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Type of Document	Dissertation
Author	Jaksic, Vojkan
URN	etd-09122005-162352
Persistent URL	http://resolver.caltech.edu/CaltechETD:etd-09122005-162352
Title	Solutions to some problems in mathematical physics
Degree	PhD
Option	Mathematics
Advisory Committee	Advisor Name Title Barry Simon Committee Chair
Keywords	none
Date of Defense	1991-06-20
Availability	restricted
Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. 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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