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arXiv:1309.5065 · Physical Review A

From self-adjoint to non-self-adjoint harmonic oscillators: Physical consequences and mathematical pitfalls

Fabio Bagarello

Published Sep 30, 2013• 35 citations
arXivHTMLPDF

Abstract

Using as a prototype example the harmonic oscillator we show how losing self-adjointness of the Hamiltonian $H$ changes drastically the related functional structure. In particular, we show that even a small deviation from strict self-adjointness of $H$ produces two deep consequences, not well understood in the literature: First of all, the original orthonormal basis of $H$ splits into two families of biorthogonal vectors. These two families are complete but, contrarily to what often claimed for similar systems, none of them is a basis for the Hilbert space $\mathcal{H}$. Second, the so-called metric operator is unbounded, as well as its inverse. In the second part of the paper, after an extension of some previous results on the so-called $\mathcal{D}$ pseudobosons, we discuss some aspects of our extended harmonic oscillator from this different point of view.

DOI
10.1103/physreva.88.032120

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Using as a prototype example the harmonic oscillator we show how losing self-adjointness of the Hamiltonian HHH changes drastically the related functional structure. In particular, we show that even a small deviation from strict self-adjointness of HHH produces two deep consequences, not well understood in the literature: First of all, the original orthonormal basis of HHH splits into two families of biorthogonal vectors. These two families are complete but, contrarily to what often claimed for similar systems, none of them is a basis for the Hilbert space H\mathcal{H}H. Second, the so-called metric operator is unbounded, as well as its inverse. In the second part of the paper, after an extension of some previous results on the so-called D\mathcal{D}D pseudobosons, we discuss some aspects of our extended harmonic oscillator from this different point of view.
Abstract Non-self-adjoint operators with a real discrete spectrum exhibit, in general, a quite different behavior from the one of self-adjoint operators, with the same spectrum. This fact has serious mathematical and physical implications, namely in the rigorous formulation of quantum mechanics (QM), where self-adjointness plays a central role, in particular in the spectral analysis of the involved operators. In this framework, a non-self-adjoint 2D-harmonic oscillator is considered. Its eigenvalues and eigenfunctions, as well as those of its adjoint, are explicitly determined. Although the eigensystems are complete they may not form bases, a drawback for the mathematical formulation of the QM of the system. The existence of a metric operator defining an inner product which renders the Hamiltonian self-adjoint, is discussed.
A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally, we discuss their application in the so-called pseudo-Hermitian quantum mechanics.
Abstract It is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by the eigenvectors of the Hamiltonians. In some recent papers, this expansion has been extended to the case in which these eigenvectors form a Riesz basis or, more recently, a D{\mathcal{D}}D D -quasi basis (Bagarello and Bellomonte in J. Phys. A 50:145203, 2017, Bagarello et al. in J. Math. Phys. 59:033506, 2018), rather than an orthonormal basis. Here we discuss what can be done when these sets are replaced by Parseval frames. This interest is motivated by physical reasons, and in particular by the fact that the mathematical Hilbert space where the physical system is originally defined, contains sometimes also states which cannot really be occupied by the physical system itself. In particular, we show what changes in the spectrum of the observables, when going from orthonormal bases to Parseval frames. In this perspective we propose the notion of EEE E -connection for observables. Several examples are discussed.
In the theory of quantum mechanics the Hamiltonian H is typically selfadjoint, i.e., H = H *. The self-adjointness ensures that the spectrum of the Hamiltonian, representing the energy spectrum of H , is real but it is not a necessary condition. In the literature on so-called PT -symmetric quantum mechanics (see, e.g., [BB98], [BBM99], [BBJ03], [Ben04b] and [Ben07]), it is believed that self-adjointness is rather a mathematical requirement than a physically established fact. Therefore, it was considered a surprise that operators exist which are not self-adjoint in the given quantum mechanical Hilbert space, but have real spectrum and that – if e.g. complex eigenvalues were present – they occurred only in complex conjugate pairs.
In this article, we review the quantum mechanical setting associated with a non-self-adjoint Hamiltonian with a real spectrum. The spectral properties of the Hamiltonian of a Swanson-like model are investigated. The eigenfunctions associated with the real simple eigenvalues are shown to form complete systems but not a (Riesz) basis, which gives rise to difficulties in the rigorous mathematical formulation of quantum mechanics. A new inner product, appropriate for the physical interpretation of the model, has been consistently introduced. The dynamics of the system is described. Some specificities of the theory of non-self-adjoint operators with implications in quantum mechanics are discussed.