Spectral Analysis

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9707661 Simon The spectral analysis of Schrodinger operators with slowly decaying or ergodic potentials will be studied. Making use of recent results of Last-Simon, Kiselev-Last-Simon, and Jitomirskaya-Last, the project examines fractal properties of spectral measures and related issues of long time for quantum dynamics. A major thrust will be to attempt to extend to general dimension the ideas, techniques, and results developed for one-dimensional systems. Among open questions which will be examined is the existence of mixed singular continuous and absolutely continuous spectra in explicit models. Spectral analysis of differential operators has been a major theme of twentieth century analysis and theoretical physics. Recent studies have shown that potentialities that were allowed by classical works of Weyl, Titchmarsh, Hartman, and Wintner not only occur, but are not pathological as was suggested by these classical works. Links were made with ideas of chaos science, particularly with notions of Hausdorff dimension. It is proposed to study these ideas further, to extend them from the model one space dimension of earlier works to physically relevant three dimensions, and to explore the consequences for the long-time dynamics of quantum systems.