Volume 21

Exploring Spectral Distribution for Schrödinger Operators on Finite and Infinite Intervals

Christopher J. Winfield
Published October 31, 2019

We study the distribution of eigenspectra for operators of the form -y''+q(x)y with self-adjoint boundary conditions on both bounded and unbounded interval domains. With integrable potentials q, we explore computational methods for calculating spectral density functions involving cases of discrete and continuous spectra where discrete eigenvalue distributions approach a continuous limit as the domain becomes unbounded. We develop methods from classic texts in ODE analysis and spectral theory in a concrete, visually oriented way as a supplement to introductory literature on spectral analysis. As a main result of this study, we develop a routine for computing eigenvalues as an alternative to , resulting in fast approximations to implement in our demonstrations of spectral distribution. Read More »

H. S. M. Coxeter, George Beck
Published August 4, 2019

H. S. M. Coxeter wrote several geometry film scripts that were produced between 1965 and 1971 [1]. In 1992, Coxeter gave George Beck mimeographs of two scripts that had not been made. Beck wrote Mathematica code for the stills and animations. This material was added to the third edition of Coxeter’s The Real Projective Plane [2]. This article updates the Mathematica code. Read More »

C. Christopher Reed, Alvar M. Kabe
Published April 19, 2019
A comprehensive discussion is presented of the closed-form solutions for the responses of single-degree-of-freedom systems subject to swept-frequency harmonic excitation. The closed-form solutions for linear and octave swept-frequency excitation are presented and these are compared to results obtained by direct numerical integration of the equations of motion. Included is an in-depth discussion of the numerical difficulties associated with the complex error functions and incomplete gamma functions, which are part of the closed-form solutions, and how these difficulties were overcome by employing exact arithmetic. The closed-form solutions allowed the in-depth study of several interesting phenomena. These include the scalloped behavior of the peak response (with multiple discontinuities in the derivative), the significant attenuation of the peak response if the sweep frequency is started at frequencies near or above the natural frequency, and the fact that the swept-excitation response could exceed the steady-state harmonic response. Read More »