Explicit energy expansion for general odd degree polynomial potentials

In this paper we derive an almost explicit analytic formula for asymptotic eigenenergy expansion of arbitrary odd degree polynomial potentials of the form $V(x)=(ix)^{2N+1}+\beta _{1}x^{2N}+\beta _{2}x^{2N-1}+\cdot \cdot \cdot \cdot \cdot +\beta _{2N}x$ where $\beta _{k}^{\prime }$s are real or complex for $1\leq k\leq 2N$. The formula can be used to find semiclassical analytic expressions for eigenenergies up to any order very efficiently. Each term of the expansion is given explicitly as a multinomial of the parameters $\beta _{1},\beta _{2}....$ and $\beta _{2N}$ of the potential. Unlike in the even degree polynomial case, the highest order term in the potential is pure imaginary and hence the system is non-Hermitian. Therefore all the integrations have been carried out along a contour enclosing two complex turning points which lies within a wedge in the complex plane. With the help of some examples we demonstrate the accuracy of the method for both real and complex eigenspectra.Comment: 10 page

Effects of complex parameters on classical trajectories of Hamiltonian systems

Anderson $\textit{et al}$ have shown that for complex energies, the classical trajectories of $\textit{real}$ quartic potentials are closed and periodic only on a discrete set of eigencurves. Moreover, recently it was revealed that, when time is complex $t$ $(t=t_{r}e^{i\theta _{\tau }}),$ certain real hermitian systems possess close periodic trajectories only for a discrete set of values of $\theta _{\tau }$. On the other hand it is generally true that even for real energies, classical trajectories of non $\mathcal{PT}$- symmetric Hamiltonians with complex parameters are mostly non-periodic and open. In this paper we show that for given real energy, the classical trajectories of $\textit{complex}$ quartic Hamiltonians $H=p^{2}+ax^{4}+bx^{k}$, (where $a$ is real, $b$ is complex and $k=1$ $or$ $2$) are closed and periodic only for a discrete set of parameter curves in the complex $b$-plane. It was further found that given complex parameter $b$, the classical trajectories are periodic for a discrete set of real energies (i.e. classical energy get discretized or quantized by imposing the condition that trajectories are periodic and closed). Moreover, we show that for real and positive energies (continuous), the classical trajectories of $\textit{complex}$ Hamiltonian $H=p^{2}+\mu x^{4}, (\mu=\mu _{r}e^{i\theta })$ are periodic when $\theta =4 tan^{-1}[(n/(2m+n))]$ for $\forall$ $n$ and $m\in \mathbb{Z}$.Comment: 9 pages, 2 tables, 6 figure

Hamilton-Jacobi/action-angle-variable theory of scattering

The classical and quantum action variable theory is extended to scattering by defining the action variable for the scattering region such that the new definition is consistent with the definition given earlier for the eigenstates. The forms of both the classical and the quantum action variables are maintained for both off-eigenstates and scattering states. The location of the turning points is studied in order to define the classical action variable and the location of the poles of the quantum momentum function is studied in order to define the quantum action variable. Two types of families of states are defined classically and quantum mechanically; radial momentum families and angular momentum families. Phase integral methods are used as one method to find the location of the poles of the quantum momentum function. The Coulomb potential is used as the main example and general methods are described for the other potentials