2005/08/31 by Ali Mostafazadeh · 4 citations
Mathematics · Physics and Astronomy · #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #hep-th #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1088/0305-4470/39/32/s18
published as J.Phys. A39 (2006) 10171-10188 · 23 pages, 1 figure, slightly revised (published) version
arxiv created 2006/07/26 · openalex publication_date 2006/07/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We present a systematic perturbative construction of the most general metric operator (and positive-definite inner product) for quasi-Hermitian Hamiltonians of the standard form, H= p2/2 + v(x), in one dimension. We show that this problem is equivalent to solving an infinite system of iteratively decoupled hyperbolic partial differential equations in (1+1)-dimensions. For the case that v(x) is purely imaginary, the latter have the form of a nonhomogeneous wave equation which admits an exact solution. We apply our general method to obtain the most general metric operator for the imaginary cubic potential, v(x)=i εx3. This reveals an infinite class of previously unknown CPT- as well as non-CPT-inner products. We compute the physical observables of the corresponding unitary quantum system and determine the underlying classical system. Our results for the imaginary cubic potential show that, unlike the quantum system, the corresponding classical system is not sensitive to the choice of the metric operator. As another application of our method we give a complete characterization of the pseudo-Hermitian canonical quantization of a free particle moving in real line that is consistent with the usual choice for the quantum Hamiltonian. Finally we discuss subtleties involved with higher dimensions and systems having a fixed length scale.