Quantum Mechanics of Complex Octic Potential in One Dimension

Ram Mehar Singh^{*}

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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.

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