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I'm sorry my title is not descriptive; the function I am interested in is too long to put in there. What I am looking at is the roots of the following function: $f(\epsilon) = (\Delta^2-\epsilon^2)(\epsilon^2 - (\Gamma_1+\Gamma_2)^2/4) + \Delta^2 \Gamma_1 \Gamma_2 \sin^2(\phi/2) + (\Gamma_1+\Gamma_2)\,\epsilon^2\,\sqrt{\Delta^2-\epsilon^2}$

Here all parameters are real numbers.

Now, this function is readily evaluated with numerical methods, but I would like to know if there is some analytics we can do to end up with expressions for how things scale.

Specifically, I am interested in the limiting behavior of $\epsilon$ (for which $f(\epsilon) = 0$) around $\phi = \pi$, and its scaling with $\delta = \vert{\Gamma_1-\Gamma_2}\vert$. Indeed, one finds that for $\Gamma_1 = \Gamma_2$, $\epsilon = 0$ at $\phi = \pi$. I'd like to know how $\epsilon$ approaches 0 as $\delta$ goes to 0. Is that something that is possible analytically?

Some assumptions we can tack on, if needed, and it probably is, is that either $\Delta \gg \max{(\Gamma_1,\Gamma_2)}$, or $\min{(\Gamma_1,\Gamma_2) \gg \Delta}$, where the former has my preference as its closer to the situation I want to investigate, but the latter also has some value.

For context, the equation comes from Resonant Josephson Current through a Quantum Dot by Beenakker and van Houten from 1992 and describes the energy-phase relationship of a bound state in the system.

user3183724
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  • This is equivalent to a quartic equation in $\epsilon^2$, so an analytical solution is possible, though complicated. –  Feb 17 '19 at 17:07
  • That makes sense. In that case I should probably focus the question towards if the general solution (which does indeed appear to exist, Mathematica comes up with a very long one but it is closed form) can be simplified in some useful manner given the limits/scaling behaviour I am interested in. – user3183724 Feb 17 '19 at 17:19

1 Answers1

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To make the equation more workable, let us define first $$A=\Gamma_1+\Gamma_2 \qquad B=\Delta^2\Gamma_1\Gamma_2\qquad \epsilon=\sqrt{\Delta ^2-t^2}$$ which make $$f=-B \sin ^2\left(\frac{\phi }{2}\right) -A \Delta ^2 t+ \left(\frac{A^2}{4}-\Delta ^2\right)t^2+A t^3+t^4$$ Continuing $$C=-A \Delta ^2\qquad D=\frac{A^2}{4}-\Delta^2\qquad E=-B \sin ^2\left(\frac{\phi }{2}\right)$$ to make $$f=E+C t+Dt^2+At^3+t^4$$ which has explicit solutions in $t$ (even if nasty.

From the solutions, you could "easily" perform sensitivity analysis (the chain rule will obviously be very useful).

Claude Leibovici
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  • Thank you, that is indeed a very good starting point! I'll try and see if I can come up with how to proceed in performing the sensitivity analysis you mention. If I cannot, perhaps I should make a separate question focusing on that aspect itself, as it might be beyond the scope of a single one – user3183724 Feb 18 '19 at 12:48
  • Would you perhaps be able to go into the sensitivity analysis one could perform (without asking you to actually do it for me)? – user3183724 Feb 18 '19 at 18:11