The Pythagorean Angle Lattice as a Quantum System

A geometric parameterization of the constant-potential discrete Schrödinger operator by Omni-Metallic numbers arising from the Pythagorean Angle Lattice. The Pythagorean Angle Lattice is a countable set of real numbers defined by L = { cot(arg(Z)/D) : Z = W² or iW², W ∈ Z[i]′_prim, D ∈ N }. It arises from primitive Pythagorean triples via Gaussian integer squares and cotangent normalization. This paper describes a connection between this lattice and a family of exactly solvable quantum systems. The lattice has a discrete angle spectrum θ_n = 180° − 2 arctan(n/2), n = 5, 6, 7, … For each n, the Omni-Metallic number ℧_{2,1}(n,4) = n + √(n² + 4) satisfies ℧_{2,1}(n,4) = 2 cot(θ_n / 4). We consider the constant-potential discrete Schrödinger operator (H_n ψ)_k = ψ_{k+1} + ψ_{k−1} + ℧_{2,1}(n,4) ψ_k. For this operator, the spectrum, density of states, and Green’s function are known. We record them here for completeness and to show how they depend on the parameter ℧_{2,1}(n,4). The three families of primitive Pythagorean triples (Socrates, Plato, Pythagoras) give three sequences of spectral bands. The purpose of this note is to document the parameterization of a known quantum operator by numbers arising from Pythagorean geometry.