Solutions and Spectrum of Schrodinger Operators

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Proposal: DMS-9801530 Principal Investigator: Alexander Kiselev Abstract: The research will focus mainly on two objectives. The first is to study the general relationships between the behavior of solutions of the generalized eigenfunction equation, spectral properties and quantum dynamics of Schroedinger (and more generally elliptic) operators on infinite domains in any dimension. The second objective is the investigation of spectral and dynamical properties of concrete quantum mechanical systems. The goal is the development of a new spectral analysis technique for studying fine structure in the essential spectra of Schroedinger operators (that is, the decomposition of the essential spectrum into absolutely continuous, singular continuous and pure point components). In addition to spectral information, the new methods may yield valuable results about quantum dynamics of various systems via new relations between the behavior of solutions of generalized eigenfunction equation, spectral measures, and quantum dynamics. This technique may have potential applications to and will provide further insight into many important problems, in particular such as models of quasicrystals, the Anderson model and other models involving random or ergodic potentials, and Schroedinger operators with slowly decaying potentials. The spectral and dynamical theory of Schroedinger operators is the cornerstone of Quantum Mechanics. This theory describes the laws which govern the behavior of quantum particles, such as electrons, atoms and molecules. Much of the fundamental scientific knowledge about many important physical processes (such as, for example, chemical reactions or conduction properties of various materials) comes from the theory of Schroedinger operators. This proposal focuses on the development of the new methods in spectral and dynamical theory of Schroedinger operators which may allow a new approach to some long-standing problems in Quantum Mechanics. These problems concern, in particular, the cond uctance properties of materials with impurities and of quasicrystals, and have direct applications to modern engineering devices, wave guides and transistors to name two.