From Circle Laplacian to Berry-Keating: Unitary Equivalence and ϕ(n)-Residue Structure

We establish a precise unitary equivalence between the circle Laplacian −π∂²θ on S¹ and a compactified version of the Berry-Keating operator −π(x∂x)² on the interval [1, e²π) with measure dx/x. Both operators share the discrete spectrum {πk²: k ∈ Z}. The ϕ(n)-averaging operators, defined using coprime residue groups (Z/nZ)×, translate between the two settings via the logarithmic coordinate transformation, providing a precise arithmetic characterisation of the operator domains