Pseudo-Boson Coherent and Fock States

TL;DR

This paper introduces pseudo-boson coherent and Fock states for non-Hermitian systems, providing explicit constructions and revealing bi-orthogonal polynomials that generalize Hermite polynomials.

Contribution

It presents a novel framework for pseudo-boson states, including explicit constructions and new bi-orthogonal polynomials generalizing Hermite polynomials.

Findings

Pseudo-boson coherent states form bi-normalized, bi-overcomplete systems.

Eigenstates are proportional to new bi-orthogonal polynomials.

Explicit constructions are provided for a family of pseudo-boson operators.

Abstract

Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.