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For $a$ being positive what are the quantization conditions for an exponential potential?

$$ - \frac{d^{2}}{dx^{2}}y(x)+ ae^{|x|}y(x)=E_{n}y(x) $$ with boundary conditions $$ y(0)=0=y(\infty) $$ I believe that the energies $ E_{n} $ will be positive and real

I have read a similar paper: P. Amore, F. M. Fernández. Accurate calculation of the complex eigenvalues of the Schrödinger equation with an exponential potential. Physics Letters A 372 (2008), pp. 3149–3152. doi:10.1016/j.physleta.2008.01.053, arXiv:0712.3375 [math-ph].

But I get this strange quantization condition

$$ J_{2i\sqrt{E_{n}}}(\sqrt{-a})=0 $$

However in case $ a >0 $ how can I handle with this?

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Maybe you can show how you got to your quantization condition? – Bernhard Dec 18 '12 at 11:50
the quantization condition is explained in the paper, due to the condition $ y(0)=0$ you get the quantizaton condition, in a similar way to the Airy function for the potential $ V(x)=x$ – Jose Javier Garcia Dec 18 '12 at 11:53
Bessel functions of imaginary order and argument are relatively hard to manage but this DLMF section may be of help. If everything is done correctly then I would not be surprised by imaginary order and imaginary argument yielding real roots for $E_n$. – Emilio Pisanty Dec 18 '12 at 13:11
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The potential must be attractive to have positive $E_n$. For a repulsive potential one can get quantized $E_n$, but they may become negative and unbounded from below. – Vladimir Kalitvianski Dec 18 '12 at 15:29

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