A Swanson-like Hamiltonian and the inverted harmonic oscillator

• URL: http://arxiv.org/abs/2204.09968v1
• Date: Thu, 21 Apr 2022 08:46:42 GMT
• Title: A Swanson-like Hamiltonian and the inverted harmonic oscillator
• Authors: Fabio Bagarello
• Abstract summary: We deduce the eigenvalues and the eigenvectors of a parameter-dependent Hamiltonian $H_theta$. We show that there is no need to introduce a different scalar product using some ad hoc metric operator.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We deduce the eigenvalues and the eigenvectors of a parameter-dependent Hamiltonian $H_\theta$ which is closely related to the Swanson Hamiltonian, and we construct bi-coherent states for it. After that, we show how and in which sense the eigensystem of the Hamiltonian $H$ of the inverted quantum harmonic oscillator can be deduced from that of $H_\theta$. We show that there is no need to introduce a different scalar product using some ad hoc metric operator, as suggested by other authors. Indeed we prove that a distributional approach is sufficient to deal with the Hamiltonian $H$ of the inverted oscillator.

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