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arXiv:2204.09968 · Journal of Physics A Mathematical and Theoretical

A Swanson-like Hamiltonian and the inverted harmonic oscillator

Fabio Bagarello

Published Apr 26, 2022• 14 citations
arXivHTMLPDF

Abstract

Abstract We deduce the eigenvalues and the eigenvectors of a parameter-dependent Hamiltonian H θ 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 θ . 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.

DOI
10.1088/1751-8121/ac6a92

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Abstract We deduce the eigenvalues and the eigenvectors of a parameter-dependent Hamiltonian H θ 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 θ . 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.
We treat the quantum dynamics of a harmonic oscillator as well as its inverted counterpart in the Schrödinger picture. Generally in the most papers of the literature, the inverted harmonic oscillator is formally obtained from the harmonic oscillator by the replacement of ω to iω, this leads to unbounded eigenvectors. This explicitly demonstrates that there are some unclear points involved in redefining the variables in the harmonic oscillator inversion. To remedy this situation, we introduce a scaling operator (Dyson transformation) by connecting the inverted harmonic oscillator to an anti-PT-symmetric harmonic oscillator, we obtain the standard quasi-Hermiticity relation which would ensure the time invariance of the eigenfunction's norm. We give a complete description for the eigenproblem. We show that the wavefunctions for this system are normalized in the sense of the pseudo-scalar product. A Gaussian wave packet of the inverted oscillator is investigated by using the ladder operators method. This wave packet is found to be associated with the generalized coherent state that can be crucially utilized for investigating the mean values of the space and momentum operators. We find that these mean values reproduce the classical motion.
Abstract We use the invariant operator method to investigate the quantum characteristics of the inverted oscillator. We introduce a unitary transformation that maps a time-dependent general Hermitian linear invariant operator to a time-independent one, which describes a harmonic oscillator with a unit mass and a constant frequency. Our investigation includes three distinguished cases: negative frequency, zero frequency, and positive frequency. Our most interesting result concerns the last case. For the case of negative frequency, we propose a new inner product by the introduction of the new metric operator η. Coherent state for this case will be constructed.
It is known that the standard and the inverted harmonic oscillator are different. Replacing thus ω by ±iω in the regular oscillator is necessary going to give the inverted oscillator H^{r}. This replacement would lead to anti- PT-symmetric harmonic oscillator Hamiltonian (∓iH^{os}). The pseudo-hermiticity relation has been used here to relate the anti-PT-symmetric harmonic Hamiltonian to the inverted oscillator. By using a simple algebra, we introduce the ladder operators describing the inverted harmonic oscillator to reproduce the analytical solutions.We construct the inverted coherent states which minimize the quantum mechanical uncertainty between the position and the momentum. This paper is dedicated to the memory of Omar Djemli and Nouredinne Mebarki who died due to covid 19.
We propose an unified view to three well known quadratic Hamiltonians widely considered in quantum mechanics, i.e. those of the harmonic oscillator, of the inverted harmonic oscillator, and the Berry-Keating Hamiltonian. We describe an unified approach to determine the discrete spectrum of these operators, and to deduce their (generalized) eigenstates. We also look at these Hamiltonians as the generators of su(1,1)\mathfrak{su}(1,1)su(1,1), and of a generalized version of it relevant in presence of non Hermitian generators.
The generalized Swanson Hamiltonian with can be transformed into an equivalent Hermitian Hamiltonian with the help of a similarity transformation. It is shown that the equivalent Hermitian Hamiltonian can be further transformed into the harmonic oscillator Hamiltonian so long as constant. However, the main objective of this communication is to show that though the commutator of and is constant, the generalized Swanson Hamiltonian is not necessarily isospectral to the harmonic oscillator. The reason for this anomaly is discussed in the framework of position-dependent mass models by choosing A ( x ) as the inverse square root of the mass function.