Solutions to some problems in mathematical physics

Citation

Jaksic, Vojkan (1992) Solutions to some problems in mathematical physics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/k4n5-at08. https://resolver.caltech.edu/CaltechETD:etd-09122005-162352

Abstract

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In Part I, we study the adiabatic limit for Hamiltonians with certain complex-analytic dependence on the time variable. We show that the transition probability from a spectral band that is separated by gaps is exponentially small in the adiabatic parameter. We find sufficient conditions for the Landau-Zener formula, and its generalization to nondiscrete spectrum, to bound the transition probability.

Part II is concerned with eigenvalue asymptotics of a Neumann Laplacian [...] in unbounded regions [...] of [...] with cusps at infinity (a typical example is [...]. We prove that [...], where [...] is the canonical, one-dimensional Schrodinger operator associated with the problem. We also establish a similar formula for manifolds with cusps and derive the eigenvalue asymptotics of a Dirichlet Laplacian [...] for a class of cusp-type regions of infinite volume.

In Part III we study the spectral properties of random discrete Schrodinger operators [...], of the form [...], acting on [...], where [...] are independent random variables uniformly distributed on [0, 1]. We show, for typical [...], that [...], has a discrete spectrum if [...], and we calculate its eigenvalue asymptotics. If [...] for positive integer k, we prove that for typical [...] and non-random strictly decreasing sequence [...], [...]. The large k asymptotic of sequence [...] is studied. We also investigate the continuous analog of the above model.

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