Quantum Mechanics of Complex Octic Potential in One Dimension

For gaining further insight into the nature of the eigenspectra of a complex octic potential [say ], we investigate the quasi exact solutions of the Schrödinger equation in an extended complex phase space characterized by   8 0 i i i V x a x    1 2 1 , 2 x x ip p p ix     . 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.   i E


Introduction
Though complex potentials are in practice for a long time, the quantum mechanics of such potentials has not been studied to a desired level.It is only in the last few years that study of complex potentials has become important for better theoretical understanding of some newly discovered phenomena in Physics and Chemistry like phenomena pertaining to resonance scattering in atomic, molecular, and nuclear Physics and to some chemical reactions [1,2].Besides some general studies of a complex Hamiltonian in nonlinear domain [3][4][5], efforts have been made to study both classical as well as quantum aspects [6][7][8] of the one-dimensional complex Hamiltonian systems.In the classical context,   , H x p becomes the function of two complex variables and analyticity property of the Hamiltonian leading to a class of integrable systems.In the quantum context, the analyticity of   , H x p is translated into the complex potential   V x P .It is observed that complex Hamiltonian is no longer hermitian and ordinarily does not guarantee for real eigenvalues; however, in its PT-symmetric form [9][10][11], the system is found to exhibit real eigenvalue spectrum [12].The reality of the spectrum is a direct consequence of the combined action of the parity and time reversal invariance of Hamiltonian [13].The parity operator and the time reversal operator are defined by the action of position and momentum operator as and .The non-hermitian PT-symmetric Hamiltonians play an important role in various fields like superconductivity, population biology, quantum cosmology, condensed matter Physics and quantum field theory etc [5,8] , .
In literature, the complexity of the Hamiltonian is introduced in several ways but here we use the scheme given by Xavier and de Aguir [10], which is used to develop an algorithm for the computation of semi classical coherent state propagator, to transform potentials using extended complex phase space approach (ECPSA).In this approach, the transformations of the positions and the momenta variables are defined as Here, the variables   , , , x p x p are considered as some sort of coordinate-momentum interactions for a dynamical system.For the dimensional considerations, there should be a constant "d" in (1) i.e. , etc and for simplicity, we select expand the domain of applications, we find the quasiexact solutions of the SE for a coupled complex octic potential and its variant.
The paper is organized as follows: in Section 2, we carry out the mathematical formulation of the ECPSA to compute eigenvalue spectra of the SE in one dimension.Under such mathematical prescription, the ground state solutions are described in Section 3, whereas excited state solutions are addressed in Section 4. Finally, concluding remarks are presented in Section 5.

The Methods
For, one-dimensional complex Hamiltonian system   , H x p , the ASE is given by where After employing transformation (1), one gets Under the transformation (1), momentum operator becomes : The real and imaginary parts of     ,

V x x
 and E are written as where the subscript "r" and "i" denotes the real and imaginary parts of the corresponding quantities and other subscripts to these quantities separated by comma denotes the partial derivatives of the quantity concerned.On employing Equations ( 1), (3) and (5a)-(5e) in Equation (2), then after separating real and imaginary parts in the final result, the following pair of partial differential equations is obtained Under Cauchy-Riemann condition, the analyticity property of the wavefunction   By imposing the analyticity condition (7), on Equations (6a) and (6b), we have The Ansatz for the wavefunction   x  is assumed as [14]  where,   x  and   g x are written as on substituting Equations (5b), (10a) and (10b) in Equation (9), the real and imaginary parts of the wavefunction are expressed as Then analyticity condition for r g and i g becomes On implying Equations (11a) and (11b) in Equations (8a) and (8b), one gets 2 2 2 0 , For the ground state solutions,   x  is taken as constant, then Equations (13a) and (13b) reduces to   By assuming the appropriate ansatz for a given potential, rationalization of Equations (14a) and (14b) provides the ground state solutions, whereas Equations (13a) and (13b) yield the excited state solutions.

Ground State Solutions
In this section, we are devoted with ground state solutions of a complex octic potential and its variants as

Generalized Octic Potential
Consider a generalized octic potential of the form , V x a a x a x a x a x a x a x a x a x where, the coupling parameters   ,1 8 are complex constants.
Under the transformation (1), the real and imaginary component of the potential (15) are written as The functional forms of , g x p in consonance with condition (12) are written as where i  and i  are real parameters.Now, substituting Equations (18a) and (18b) in Equations ( 14a) and (14b), then equating the coefficients of various terms to zero, one gets the following set of 18 non-repeating equations     , in the same way, one can obtain four constraining relations from Equations (19i)-(19l) also.On utilizing the above values of ij  's and ij  's in (19b) and (19c), the real and imaginary compone of the energy eigenvalues for ground state are written as The corresponding eigenfunctions becomes  x a a x a x a x a x a x a x a x a x In order to extract the information regarding the nature of functions turn out to be V the energy eigenvalues and eigenfunctions for the potential (27), we follow the same prescription as laid down in general case.Then the energy eigenvalues and eigenas

Variant of Octic Potential
ional octic potential Here, we consider the one-dimens along with inverse harmonic term as where, the parameters   ,1 8 are complex c stants.
ormation (1) rts of the potential (30) becomes on-Again under the transf , the real and imaginary pa where, and are same as given by E and (1 Th ional form of 1 r V 7).31a), (31b), (32a) and E s ( uations (18a) 8b).A (32b) in quation 14a) and (14b), the rationalization of the final expression yields the following 10 non-repeating equations Now, Equations (33g) and (33h) lead to (33j)   The corresponding eigenfunctions becomes After using the condition x p p x i x p p x i   solutions for the potential (30) are give  , the PT-symmetric n by

Excited State Solutions
To compute the eigenvalues and eigenfunctions for the first excited state, we follow the same prescription as adopted in earlier section.The functional form of   x  for the first excited state is taken as where,  and  are considered as real constants.
Then under the transformation (1), Equation (40) reduce to s rgy eigenvalues and associated eigenfunctions for the first excite tential (15), we use the same func

Generalized Octic Potential
In order to compute the ene d state of the potional forms of r g and i g mentioned in Equations (18a) and (18b).Then inserting Equations (18a), (18b) and (41) in Equations (13 as a) and (13b), the rationalization of the resultant expression yields the following non-repeating equ addition to Equations (19c), (19d) and (19i)-(19 ations in r) as   The Equations (42c) and (42d) g relations as ives the constraining whereas, Equations (42e) and (42f) can be solved for   After inserting the values of various Ansatz parameters in Equations (42a) and (42b), the energy eigenvalues for the first excited state are (44b) and the corresponding eigenfunctions becomes where, the Equations (48i) and (48j) (49b) can be solved for  and  .For convenience, we set   The corresponding eigenfunctions becomes he potential (30) turn out to be The PT-symmetric excited state solutions for t

Conclusion
In the present work, we have computed the quasi-exact solutions of the SE for one-dimensional octic potential and its variants by using ECPS approach.The non-hermiticity arising is not only due to the potential parameters but the underlying phase space is also considered as complex.The PT-symmetry discussed here is of ized nature which in certain limit (for real generalx and p ) om-reduces to conventional PT-symmetry, besides the c plexity of the phase space produced by (1), the complexity of the potential parameters is also taken into account.It is also emphasized that solutions of the ASE in the above mentioned cases are obtained only in the presence ning relation among the potential 23-540.2)90579-Z of certain constrai parameters, such constraining relations give rise to bound state energies of a system.It is found that the imaginary part of the energy eigenvalues always vanishes for the solvable cases of ASE as long as all the potential parameters are real [14][15][16].However, for the PT-symmetric potentials, the energy eigenvalues are found real, even if the concerned potential possesses complex couplings parameters.The result obtained in this manner coincides with those computed from the invariant of Hamiltonian under PT operation.The interesting aspect of this method is an account of the complex coupling parameters in the potential in addition to the complex phase space, such feature of the method lead to complex spectra.If PT-symmetric version of the concerned potential is taken into account, then it is found that imaginary part of the complex eigenvalue vanishes after imposing certain restrictions on the complex coupling parameters of the potential.It is observed that real and imaginary parts of the eigenvalues follow just opposite ordering for the discrete energy level by retaining the conventional ordering for the magnitude of the eigenvalues.