Theses/Dissertations - Mathematics

Orbit structures of homeomorphisms.

2012-11-29T16:22:23Z

Orbit structures of homeomorphisms. Sherman, Casey L. In this dissertation we answer the following question: If $X$ is a Cantor set and $T:X\to X$ is a homeomorphism, what possible orbit structures can $T$ have? The answer is given in terms of the orbit spectrum of $T$. If $X$ is a Cantor set, then there is a homeomorphism $T:X\to X$ with $\sigma(T)=(0,\zeta,\sigma_1,\sigma_2,\sigma_3,\dots)$ if and only if one of the following holds: \begin{enumerate} \item $\zeta=0$, there exists $k\in\N$ and a set $\{n_1,\dots,n_k\}$ with $\sigma_{n_i}>0$ for each $1\leq i\leq k$ such that if $\sigma_j>0$ then there exists $i\in\{1,2,\dots,k\}$ with $n_i|j$ and there is an $m\in\N$ with $\sigma_{mj}=\cont$, \item $1\leq\zeta<\cont$, $\{n:\sigma_n=\cont\}$ is infinite, and $\sum\{\sigma_n:\sigma_{mn}<\cont\text{ for all }m\in\N\}\leq\zeta$, or \item $\zeta=\cont$. \end{enumerate}

Spectral functions for generalized piston configurations.

2012-11-29T16:14:04Z

Spectral functions for generalized piston configurations. Morales-Almazán, Pedro Fernando. In this work we explore various piston configurations with different types of potentials. We analyze Laplace-type operators $P=-g^{ij}\nabla^E_i\nabla^E_j+V$ where $V$ is the potential. First we study delta potentials and rectangular potentials as examples of non-smooth potentials and find the spectral zeta functions for these piston configurations on manifolds $I\times\mathcal{N}$, where $I$ is an interval and $\mathcal{N}$ is a smooth compact Riemannian $d-1$ dimensional manifold. Then we consider the case of any smooth potential with a compact support and develop a method to find spectral functions by finding the asymptotic behavior of the characteristic function of the eigenvalues for $P$. By means of the spectral zeta function on these various configurations, we obtain the Casimir force and the one-loop effective action for these systems as the values at $s=-1/2$ and the derivative at $s=0$. Information about the heat kernel coefficients can also be found in the spectral zeta function in the form of residues, which provide an indirect way of finding this geometric information about the manifold and the operator.

Global SL(2,R) representations of the Schrödinger equation with time-dependent potentials.

2012-12-20T21:37:14Z

Global SL(2,R) representations of the Schrödinger equation with time-dependent potentials. Franco, Jose A. We study the representation theory of the solution space of the one-dimensional Schrödinger equation with time-dependent potentials that possess sl2-symmetry. We give explicit local intertwining maps to multiplier representations and show that the study of the solution space for potentials of the form V (t, x) = g2(t)x^2+g1(t)x+g0(t) reduces to the study of the potential free case. We also show that the study of the time-dependent potentials of the form V (t, x) = [lamba] x^-2+g2(t)x2+g0(t) reduces to the study of the potential V (t, x) = [lamba] x^-2. Therefore, we study the representation theory associated to solutions of the Schrödinger equation with this potential only. The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category.

Comparison of smallest eigenvalues and extremal points for third and fourth order three point boundary value problems.

2012-12-20T21:38:17Z

Comparison of smallest eigenvalues and extremal points for third and fourth order three point boundary value problems. Neugebauer, Jeffrey T. The theory of u₀-positive operators with respect to a cone in a Banach space is applied to the linear differential equations u⁽⁴⁾ + λ₁p(x)u = 0 and u⁽⁴⁾ + λ₂q(x)u = 0, 0 ≤ x ≤ 1, with each satisfying the boundary conditions u(0) = u′(r) = u″(r) = u‴(1) = 0, 0 < r < 1. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained. These results are then extended to the nth order problem using two different methods. One method involves finding the Green's function for –u⁽ⁿ⁾ = 0 satisfying the higher order boundary conditions, and the other involves making a substitution that allows us to work with a variation of the fourth order problem. Extremal points via Krein-Rutman theory are then found. Analogous results are then obtained for the eigenvalue problems u‴ + λ₁p(x)u = 0 and u‴ + λ₂q(x)u = 0, with each satisfying u(0) = u′(r) = u″(1) = 0, 0 < 1/2 < r < 1.