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arXiv:math-ph/0604055 · Journal of Physics A Mathematical and General

Closed formula for the metric in the Hilbert space of a -symmetric model

David Krejčiřı́k, Hynek Bíla, Miloslav Znojil

Published Jul 26, 2006• 94 citations
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Abstract

We introduce a very simple, exactly solvable PT-symmetric non-Hermitian model with real spectrum, and derive a closed formula for the metric operator which relates the problem to a Hermitian one.
DOI
10.1088/0305-4470/39/32/s15

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A transformation of the form x to iy; x,y in R, or an equivalent similarity transformation with a metric operator ηηη are shown to transform non-Hermitian PT-symmetric Hamiltonians into Hermitian partner Hamiltonians in Hilbert space. Isospectrality and mass signature are discussed.
The complete menu of eligible metrics for a family of toy Hamiltonians H≠H†H \neq H^\daggerH=H† with real spectra
An elementary set of non-Hermitian NNN by NNN matrices H(N)(λ)≠[H(N)(λ)]†H^{(N)}(λ) \neq [ H^{(N)}(λ)]^\daggerH(N)(λ)=[H(N)(λ)]† with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc} Hilbert space of states H(physical){\cal H}^{(physical)}H(physical). The problem of an explicit specification of all of these spaces (i.e., in essence, of all of the eligible {\it ad hoc} inner products and metric operators ΘΘΘ) is addressed. The problem is shown exactly solvable and, for every size N=2,4,...N=2,4,...N=2,4,... and parameter λ∈(−1,1)λ\in (-1,1)λ∈(−1,1) in matrix H(N)(λ)H^{(N)}(λ)H(N)(λ), the complete N−N-N−parametric set of metrics Θα1,α2,...,αN(N)(λ)Θ^{(N)}_{α_1,α_2,...,α_N}(λ)Θα1​,α2​,...,αN​(N)​(λ) is recurrently defined by closed formula.
The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for Swanson’s PT-symmetric Hamiltonian, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized Swanson’s non-Hermitian (not necessarily PT-symmetric) Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed. Short title: Swanson’s non-Hermitian Hamiltonian and su(1,1)