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arXiv · 2204.09968

A Swanson-like Hamiltonian and the inverted harmonic oscillator

Fabio Bagarello
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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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Fabio Bagarello. 2022-04-21. A Swanson-like Hamiltonian and the inverted harmonic oscillator. https://doi.org/10.1088/1751-8121%2Fac6a92

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