Novel Phenomena in Dilute Electron Systems in Two Dimensions

We review recent experiments that provide evidence for a transition to a conducting phase in two dimensions at very low electron densities. The nature of this phase is not understood, and is currently the focus of intense theoretical and experimental attention.Comment: To appear as a Perspective in the Proceedings of the National Academy of Sciences. Reference to Chakravarty, Kivelson, Nayak, and Voelker's paper added (Phil. Mag., in press

Quaternionic factorization of the Schroedinger operator and its applications to some first order systems of mathematical physics

We consider the following first order systems of mathematical physics. 1.The Dirac equation with scalar potential. 2.The Dirac equation with electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The system describing non-linear force free magnetic fields or Beltrami fields with nonconstant proportionality factor. 5.The Maxwell equations for slowly changing media. 6.The static Maxwell system. We show that all this variety of first order systems reduces to a single quaternionic equation the analysis of which in its turn reduces to the solution of a Schroedinger equation with biquaternionic potential. In some important situations the biquaternionic potential can be diagonalized and converted into scalar potentials

Wave polynomials, transmutations and Cauchy's problem for the Klein-Gordon equation

We prove a completeness result for a class of polynomial solutions of the wave equation called wave polynomials and construct generalized wave polynomials, solutions of the Klein-Gordon equation with a variable coefficient. Using the transmutation (transformation) operators and their recently discovered mapping properties we prove the completeness of the generalized wave polynomials and use them for an explicit construction of the solution of the Cauchy problem for the Klein-Gordon equation. Based on this result we develop a numerical method for solving the Cauchy problem and test its performance.Comment: 31 pages, 8 figures (16 graphs