2011/05/10 by Miloslav Znojil · 2 citations
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Algebraic structures and combinatorial models #Quantum Mechanics and Non-Hermitian Physics #hep-lat #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1016/j.physleta.2011.05.027
published as Phys.Lett.A375:2503-2509,2011 · 16 pp., 4 figs
arxiv created 2011/05/10 · openalex publication_date 2011/05/20 · arxiv updated 2011/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A toy-model quantum system is proposed. At a given integer N it is defined by the pair of N by N real matrices (H,Θ) of which the first item H specifies an elementary, diagonalizable non-Hermitian Hamiltonian H ≠ H^† with the real and explicit spectrum given by the zeros of the N-th Chebyshev polynomial of the first kind. The second item Θ≠ I must be (and is being) constructed as the related Hilbert-space metric which specifies the (in general, non-unique) physical inner product and which renders our toy-model Hamiltonian selfadjoint, i.e., compatible with the Dieudonne equation H^† Θ= Θ H. The elements of the (in principle, complete) set of the eligible metrics are then constructed in closed band-matrix form. They vary with our choice of the N-plet of optional parameters, Θ=Θ(κ)>0 which must be (and are being) selected as lying in the positivity domain of the metric, κ ∈ \cal D(physical).