RUI: Spectral Properties of Some Random Schrodinger Operators

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Abstract Shubin The research will be concerned with problems from mathematical physics. Several of the proposed problems center around the spectral theory of random Schrodinger operators on trees or graphs or in the continuum. A problem from the study of integral equations which has applications to models from physics is also discussed. Some of the problems to be investigated arise from the field of solid state physics. To form a rough idea, start by imagining the motion of a particle such as an electron in a crystal with a perfectly regular structure. It is well-known that an electron, with a certain energy (in a band), will travel through the crystal. In other words, the crystal acts as a conductor. Consider a crystal with impurities. Such a system is called disordered. Will this disordered crystal also act as a conductor? In order to begin to answer this question, a precise definition of disorder needs to be given. However many types of disordered crystals are not conductors; rather they are insulators. Since the seminal work of Nobel Prize winner P. Anderson, disordered systems have been the object of intense mathematical investigations. Although many questions have been answered in the past forty years, many more problems remain open. In particular, disordered systems with space dimension greater than one, are not well understood.