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## Explicit energy expansion for general odd degree polynomial potentials

Asiri Nanayakkara, Thilagarajah Mathanaranjan

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.

http://arxiv.org/abs/1407.0191

Mathematical Physics (math-ph)

Hofstadter showed that the energy levels of electrons on a lattice plotted as a function of magnetic field form an beautiful structure now referred to as “Hofstadter’s butterfly”. We study a non-Hermitian continuation of Hofstadter’s model; as the non-Hermiticity parameter $$g$$ increases past a sequence of critical values the eigenvalues successively go complex in a sequence of “double-pitchfork bifurcations” wherein pairs of real eigenvalues degenerate and then become complex conjugate pairs. The associated wavefunctions undergo a spontaneous symmetry breaking transition that we elucidate. Beyond the transition a plot of the real parts of the eigenvalues against magnetic field resembles the Hofstadter butterfly; a plot of the imaginary parts plotted against magnetic fields forms an intricate structure that we call the Hofstadter cocoon. The symmetries of the cocoon are described. Hatano and Nelson have studied a non-Hermitian continuation of the Anderson model of localization that has close parallels to the model studied here. The relationship of our work to that of Hatano and Nelson and to PT transitions studied in PT quantum mechanics is discussed.