We are interested in rigorous asymptotic results pertaining to three different differential equations which lie in the Heun class (see [1] §31). The Heun class contains those ordinary linear second-order differential equations with four regular singularities. We first investigate the Lamé equation d²w/dz² +(h − v(v + 1)k² sn²(z, k)) w = 0, z ∈ [−K,K], where 0 < k < 1, sn(z, k) is a Jacobi elliptic function, and K = ∫ 1 0 dz/√(1 − z²)(1 − k²z²) is the complete elliptic integral of the first kind. We obtain rigorous uniform asymptotic approximations complete with error bounds for the Lamé functions Ecm/v (z, k²) and Esm/v+¹ (z, k²) for z ∈ [0,K] and m ∈ N0, and rigorous approximations for their respective eigenvalues am/v and bm/v+¹, as v →∞. Then we obtain asymptotic expansions for the Lamé functions complete with error bounds, which hold only in a shrinking neighbourhood of the origin as v →∞. We also find corresponding expansions for the eigenvalues complete with order estimates for the errors. Then finally we give rigorous result for the exponentially small difference between the eigenvalues bm/v+¹ and am/v as v →∞. Second we investigate Mathieu’s equation d²w/dz² + (λ − 2h² cos 2z) w = 0, z ∈ [0, π], and obtain analogous results for the Mathieu functions cem(h, z) and sem+1(h, z) and their corresponding eigenvalues am and bm+1 for m ∈ N0 as h→∞, which are derived from those of Lamé ’s equation by considering a limiting case. Lastly we investigate the spheroidal wave equation d /dz ((1 − z²) dw/dz) + ( λ+ γ²(1 − z²) − μ²/1 − z²) w = 0, z ∈ [−1, 1], and consider separately the cases where γ² > 0 and γ² < 0. In the first case we give similar results to those previously for the prolate spheroidal wave functions Ps(z, γ²) and their corresponding eigenvalues λm/n for m, n ∈ N0 and n ≥ m as γ² →∞, and in the second we discuss the gap in theory which makes it difficult to obtain rigorous results as γ² → -∞, and how one would bridge this gap to obtain these.

We use the methods of exponential asymptotics to study the solutions of a one dimensional wave equation with a non-constant wave speed c(x,t) modelling, for example, a slowly varying spatio-temporal topography. The equation reads htt(x,t) = (c2(x,t)hx(x,t))x' (1) where the subscripts denote differentiation w.r.t. the parameters x and t respectively. We focus on the exponentially small reflected wave that appears as a result of a Stokes phenomenon associated with the complex singularities of the speed. This part of the solution is not captured by the standard WKBJ (geometric optics) approach. We first revisit the time-independent propagation problem using resurgent analysis. Our results recover those obtained using Meyers integral-equation approach or the Kruskal-Segur (K-S) method. We then consider the time-dependent propagation of a wavepacket, assuming increasingly general models for the wave speed: time independent, c(x), and separable, c1(x)c2(t). We also discuss the situation when the wave speed is an arbitrary function, c(x,t), with the caveat that the analysis of this setup has yet to be completed. We propose several methods for the computation of the reflected wavepacket. An integral transform method, using the Dunford integral, provides the solution in the time independent case. A second method exploits resurgence: we calculate the Stokes multiplier by inspecting the late terms of the dominant asymptotic expansion. In addition, we explore the benefits of an integral transform that relates the coefficients of the dominant solution in the time-dependent problem to the coefficients of the dominant solution in the time-independent problem. A third method is a partial differential equation extension of the K-S complex matching approach, containing details of resurgent analysis. We confirm our asymptotic predictions against results obtained from numerical integration.

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Thesis (Ph. D.)--Lehigh University, 2000. / Includes vita. Includes bibliographical references (leaves 142-147).

We study the behavior of truncated Rayleigh-Schröodinger series for the low-lying eigenvalues of the time-independent Schröodinger equation, when the Planck's constant is considered in the semiclassical limit. Under certain hypotheses on the potential energy, we prove that, for any given small value of the Planck's constant, there is an optimal truncation of the series for the approximate eigenvalues, such that the difference between an approximate and actual eigenvalue is smaller than an exponentially small function of the Planck's constant. We also prove the analogous results concerning the eigenfunctions. / Ph. D.

This thesis investigates the small time asymptotics of solutions of stochastic equations in infinite dimensions. In this abstract H denotes a separable Hilbert space, A denotes a linear operator on H generating a strongly continuous semigroup and (W(t))t???0 denotes a separable Hilbert space-valued Wiener process. In chapter 2 we consider the mild solution (Xx(t))t???[0,1] of a stochastic initial value problem dX = AX dt + dW t ??? (0, 1] X(0) = x ??? H , where the equation has an invariant measure ??. Under some conditions L(Xx(t)) has a density k(t, x, ??) with respect to ?? and we can find the limit limt???0 t ln k(t, x, y). For infinite dimensional H this limit only provides the lower bound of a large deviation principle (LDP) for the family of continuous trajectory-valued random variables { t ??? [0, 1] ??? Xx(??t) : ?? ??? (0, 1]}. In each of chapters 3, 4 and 5 we find an LDP which describes the small time asymptotics of the continuous trajectories of the solution of a stochastic initial value problem. A crucial role is played by the LDP associated with the Gaussian trajectory-valued random variable of the noise. Chapter 3 considers the initial value problem dX(t) = (AX(t) + F(t,X(t))) dt + G(X(t)) dW(t) t ??? (0, 1] X(0) = x ??? H, where drift function F(t, ??) is Lipschitz continuous on H uniformly in t ??? [0, 1] and diffusion function G is Lipschitz continuous, taking values that are Hilbert-Schmidt operators. Chapter 4 considers an equation with dissipative drift function F defined on a separable Banach space continuously embedded in H; the solution has continuous trajectories in the Banach space. Chapter 5 considers a linear initial value problem with fractional Brownian motion noise. In chapter 6 we return to equations with Wiener process noise and find a lower bound for liminft???0 t ln P{X(0) ??? B,X(t) ??? C} for arbitrary L(X(0)) and Borel subsets B and C of H. We also obtain an upper bound for limsupt???0 t ln P{X(0) ??? B,X(t) ??? C} when the equation has an invariant measure ??, L(X(0)) is absolutely continuous with respect to ?? and the transition semigroup is holomorphic.

<p>Stochastic microlensing is a central tool in probing dark matter on galactic scales. From first principles, we initiate the development of a mathematical theory </p><p>of stochastic microlensing. We first construct a natural probability space for stochastic microlensing and characterize the general behaviour of the random time </p><p>delay functions' random critical sets. Next we study stochastic microlensing in two distinct random microlensing scenarios: The uniform stars' distribution with</p><p> constant mass spectrum and the spatial stars' distribution with general mass spectrum. For each scenario, we determine exact and asymptotic (in the large number</p><p> of point masses limit) stochastic properties of the random time delay functions and associated random lensing maps and random shear tensors, including their </p><p>moments and asymptotic density functions. We use these results to study certain random observables, such as random fixed lensed images, random bending angles, </p><p>and random magnifications. These results are relevant to the theory of random </p><p>fields and provide a platform for further generalizations as well as analytical limits for checking astrophysical studies of stochastic microlensing.</p><p>Continuing our development of a mathematical theory of stochastic microlensing, we study the stochastic version of the Image Counting Problem, first considered </p><p>in the non-random setting by Einstein and generalized by Petters. In particular, we employ the Kac-Rice formula and Morse theory to deduce general formulas for </p><p>the expected total number of images and the expected number of saddle images for a general random lensing scenario. We further </p><p>generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of</p><p> global expected number of positive parity images due to a general lensing map. Applying the result to the uniform stars' distribution random microlensing </p><p>scenario, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars. This global expectation is bounded, </p><p>while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars.</p><p>Finally, we outline a framework for the study of stochastic microlensing in the neighbourhood of lensed images. This framework is related to the study of the </p><p>local geometry of a random surface. In our case, the surface is non-Gaussian, and therefore standard literature on the subject does not apply. We explore the case</p><p> of a random gravitational field caused by a random star.</p> / Dissertation

We study the leading order asymptotics of a Random Matrix theory partition function related to colored triangulations. This partition function comes from a three Hermitian matrix model that has been introduced in the physics literature. We provide a detailed and precise description of the combinatorial objects that the partition function counts that has not appeared previously in the literature. We also provide a general framework for studying the leading order asymptotics of an N dimensional integral that one encounters studying the partition function of colored triangulations. The results are obtained by generalizing well know results for integrals coming from Hermitian matrix models with only one matrix that give the leading order asymptiotics in terms of a finite dimensional variational problem. We apply these results to the partition function for colored triangulations to show that the minimizing density of the variational problem is unique, and agrees with the one proposed in the physics literature. This provides the first complete mathematically rigorous description of the leading order asymptotics of this matrix model for colored triangulations.

This thesis investigates the small time asymptotics of solutions of stochastic equations in infinite dimensions. In this abstract H denotes a separable Hilbert space, A denotes a linear operator on H generating a strongly continuous semigroup and (W(t))t???0 denotes a separable Hilbert space-valued Wiener process. In chapter 2 we consider the mild solution (Xx(t))t???[0,1] of a stochastic initial value problem dX = AX dt + dW t ??? (0, 1] X(0) = x ??? H , where the equation has an invariant measure ??. Under some conditions L(Xx(t)) has a density k(t, x, ??) with respect to ?? and we can find the limit limt???0 t ln k(t, x, y). For infinite dimensional H this limit only provides the lower bound of a large deviation principle (LDP) for the family of continuous trajectory-valued random variables { t ??? [0, 1] ??? Xx(??t) : ?? ??? (0, 1]}. In each of chapters 3, 4 and 5 we find an LDP which describes the small time asymptotics of the continuous trajectories of the solution of a stochastic initial value problem. A crucial role is played by the LDP associated with the Gaussian trajectory-valued random variable of the noise. Chapter 3 considers the initial value problem dX(t) = (AX(t) + F(t,X(t))) dt + G(X(t)) dW(t) t ??? (0, 1] X(0) = x ??? H, where drift function F(t, ??) is Lipschitz continuous on H uniformly in t ??? [0, 1] and diffusion function G is Lipschitz continuous, taking values that are Hilbert-Schmidt operators. Chapter 4 considers an equation with dissipative drift function F defined on a separable Banach space continuously embedded in H; the solution has continuous trajectories in the Banach space. Chapter 5 considers a linear initial value problem with fractional Brownian motion noise. In chapter 6 we return to equations with Wiener process noise and find a lower bound for liminft???0 t ln P{X(0) ??? B,X(t) ??? C} for arbitrary L(X(0)) and Borel subsets B and C of H. We also obtain an upper bound for limsupt???0 t ln P{X(0) ??? B,X(t) ??? C} when the equation has an invariant measure ??, L(X(0)) is absolutely continuous with respect to ?? and the transition semigroup is holomorphic.

No description available.