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arXiv:1609.07579 · Journal of Mathematical Physics

Intertwining operators for non-self-adjoint Hamiltonians and bicoherent states

F. Bagarello

Published Oct 1, 2016• 20 citations
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Abstract

This paper is devoted to the construction of what we will call exactly solvable models, i.e., of quantum mechanical systems described by an Hamiltonian H whose eigenvalues and eigenvectors can be explicitly constructed out of some minimal ingredients. In particular, motivated by PT-quantum mechanics, we will not insist on any self-adjointness feature of the Hamiltonians considered in our construction. We also introduce the so-called bicoherent states, we analyze some of their properties and we show how they can be used for quantizing a system. Some examples, both in finite and in infinite-dimensional Hilbert spaces, are discussed.

DOI
10.1063/1.4964128

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•Jan 1, 2011•0 citations
This paper is devoted to the construction of what we will call exactly solvable models, i.e., of quantum mechanical systems described by an Hamiltonian H whose eigenvalues and eigenvectors can be explicitly constructed out of some minimal ingredients. In particular, motivated by PT-quantum mechanics, we will not insist on any self-adjointness feature of the Hamiltonians considered in our construction. We also introduce the so-called bicoherent states, we analyze some of their properties and we show how they can be used for quantizing a system. Some examples, both in finite and in infinite-dimensional Hilbert spaces, are discussed.
Abstract Motivated by what one observes dealing with PT-symmetric quantum mechanics, we discuss what happens if a physical system is driven by a diagonalizable Hamiltonian with not all real eigenvalues. In particular, we consider the functional structure related to systems living in finite-dimensional Hilbert spaces, and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators. We also analyze a simple model, computing the transition probabilities in the broken and in the unbroken regime.
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.
Non-self-adjoint quantum mechanical operators do not necessarily possess eigenvalues. Finite N ? N matrix representations of these operators, however, can be hermitian and therefore have a finite set of N real eigenvalues. Using the momentum operator, the kinetic energy operator, and the relativistic Hamiltonian of the Coulomb problem for the Klein?Gordon equation as examples, we examine analytically and also numerically the properties of the spectrum and eigenvectors in finite dimensional Hilbert spaces. We study the limit of N ? ? for which some eigenvalues cease to exist as the corresponding operators are not self-adjoint.
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.
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.
We study selected non-Hermitian quantum models, indecomposable representations of the de Sitter group, and coherent state quantization in infinite well Systems. The first part is focused on the investigation of spectral and bases properties of PT-symmetric operators. At first, we consider Schrodinger operators subjected to non-self-adjoint boundary conditions. The existence and the structure of similarity transformations to normal or self-adjoint operators is examined and general results are illustrated on explicitly solvable models. A physical motivation for the study of Schrodinger operators with non-self-adjoint Robin-type boundary conditions is established in a reflectionless scatterin. Further, irregular boundary conditions are considered and operators with very non-normal spectral ; properties are found. Finally, an operator with PT-symmetric version of Coulomb potential is investigate an effective behavior of the kinetic term is discussed and the reflection and transmission coefficients for this System are calculated. The second part deals with elements of the discrete scalar series of representations of the de Sitter group Physical motivations and origins of the so-called zero mode problem are explained. The generalization of the Gupta-Bleuler-like quantization of the massless minimally coupled field, based on the construction an indecomposable representation, is worked out and put into the mathematical framework of the cohomology of representations and Gupta-Bleuler triplets. The third part is oriented on the construction of coherent states for the Poschl-Teller. The obtained stat exhibit remarkable semi-classical properties.