From Circle Laplacian to Berry-Keating: Unitary Equivalence and ϕ(n)-Residue Structure
Year: 2026
Type: article
Abstract: 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
Author: Palelei, Trent
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7118174583
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