| … | … | @@ -7525,6 +7525,79 @@ with |
| 7525 | 7525 | Who knows. |
| 7526 | 7526 | Assuming it is, |
| 7527 | 7527 | the radial equations become |
| 7528 | \begin_inset Formula |
| 7529 | \begin{align*} |
| 7530 | 2AEu & =\left(1-r^{2}\right)v^{\prime}-rv\\ |
| 7531 | 2AEv & =-\left(1-r^{2}\right)u^{\prime}-\frac{2+r^{2}}{r}u |
| 7532 | \end{align*} |
| 7533 | |
| 7534 | \end_inset |
| 7535 | |
| 7536 | with |
| 7537 | \begin_inset Formula $u^{\prime}=du/dr$ |
| 7538 | \end_inset |
| 7539 | |
| 7540 | etc. |
| 7541 | Substituting, |
| 7542 | Claude claims this gives |
| 7543 | \begin_inset Formula |
| 7544 | \[ |
| 7545 | u^{\prime\prime}+\frac{2}{r}u^{\prime}+\frac{\left(2AEr\right)^{2}-\left(2r^{4}-r^{2}+2\right)}{\left(1-r^{2}\right)^{2}r^{2}}u=0 |
| 7546 | \] |
| 7547 | |
| 7548 | \end_inset |
| 7549 | |
| 7550 | |
| 7551 | \end_layout |
| 7552 | |
| 7553 | \begin_layout Standard |
| 7554 | The geodesic distance from the origin is |
| 7555 | \begin_inset Formula $\rho=\tanh^{-1}r$ |
| 7556 | \end_inset |
| 7557 | |
| 7558 | . |
| 7559 | Viz, |
| 7560 | |
| 7561 | \begin_inset Formula $r$ |
| 7562 | \end_inset |
| 7563 | |
| 7564 | was the coordinate chart coordinate, |
| 7565 | but the geodesic distance gets infinite as |
| 7566 | \begin_inset Formula $r\to1$ |
| 7567 | \end_inset |
| 7568 | |
| 7569 | which was exactly what makes this model interesting. |
| 7570 | \end_layout |
| 7571 | |
| 7572 | \begin_layout Standard |
| 7573 | Plugging through gives |
| 7574 | \begin_inset Formula |
| 7575 | \[ |
| 7576 | u^{\prime\prime}+u^{\prime}\,2\coth\rho+\left(\epsilon^{2}-3-\frac{8}{\sinh^{2}2\rho}\right)u=0 |
| 7577 | \] |
| 7578 | |
| 7579 | \end_inset |
| 7580 | |
| 7581 | with |
| 7582 | \begin_inset Formula $\epsilon=2AE$ |
| 7583 | \end_inset |
| 7584 | |
| 7585 | and prime is now |
| 7586 | \begin_inset Formula $d/d\rho$ |
| 7587 | \end_inset |
| 7588 | |
| 7589 | . |
| 7590 | Claude claims that this has the form of a |
| 7591 | \begin_inset Quotes eld |
| 7592 | \end_inset |
| 7593 | |
| 7594 | Pöschl-Teller potential |
| 7595 | \begin_inset Quotes erd |
| 7596 | \end_inset |
| 7597 | |
| 7598 | . |
| 7599 | So it knows about this. |
| 7600 | Someone must have figured this out. |
| 7528 | 7601 | \end_layout |
| 7529 | 7602 | |
| 7530 | 7603 | \begin_layout Paragraph* |