What question did this study set out to answer?

The aim is to establish a unitary equivalence between the circle Laplacian and the Berry-Keating operator.

January 6, 20260 citationsOpen Access

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

Key Points

The aim is to establish a unitary equivalence between the circle Laplacian and the Berry-Keating operator.
Defined the circle Laplacian on S¹ and the Berry-Keating operator on the interval [1, e²π) with a specific measure.
Utilized ϕ(n)-averaging operators based on coprime residue groups for the connection.
Applied a logarithmic coordinate transformation to relate the two operators.
Both operators have a shared discrete spectrum given by {πk²: k ∈ Z}.
Defined a precise arithmetic characterization of operator domains through the defined operators.

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

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