A Swanson-like Hamiltonian and the inverted harmonic oscillator

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.

arXiv commentin press in J. Phys. A: Math. and Theor

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Cited by in corpus (1)

• Integral Transforms and $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians