Continuous and Discrete Spectra

This chapter explores the spectra of audio signals first from a continuous time and continuous frequency perspective. It starts by reviewing Fourier's theorem and then moves on to put it into its most general form, the Fourier transform. The spectra of simple signals are explored and determined. The operation of convolution is introduced, and through its discrete form, it is applied to the concepts of spectrum and waveform as mediated by the Fourier transform. The chapter is completed with a study of the discrete spectra of classic synthesis waveforms. A revised notion of spectrum is presented at the conclusion.

Special Orthogonal Polynomials in Quantum Mechanics

Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. One of these is a four-parameter polynomial with a discrete spectrum. Another that appeared while solving a Heun-type equation has a mix of continuous and discrete spectra. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of these polynomials are defined only by their three-term recursion relations and initial values. However, their other properties like the weight function, generating function, orthogonality, Rodrigues-type formula, etc. are yet to be derived analytically. This is an open problem in orthogonal polynomials.

SUPERSYMMETRIC YANG-MILLS QUANTUM MECHANICS IN VARIOUS DIMENSIONS

Recent analytical and numerical solutions of above systems are reviewed. Discussed results include: a) exact construction of the supersymmetric vacua in two space-time dimensions, and b) precise numerical calculations of the coexisting, continuous and discrete, spectra in the four-dimensional system, together with the identification of dynamical supermultiplets and SUSY vacua. New construction of the gluinoless SO (9) singlet state, which is vastly different from the empty state, in the ten-dimensional model is also briefly summarized.

General stability analysis of force-free magnetic fields. Part 2. Application to continuous and discrete spectra of α

Theorems and criteria on stability of force-free fields, developed in part 1 (Krüger 1976) are illustrated for continuous, as well as discontinuous, spectra of α. Previous and new results are understood from a unified viewpoint.