Exponential Equations Related to the Quantum 'ax + b' Group

We study pairs b, β of unbounded selfadjoint operators, satisfying commutation rules inspired by the quantum 'ax + b' group [19]: bβ = -βb and β²= id except for kerb, on which β ² = 0. We find all measurable, unitary-operator valued functions F satisfying the exponential equation: F(b, β)F(d, δ) = F((b, β) circled inside astrik sign (d, δ)), where d, δ satisfy the same commutation rules as b, β, and cirlcle astrik sign is modeled after the comultiplication of the quantum 'ax + b' group. This result is crucial for classification of all unitary representations of the quantum 'ax + b' group, which is achieved in our forthcoming paper.