Exercises and Problems in Mathematical Methods of Physics

Giampaolo Cicogna worked at the University of Pisa in Italy from 1966 to 2012, as Assistant Professor of Geometry for Physicists and of Complementary Mathematics for Engineers (1966-80) and as Associated Professor of Mathematical Methods of Physics (1967-2012). He joined the university's staff shortly after graduating in Physics "magna cum laude" from the Department of Physics of Pisa University in 1964. He worked as a collaborator with the Italian National Institute for Nuclear Physics (INFN) until 2015. Following his retirement, he continues to collaborate in research and didactic activities. Author of more than 120 publications in international scientific journals, mainly addressing the mathematical aspects (with special emphasis on the role of symmetries) of several fields of physics, he has acted as a referee for various journals and as a reviewer for academic and research institutions and for CINECA, a nonprofit consortium comprising Italian universities and the Italian Ministry of Education, Universities and Research (MIUR).

1 Hilbert spaces.- 1.1 Complete sets, Fourier expansions.- 1.1.1 Preliminary notions. Subspaces. Complete sets.- 1.1.2 Fourier expansions.- 1.1.3 Harmonic functions; Dirichlet and Neumann Problems.- 1.2 Linear operators.- 1.2.1 Linear operators defined giving T en = vn, and related Problems.- 1.2.2 Operators of the form T x = v(w;x) and T x = ån vn(wn;x).- 1.2.3 Operators of the form T f (x) = j(x) f (x).- 1.2.4 Problems involving differential operators.- 1.2.5 Functionals.- 1.2.6 Time evolution Problems. Heat equation.- 1.2.7 Miscellaneous Problems.- 2 Functions of a complex variable.- 2.1 Basic properties of analytic functions.- 2.2 Evaluation of integrals by complex variable methods.- 2.3 Harmonic functions and conformal mappings.- 3 Fourier and Laplace transforms. Distributions.- 3.1 Fourier transform in L1(R) and L2(R).- 3.1.1 Basic properties and applications.- 3.1.2 Fourier transform and linear operators in L2(R).- 3.2 Tempered distributions and Fourier transforms.- 3.2.1 General properties.- 3.2.2 Fourier transform, distributions and linear operators.- 3.2.3 Applications to ODE's and related Green functions.- 3.2.4 Applications to general linear systems and Green functions.- 3.2.5 Applications to PDE's.- 3.3 Laplace transforms.- vvi Contents.- Groups, Lie algebras, symmetries in physics.- 4.1 Basic properties of groups and representations.- 4.2 Lie groups and algebras.- 4.3 The groups SO3; SU2; SU3.- 4.4 Other direct applications of symmetries to physics.- Answers and Solutions.