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arXiv · 1407.0191

Explicit energy expansion for general odd degree polynomial potentials

Asiri Nanayakkara·Thilagarajah Mathanaranjan
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Abstract

In this paper we derive an almost explicit analytic formula for asymptotic eigenenergy expansion of arbitrary odd degree polynomial potentials of the form $V(x)=(ix)^{2N+1}+β_{1}x^{2N}+β_{2}x^{2N-1}+\cdot \cdot \cdot \cdot \cdot +β_{2N}x$ where $β_{k}^{\prime }$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 $β_{1},β_{2}....$ and $β_{2N}$ 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.

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Asiri Nanayakkara, Thilagarajah Mathanaranjan. 2014-07-01. Explicit energy expansion for general odd degree polynomial potentials. https://doi.org/10.1088/0031-8949%2F88%2F05%2F055004

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