The Generalised Born Oscillator and the Berry-Keating Hamiltonian

In this study, we introduce and investigate a family of quantum mechanical models in 0+1 dimensions, known as generalized Born quantum oscillators. These models represent a one-parameter deformation of a specific system obtained by reducing the Nambu-Goto theory to 0+1 dimensions. Despite these systems showing significant similarities with $\mathrm{T}\overline{\mathrm{T}}$-type perturbations of two-dimensional relativistic models, our analysis reveals their potential as interesting regularizations of the Berry-Keating theory. We quantize these models using the Weyl quantization scheme up to very high orders in $\hbar$. By examining a specific scaling limit, we observe an intriguing connection between the generalized Born quantum oscillators and the Riemann-Siegel $\theta$ function.