Research on Problems Related to Random Schrodinger Operators

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The aim of this project is to study problems related to random Schrodinger operators from various viewpoints. The main problem is related to the Anderson transition. For this, we gave a Wegner type estimate for a class of random Schrodinger operators with electromagnetic potentials given by Gaussian random fields, and proved the Anderson localization at sufficiently low energy for these operators. This is the first proof of the strong dynamical localization for Gaussian random potentials. For the proof, we extend Klopp's vector field method for models with lattice structures to our models without lattice structures and extend Germinet-Klein theory on the Anderson localization for operators bounded below to our models unbounded below. On the other hand, the other attractive problem is to study the asymptotic behavior of the integrated density of states. For this, we study that of the Pauli Hamiltonian with a random magnetic field whose mean is zero, and we gave a lower bound on a neighborhood of zero in terms of logarithmic functions in a basic case where the magnetic field is uniform in one direction. This shows that our integrated density of states increases rapidly at zero and our Pauli Hamiltonian has many states whose energies are close to zero. This phenomenon is contrary to the phenomenon known as Lifschitz tail for other random Schrodinger operators where the integrated density of states increases slowly. The speed of the increment we showed is the fast one in the known results except for the case that the integrated density of states has a jump. We showed this by applying the Aharonov-Casher theory rigorously.