Jaksic, Vojkan (1992) Solutions to some problems in mathematical physics. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-09122005-162352
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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.
|Item Type:||Thesis (Dissertation (Ph.D.))|
|Degree Grantor:||California Institute of Technology|
|Division:||Physics, Mathematics and Astronomy|
|Thesis Availability:||Restricted to Caltech community only|
|Defense Date:||20 June 1991|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Imported from ETD-db|
|Deposited On:||12 Sep 2005|
|Last Modified:||26 Dec 2012 03:00|
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