Algebraic construction of associated functions of nondiagonalizable models with anharmonic oscillator complex interaction

TL;DR

This paper develops an algebraic method to construct associated functions for a complex, nonseparable 2D anharmonic oscillator model, improving efficiency and extending previous results for excited states.

Contribution

It introduces new operators to complete the algebraic structure, enabling a more efficient construction of associated functions and extending prior findings.

Findings

Constructed associated functions algebraically using polynomial operators

Extended results to higher excited states and more complex Hamiltonians

Provided a more efficient approach than previous methods

Abstract

A shape invariant nonseparable and nondiagonalizable two-dimensional model with anharmonic complex interaction, first studied by Cannata, Ioffe, and Nishnianidze, is re-examined with the purpose of providing an algebraic construction of the associated functions to the excited-state wavefunctions, needed to complete the basis. The two operators $A^+$ and $A^-$, coming from the shape invariant supersymmetric approach, where $A^+$ acts as a raising operator while $A^-$ annihilates all wavefunctions, are completed by introducing a novel pair of operators $B^+$ and $B^-$, where $B^-$ acts as the missing lowering operator. It is then shown that building the associated functions as polynomials in $A^+$ and $B^+$ acting on the ground state provides a much more efficient approach than that used in the original paper. In particular, we have been able to extend the previous results obtained for the first two excited states of the quartic anharmonic oscillator either by considering the next three excited states or by adding a cubic or a sextic term to the Hamiltonian.