An exactly solvable quantum-lattice model with a tunable degree of nonlocality

TL;DR

This paper introduces an exactly solvable quantum lattice model with adjustable nonlocality, based on Laguerre polynomials, and discusses the construction of inner products that make the Hamiltonian Hermitian in different Hilbert spaces.

Contribution

It presents a novel quantum lattice model with tunable nonlocality and provides explicit constructions of inner products for different parameterizations, advancing understanding of cryptohermitian Hamiltonians.

Findings

Spectrum equals zeros of Laguerre polynomial

Explicit inner products for k=0,1,2,3

Parameter k measures lattice coordinate smearing

Abstract

An array of N subsequent Laguerre polynomials is interpreted as an eigenvector of a non-Hermitian tridiagonal Hamiltonian $H$ with real spectrum or, better said, of an exactly solvable N-site-lattice cryptohermitian Hamiltonian whose spectrum is known as equal to the set of zeros of the N-th Laguerre polynomial. The two key problems (viz., the one of the ambiguity and the one of the closed-form construction of all of the eligible inner products which make $H$ Hermitian in the respective {\em ad hoc} Hilbert spaces) are discussed. Then, for illustration, the first four simplest, $k-$parametric definitions of inner products with $k=0,k=1,k=2$ and $k=3$ are explicitly displayed. In mathematical terms these alternative inner products may be perceived as alternative Hermitian conjugations of the initial N-plet of Laguerre polynomials. In physical terms the parameter $k$ may be interpreted as a measure of the "smearing of the lattice coordinates" in the model.