Citation
Jaksic, Vojkan (1992) Solutions to some problems in mathematical physics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/k4n5at08. https://resolver.caltech.edu/CaltechETD:etd09122005162352
Abstract
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In Part I, we study the adiabatic limit for Hamiltonians with certain complexanalytic 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 LandauZener 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, onedimensional 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 cusptype 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 nonrandom 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 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  20 June 1991 
Record Number:  CaltechETD:etd09122005162352 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd09122005162352 
DOI:  10.7907/k4n5at08 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  3494 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  12 Sep 2005 
Last Modified:  16 Apr 2021 23:06 
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