Renormalization group and spectra of the generalized Pöschl–Teller potential

Camara da Silva, Ulysses; Pereira, Carlos F.S.; Alves Lima, Andre

doi:10.1016/j.aop.2023.169549

Cited by 4 publications

(1 citation statement)

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“…A key feature of the ISP is that the non-relativistic Schrödinger equation with this potential is scale invariant as both the kinetic energy and potential terms scale as length −2 and thus there is no natural length scale present (such as the Bohr radius in the Coulomb problem, say). However, if the singular nature of the ISP at the origin is tamed by introducing a cut-off or boundary condition at short distance (in the physical examples given above the ISP only models the long wavelength behaviour), this regulator necessarily breaks the scale invariance in a simple example of a quantum anomaly [58][59][60][61][62][63][64]. Furthermore, in order to ensure that the long wavelength physics is independent of the regulator the theory's couplings should be renormalised [65].…”

Section: Introductionmentioning

confidence: 99%

Duality between the quantum inverted harmonic oscillator and inverse square potentials

Sundaram,

Burgess,

O’Dell

2024

New J. Phys.

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In this paper we show how the quantum mechanics of the inverted harmonic oscillator (IHO) can be mapped to the quantum mechanics of a particle in a super-critical inverse square potential (ISP). We demonstrate this by relating both of these systems to the Berry–Keating system with Hamiltonian H = ( x p + p x ) / 2 . It has long been appreciated that the quantum mechanics of the ISP has an ambiguity in choosing a boundary condition near the origin and we show how this ambiguity is mapped to the IHO system. Imposing a boundary condition requires specifying a distance scale where it is applied and changes to this scale come with a renormalisation group (RG) evolution of the boundary condition that ensures observables do not directly depend on the scale (which is arbitrary). Physical scales instead emerge as RG invariants of this evolution. The RG flow for the ISP is known to follow limit cycles describing the discrete breaking of classical scale invariance in a simple example of a quantum anomaly, and we find that limit cycles also occur for the IHO. However, unlike the ISP where the continuous scaling symmetry is explicit, in the case of the IHO it is hidden and occurs because the Hamiltonian is part of a larger su(1,1) spectrum generating algebra. Our map does not require the boundary condition to be self-adjoint, as can be appropriate for systems that involve the absorption or emission of particles.

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