N-site-lattice analogues of $V(x)=i{x}^3$

Abstract

Two discrete N-level alternatives to the popular imaginary cubic oscillator are proposed and studied. In a certain domain $\mathcal{D}$ of parameters $a$ and $z$ of the model, the spectrum of energies is shown real (i.e., potentially, observable) and the unitarity of the evolution is shown mediated by the construction of a (non-unique) physical, ad hoc Hilbert space endowed with a nontrivial, Hamiltonian-dependent inner-product metric $\Theta$. Beyond $\mathcal{D}$ the complex-energy curves are shown to form a "Fibonacci-numbered" geometric pattern and/or a "topologically complete" set of spectral loci. The dynamics-determining construction of the set of the eligible metrics is shown tractable by a combination of the computer-assisted algebra with the perturbation and extrapolation techniques. Confirming the expectation that for the local potentials the effect of the metric cannot be short-ranged.

28 pp, 17 figures, submitted (and presented also during the conference "Tercentenary of the Laplace-Runge-Lenz vector", Durban, November 23-27, 2011, http://math.ukzn.ac.za/~pllrl/)

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• Solvable non-Hermitian discrete square well with closed-form physical inner product
• Crypto-Hermitian Approach to the Klein-Gordon Equation
• Asymptotic non-Hermitian degeneracy phenomenon and its exactly solvable simulation
• Coulomb potential and the paradoxes of PT-symmetrization