Quantum / CONCEPT

Zeros of the end amplitude numerator of a seven-site chain

For natural numbers a < b < c with a and c odd, b even and gcd(a, b, c) = 1, the cosine sum N(t) = (b^2 - a^2)(c^2 - a^2)(c^2 - b^2) + b^2 c^2 (c^2 - b^2) cos(at) + a^2 c^2 (c^2 - a^2) cos(bt) + a^2 b^2 (b^2 - a^2) cos(ct) has a zero in the open interval (0, pi) if and only if a is not 1. For a = 1 it is positive on [0, pi); for a >= 3 it takes a negative value in (0, pi).