RINGS, QUADRATIC-FORMS, AND COMPLETE DEGENERACY FOR A SUBCLASS OF HIGHLY OVERMODED RECTANGULAR WAVE-GUIDES

In this article eigenvalue degeneracy formulas for a subclass of highl y overmoded rectangular waveguides are determined. This subclass consi sts of those rectangular geometries with side length ratio, a/b, equal to the square root of a rational integer, square-root sigma, sigma is -an-element-of Z, and contains an unusual amount of degeneracy even wh en the side length ratio is irrational. The complete degeneracy for me mbers of this subclass (sigma < 10) is determined by interpreting its normalized eigenvalue formula as (a) the norm of an associated ring of algebraic integers and (b) a special case of a positive definite bina ry integral quadratic form.