Pseudo-Hermitian extensions of the harmonic and isotonic oscillators

Abstract

In this work, we describe certain pseudo-Hermitian extensions of the harmonic and isotonic oscillators, both of which are exactly-solvable models in quantum mechanics. By coupling the dynamics of a particle moving in a one-dimensional potential to an imaginary-valued gauge field, it is possible to obtain certain pseudo-Hermitian extensions of the original (Hermitian) problem. In particular, it is pointed out that the Swanson oscillator arises as such an extension of the quantum harmonic oscillator. For the pseudo-Hermitian extensions of the harmonic and isotonic oscillators, we explicitly solve for the wavefunctions in the position representation and also explore their intertwining relations.

Based on talk delivered at the Xth International Workshop on New Challenges in Quantum Mechanics: Graphene, Supersymmetry, and Mathematical Physics (2024); v2: Some errors corrected; v3: Final version

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Cited by in corpus (4)

• Isochronous oscillator with a singular position-dependent mass and its quantization
• Generalized Liénard systems and isochronous connections
• The quantum harmonic oscillator and the real Hilbert space
• Landau levels of a Dirac electron in graphene from non-uniform magnetic fields