# QES solutions of a two-dimensional system with quadratic nonlinearities

## TL;DR

This paper analyzes a PT symmetric two-dimensional nonlinear system, constructs a non-Hermitian Hamiltonian with position-dependent mass, and explicitly calculates quasi-exactly solvable energy levels using polynomial methods.

## Contribution

It introduces a novel approach to mapping PT symmetric systems with quadratic nonlinearities to QES systems via canonical transformations.

## Key findings

- Periodic oscillations due to system centers
- Explicit calculation of QES energy levels
- Mapping to QES systems using canonical transformations

## Abstract

We consider a one parameter family of a PT symmetric two dimensional system with quadratic non-linearities. Such systems are shown to perform periodic oscillations due to existing centers. We describe this systems by constructing a non-Hermitian Hamiltonian of a particle with position dependent mass. We further construct a canonical transformation which maps this position dependent mass systems to a QES system. First few QES levels are calculated explicitly by using Bender-Dunne (BD) polynomial method.