Jan-Erik Björk: Mathematics by Carleman

The lecture presents some contributions by Torsten Carleman who served as director of Institute Mittag Leffler from 1927 until 1949. Special attention is given to his results from 1923 about unbounded self-adjoint operators on Hilbert spaces. An example is the Laplace operator in three dimensions plus a real-valued locally square integrable potential. One of Carleman's results assert that the operator is selfadjoint under the condition that the potential is upper bounded in a neighborhood of infinity.

Another topic in Carleman's work deals with asympotics for spectral values of specific PDE:s. Consider for example a bounded Dirichlet domain in two dimensions. The eigenfunctions of the Dirichlet Laplacian form an orthonormal basis. For any internal point the average value of squared values of eigenfunctions converge to the inverse area of the domain. The proof relies heavily upon analytic function theory using certain integrals similiar to those employed by Riemann in this study of the famous Z-function.