Timeline for Behaviour of solutions to $(A-r)f=0$ in the limit $r \to \infty$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 11, 2018 at 19:06 | vote | accept | Harto Saarinen | ||
| Jul 11, 2018 at 10:24 | answer | added | Mateusz Kwaśnicki | timeline score: 2 | |
| Jul 11, 2018 at 9:53 | comment | added | Harto Saarinen | @MateuszKwaśnicki I'm assuming that the constant $c_r=1$. Sure! I think that might be something I'm after. Does it hold for the increasing solutions under similar assumptions that $\psi_r(x) \to \infty$ as $r \to \infty$? | |
| Jul 11, 2018 at 9:20 | comment | added | Mateusz Kwaśnicki | @HartoSaarinen: The limit depends on the choice of $c_r$; if $c_r = 1$, it it zero only for $x > 0$. The same will be true for any reasonable choice of $\sigma$ and $\mu$: $\varphi_r(y) / \varphi_r(x)$ will go to zero as $r \to \infty$ whenever $y > x$. If this is what you wanted, I can write the details as an answer. | |
| Jul 11, 2018 at 5:56 | comment | added | Harto Saarinen | @MateuszKwaśnicki Well at least in that case (and many other cases) it seems that the limit is $0$ for all $x$ as $r \to \infty$. In which cases does this happen? (In general almost all results that tell something how the solutions depend on $r$ are interesting.) | |
| Jul 10, 2018 at 22:29 | comment | added | Mateusz Kwaśnicki | What kind of description would you expect? That is, what is the interesting property of the solution in the simplest case, $A f = f''$, when we have $\varphi_r(x) = c_r e^{-\sqrt{r} \, x}$? The question seems to be somehow related to Krein's theory of second-order operators (Krein's strings). I do not really know nice references; you may want to take a look at an article by Kotani and Watanabe. | |
| Jul 10, 2018 at 15:22 | history | edited | Harto Saarinen | CC BY-SA 4.0 |
added 213 characters in body
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| Jul 10, 2018 at 15:20 | comment | added | Harto Saarinen | @MichaelRenardy Well almost any basic regularity conditions can be assumed. Say continuity or Lipschitz conditions etc. I didnt state any because I really dont have idea what should be assumed to achieve some results. | |
| Jul 10, 2018 at 14:18 | comment | added | Michael Renardy | You should probably give some hypotheses about the behavior of $\sigma$ and $\mu$ as $x\to\infty$. | |
| Jul 10, 2018 at 11:37 | history | asked | Harto Saarinen | CC BY-SA 4.0 |