Disorder Effects on Quantum Spectra and Dynamics

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Disorder Effects on Quantum Spectra and Dynamics Abstract of Proposed ResearchMichael AizenmanThe research will address the effects of disorder on spectra and dynamics of operators which play a role in the dynamics of quantum systems. The incorporation of even weak but extensive disorder is known to result in delocalization of at least some of the normal modes, and in the creation of pure-point spectral regimes, characterized by a dense collection of eigenvalues corresponding to well separated eigenfunctions. The focus of the proposed research is on the less understood question of the existence of extended states in the presence of disorder, and the corresponding Lebesgue absolutely continuous spectra. Also to be investigated is a possible relation of the spectral gap statistics of local operators incorporating disorder with the known eigenvalue statistics of random matrix ensembles. As the effects of randomness are dimension dependent, the initial goal is to clarify some of the issues in high dimensions, where loop effects are in general more controllable. In the converse direction, a new -fluctuation based- method is envisioned for the derivation and analysis of localization in low dimensions, where the effect is most drastic.Disorder effects play a key role in the conduction properties of quantum systems modeled by random operators, and are of direct relevance for quantum networks. The work will capitalize on recent progress made by the PI with collaborators, and on his past work on the Anderson localization and on other effects of fluctuations, such as the rounding of fist-order phase transitions in low dimensions, in what is known as the Imry-Ma effect. The subject provides the meeting grounds for techniques involving mathematical analysis with ideas drawn from statistical mechanics, where science is made of disorder. The challenge of shedding light on issues of physics enriches also fields of mathematics.