Real Spectra in PT Symmetry Hamiltonians using Tridiagonal Representation Approach

We consider the solution of PT symmetry Hamiltonians using the technique of tridiagonal representation approach. This methodology provides more accurate results and proper depiction of the Hamiltonian energy level and wavefunctions. It is well know that PT symmetry condition of a Hamiltonian ensure that its spectra are real and positive even if the Hamiltonian is non Hermitian. Here, we introduce the method of TRA to get the eigenvalues and wavefunction of this Hamiltonian for integers values of $N$ and show an approximation solution for non-integer value of $N$. Due to the nature of the Hamiltonian, the TRA was applied in a semi-Analytic manner in the paper.