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yesterday answer added Laithy timeline score: 0
May 9 at 20:19 comment added Igor Khavkine Happy to hear it! If you summarize the key points of your solution in an answer below, it might help someone with a similar question in the future.
May 9 at 16:19 comment added Laithy Thank you @IgorKhavkine. I finally proved the required bounds for $P_{\ell}$ and $Q_{\ell}$ and completed the proof of the desired estimate. Thank you for all your help. :)
May 4 at 7:28 comment added Igor Khavkine The Frobenius method gives you asymptotic estimates as $r\to\infty$ for fixed $\ell$, so you already have them. It looks like the sticky point for you is the dependence on $\ell$, meaning that you should look for bounds like $P_\ell(r) \le C r^\ell$ and $Q_\ell(r) \le C r^{-\ell-1}$ that are uniform in $\ell$ on $r\in [1,\infty)$. While there are methods to get such bounds directly from the differential equation (e.g., WKB uniform approximation) it will probably be easier to find them in the literature. Have a look for instance at MO304693.
May 3 at 17:02 history edited Laithy CC BY-SA 4.0
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May 3 at 16:50 comment added Laithy @IgorKhavkine I have tried what you recommended. The required estimates we need on $P_{\ell}$ and $Q_{\ell}$ seem to be difficult unless I'm missing something obvious. I edited the post to show my progress.
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May 2 at 16:41 comment added Laithy I have tried using Frobenius method and got all the coefficients, but the expression was very. But I didn't know that $(r^2-1)W(r)$ is a constant! This simplifies the expression significantly since there is no sum in the denominator! I will try this today. Thank you Willie and Igor.
May 2 at 7:59 comment added Igor Khavkine In addition, by Lagrange's identity, $(r^2-1) W(r)$ is simply a constant. What you are still missing are some asymptotic estimates on $P_\ell(r), Q_\ell(r)$ for large $r$, which you can get from the Frobenius method. Otherwise, it's just as Willie said
May 2 at 7:19 comment added Willie Wong Your variation of constants formula is wrong. It should be $$ u(r) = P_\ell(r) \int_r^\infty Q_\ell(t) f(t) W(t)^{-1} (t^2 - 1)^{-1} ~dt + Q_\ell(r) \int_R^r P_\ell(t) f(t) W(t)^{-1} (t^2 - 1)^{-1} ~dt $$ With the extra decay inserted I think exactly the same calculation that Igor performed for you in your previous question should work. (I suspect you got your formula from the situation where the equation is normalized so that $u''$ has coefficient $1$.)
May 2 at 2:39 history asked Laithy CC BY-SA 4.0