided fundamental help during the conservation campaign. Certain colours and motifs that have altered over time in the painting remain more legible in the drawing