Pl. Overfelt, RINGS, QUADRATIC-FORMS, AND COMPLETE DEGENERACY FOR A SUBCLASS OF HIGHLY OVERMODED RECTANGULAR WAVE-GUIDES, Journal of mathematical physics, 34(7), 1993, pp. 2975-2989
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.