Eigenspectra of a complex coupled harmonic potential in three dimensions

Within the framework of extended complex phase space approach characterized by position and momentum coordinates, we investigate the quasi-exact solutions of the Schrödinger equation for a coupled harmonic potential and its variants in three dimensions. For this purpose ansatz method is employed and nature of the eigenvalues and eigenfunctions is determined by the analyticity property of the eigenfunctions alone. The energy eigenvalue is real for the real coupling parameters and becomes complex if the coupling parameters are complex. However, in case of complex coupling parameters, the imaginary component of energy eigenvalue reduces to zero if the PTPT-symmetric condition is satisfied. Thus a non-hermitian Hamiltonian possesses real eigenvalue if it is PTPT-symmetric.