Two-dimensional non commutative Swanson model and its bicoherent states

TL;DR

This paper extends the Swanson model to a two-dimensional non-commutative space, providing exact diagonalization, explicit eigenvalues, bi-coherent states, and a resolution of the identity, advancing the understanding of non-Hermitian quantum systems.

Contribution

It introduces a novel two-dimensional non-commutative Swanson model with exact solutions and constructs bi-coherent states, enriching the study of non-Hermitian quantum mechanics.

Findings

Eigenvalues are explicitly computed.

Bi-coherent states are constructed and shown to be eigenstates.

A resolution of the identity is established in a dense subspace.

Abstract

We introduce an extended version of the Swanson model, defined on a two-dimensional non commutative space, which can be diagonalized exactly by making use of pseudo-bosonic operators. Its eigenvalues are explicitly computed and the biorthogonal sets of eigenstates of the Hamiltonian and of its adjoint are explicitly constructed. We also show that it is possible to construct two displacement-like operators from which a family of bi-coherent states can be obtained. These states are shown to be eigenstates of the deformed lowering operators, and their projector allows to produce a suitable resolution of the identity in a dense subspace of $\Lc^2(\Bbb R^2)$.