Quantum Mechanics of Complex Octic Potential in One Dimension

Abstract

For gaining further insight into the nature of the eigenspectra of a complex octic potential [say ], we investigate the quasi exact solutions of the Schr?dinger equation in an extended complex phase space characterized by . The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunction of a system. Explicit expressions of eigenvalues and eigenfunctions for the ground state as well as for the first excited state of a complex octic potential and its variant are worked out. It is found that imaginary part of the eigenvalue turns out to be zero for real coupling parameters, whereas it becomes non-zero for complex coupling parameters. However, the PT -symmetric version of a non-hermitian Hamiltonian possesses the real eigenvalue even if coupling parameters in the potential are complex.