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

Miloslav Znojil

Two discrete N-level alternatives to the popular imaginary cubic oscillator are proposed and studied. In a certain domain $${\cal D}$$ 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 $${\cal D}$$ 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.

http://arxiv.org/abs/1111.0484
Quantum Physics (quant-ph); Mathematical Physics (math-ph)