The Generalised Born Oscillator and the Berry-Keating Hamiltonian

Abstract

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 $\hslash$. By examining a specific scaling limit, we observe an intriguing connection between the generalized Born quantum oscillators and the Riemann-Siegel $\theta$ function.

arXiv comment23+16 pages; 8 figures; Published version

References in corpus (6)

• $T\bar{T}$-deformed 2D Quantum Field Theories
• Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations
• Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras
• The H=xp model revisited and the Riemann zeros
• Landau levels and Riemann zeros
• The Riemann zeros as spectrum and the Riemann hypothesis

Cited by in corpus (1)

• Complete Minimal Form Factors for Irrelevant Deformations of Integrable Quantum Field Theory