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Title: The spacing distributions of arithmetical integrable systems
Author: Greenman, Chris
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 1995
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The level spacing distribution of the two dimensional harmonic oscillator is investigated. By obtaining an explicit expression for the spacings, it is observed that the distribution is unstable under the semi-classical limit h → 0. By defining a suitable average, a distribution stable under h → 0 is obtained. Exact expressions are obtained for values of oscillator frequency ratio including the golden mean, 1/2, 1/5 and 1/e. Comparisons are made between these analytic results and the numerical ones in the paper of Berry and Tabor [1]. For a certain class of ratio, including the case 1/e, a delta function is found for the averaged spacing distribution. This is a fractal set shown to have Hausdorff dimension ½ as a subset of possible ratios. The case of generic frequency ratio is also studied for which a closed formula is found. Comments on the distribution follow. For the harmonic oscillator of general dimension n it is shown that the initial value of the level spacing distribution is (n)-1. Reasons for the conjecture that the distribution will be (monotonically) decreasing are also given. By employing a method used in the system above, it is shown for the particle in a two dimensional box, with certain possible box dimensions, that the spacing distribution is distinct from the Poisson one associated with the system.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available