Understanding solutions of the angular Teukolsky equation in the prolate asymptotic limit

Solutions to the angular Teukolsky equation have been used to solve various applied problems in physics and are extremely important to black-hole physics, particularly in computing quasinormal modes and in the extreme-mass-ratio inspiral problem. The eigenfunctions of this equation, known as spin-weighted spheroidal functions, are essentially generalizations of both the spin-weighted spherical harmonics and the scalar spheroidal harmonics. While the latter functions are quite well understood analytically, the spin-weighted spheroidal harmonics are only known analytically in the spherical and oblate asymptotic limits. Attempts to understand them in the prolate asymptotic limit have met limited success. Here, we make use of a high-accuracy numerical solution scheme to extensively explore the space of possible prolate solutions and extract analytic asymptotic expansions for the eigenvalues in the prolate asymptotic limit. Somewhat surprisingly, we find two classes of asymptotic behavior. The behavior of one class, referred to as “normal,” is in agreement with the leading-order behavior derived analytically in prior work. The second class of solutions was not previously predicted, but solutions in this class are responsible for unexplained behavior seen in previous numerical prolate solutions during the transition to asymptotic behavior. The behavior of solutions in this “anomalous” class is more complicated than that of solutions in the normal class, with the anomalous class separating into different types based on the behavior of the eigenvalues at different asymptotic orders. We explore the question of when anomalous solutions appear and find necessary but not sufficient conditions for their existence. It is our hope that this extensive numerical investigation of the prolate solutions will inspire and inform new analytic investigations into these important functions.

Figure 1

The real and imaginary components of the first 18 sequences of ${{}_{2}A}_{\ell 3}(-i|c|)$. The plot of the real components shows linear leading-order asymptotic behavior, as anticipated by Eq. (12). The imaginary component follows a leading-order asymptotic behavior of $|c{|}^{-1}$.

Figure 2

The real and imaginary components of ${}_2{S}_{33}(x;c)$ at $c=-100i$ with an emphasis on the real zero-crossings in the outer region in the lower plot. For $s=\pm 2$, one expects to see two real zero-crossings near the points of $x=\pm 2^{-1/2}\approx \pm 0.707$.

Figure 3

The real and imaginary components of ${}_2{S}_{33}(x;c)$ at $c=-10i$ with an emphasis on the real zero-crossings in the outer region in the lower plot. At $c=-10i$, the solution is only beginning to enter the asymptotic regime. Unlike Fig. 2, the zero crossings are not well approximated by the roots of Eq. (13).

Figure 4

The real and imaginary components of the first 18 sequences of ${{}_{2}A}_{\ell 2}(-i|c|)$. Note the unusual behavior of the $L=1$ sequence, which has quadratic leading-order behavior for the real component and linear leading-order behavior for the imaginary component.

Figure 5

Close of up the same eigenvalue sequences for ${{}_{2}A}_{\ell 2}(-i|c|)$ shown in Fig. 4. Note the deflectionlike behavior that occurs between the real parts of the eigenvalue sequences ${}_2{A}_{22}(i|c|)$ and ${}_2{A}_{32}(i|c|)$ at $|c|\approx 3$.

Figure 6

Eigenvector solution for ${}_2{S}_{32}(x;-100i)={{}_2\widehat{S}}_{22}(x;-100i)$. Notice that this differs from the expected behavior described in Ref. [10]. ${}_2{S}_{32}(x;c)$ does not have a number of real zero-crossing equivalent to $L=1$ nor does the eigenvector go to zero at both end points.

Figure 7

The real and imaginary components of ${{}_{2}S}_{12,2}(x;-100i)={{}_2\overline{S}}_{11,2}(x;-100i)$. In the $c\to 0$ limit the sequence of solutions corresponds to a value of $L=10$. As such, one would expect to see ten real zero-crossings in the inner region. Here we see nine zero crossings. We account for this by noting that $\overline{L}=L-{{}_{2}N}_{12,2}=9$, since there exists an anomalous sequence ${{}_2\widehat{S}}_{32}(x;c)$.

Figure 8

Log-log plots of the magnitude of the difference between the normal asymptotic fitting function of Eq. (26) and corresponding numerically computed values of ${{}_{s}A}_{\ell m}(-i|c|)$ versus $|c|$. The $(m,s)$ pairs of $(1,-2)$, (2,2), $(-5,3)$, and (18,15) are each displayed in a separate plot showing the first 16 normal sequences. For large values of $|c|$ the slope of each line is approximately $-4$.

Figure 9

Combinations of $m$, $s$, and $L$ which yield anomalous sequences. The black dots and blue cubes designate sequences which have $L={{}_{s}N}_{\ell m}$ and are contiguous as described in the text. The gray region displays the densely filled region $\mathcal{D}$ as defined by $L\ge 0$, $14L\le 11|s|-7|m|-10$, and $25L\le 9|m|-2|s|-15$. The red tetrahedra denote the eight points within $\mathcal{D}$ that are not anomalous. Finally, the light green dots outside of $\mathcal{D}$ denote the remaining noncontiguous anomalous sequences. See Sec. 3e1 for additional details.

Figure 10

The black dots represent anomalous sequences for which $L={{}_{s}N}_{\ell m}=0$. The gray region illustrates the intersection of region $\mathcal{D}$ (see Fig. 9) with $L=0$. The dashed and dotted black lines represent the boundaries of an area defined respectively by $0\le 11|s|-7|m|-9$ and $0\le 10|m|-|s|-24$ within which all points are anomalous with $L={{}_{s}N}_{\ell m}=0$. This area is defined as $\mathcal{A}$ in the text. The points outside of the gray region are the sequences denoted by blue squares in Fig. 9.

Figure 11

The black dots, light red squares, and light blue circles represent values of $|m|$ and $|s|$ at which at least one anomalous sequence with $L\ne {{}_{s}N}_{\ell m}$ could exist. The black dots represent locations where at least one such anomalous sequence has been found within our limited dataset. The light red squares represent locations where type-1 anomalous sequences could exist, but are not present in our limited dataset. The light blue circles represent locations where type-3 anomalous sequences should exist, but are not present in our limited dataset. The dashed and dotted black lines represent the boundaries of the area $\mathcal{A}$ defined in the text (see also the caption for Fig. 10).

Figure 12

Log-log plots of the magnitude of the difference between the anomalous asymptotic base fitting function of Eq. (6) with ${{}_{s}q}_{\ell m}$ replaced by ${{}_s\widehat{q}}_{\ell m}$ as given in Eq. (27) and corresponding numerically computed values of ${{}_s\widehat{A}}_{\ell m}(-i|c|)$ versus $|c|$. The $(m,s)$ pairs of (4,4), $(-5,10)$, (7,7), and (9,10) are each displayed in a separate plot showing all known anomalous sequences. The asymptotic slope for the residual of ${{}_4\widehat{A}}_{5,4}(-i|c|)$ is $-1$. The asymptotic slopes for ${{}_{10}\widehat{A}}_{11(-5)}(-i|c|)$, ${{}_7\widehat{A}}_{9,7}(-i|c|)$, and ${{}_{10}\widehat{A}}_{13,9}(-i|c|)$ are $-3$. The asymptotic slopes for all remaining anomalous sequences are $-5$.

Figure 13

Log-log plots of the magnitudes of the real and imaginary parts of the difference between the anomalous asymptotic base fitting function of Eq. (6) with ${{}_{s}q}_{\ell m}$ replaced by ${{}_s\widehat{q}}_{\ell m}$ as given in Eq. (27) and corresponding numerically computed values of ${{}_s\widehat{A}}_{\ell m}(-i|c|)$ versus $|c|$ for the case of $m=s=4$. The asymptotic slopes of both the real and imaginary parts of the residuals of ${{}_4\widehat{A}}_{5,4}(-i|c|)$ are $-1$, while the asymptotic slopes of both the real and imaginary parts for ${{}_4\widehat{A}}_{44}(-i|c|)$ are $-5$.

Figure 14

Imaginary part of ${{}_{10}A}_{\ell 9}(-i|c|)$. The labeled curves are the four anomalous sequences. The faint curves are the first 100 normal sequences. The dashed lines show the base anomalous asymptotic fitting function for the first 6 values of $\widehat{L}$ plotted over the limited range of $50\le |c|\le 150$. The last 2, with $2_s{\widehat{q}}_{\ell |m|}+1=1$ and 5, do not correspond to valid anomalous sequences.

Figure 15

Imaginary part of ${{}_{6}A}_{\ell 6}(-i|c|)$. The labeled curves are the two anomalous sequences. The faint curves are the first 155 normal sequences. The dashed line shows the base anomalous asymptotic fitting function for $\widehat{L}=2$ which corresponds to $2_s{\widehat{q}}_{\ell |m|}+1=-1$. This line could correspond to an anomalous sequence, but there is no evidence for this for $L<157$.

Figure 16

The real and imaginary parts of ${{}_{4}A}_{\ell 4}(c)$ at $c=-200i$ for the first 180 eigenvalues when sorted by the value of the real part. In these plots, $L$ represents the sorted position at $c=-200i$. The black circles are computed using the asymptotic prolate fit in Eq. (26) with $\overline{L}=L$, and the red triangles are the numerically computed values. The base asymptotic anomalous fit for $\widehat{L}=0$ and 1 are displayed as dashed lines since their position in the list of eigenvalues cannot be predicted.

Figure 17

The real and imaginary parts of ${{}_{4}A}_{\ell 4}(c)$ at $c=-300i$ for the first 280 eigenvalues when sorted by the value of the real part. See Fig. 16 for additional information. In this figure, eigenvalues are plotted beyond $L=250$ where numerical errors are significant as can easily be seen in the plot of the imaginary part.

Figure 18

The real and imaginary parts of ${{}_{6}A}_{\ell 6}(c)$. The real part is plotted for $c=-200i$ while the imaginary part is plotted for $c=-200i$, $-300i$, $-400i$, and $-600i$. See Fig. 16 for additional information. The base asymptotic anomalous fit for $\widehat{L}=0$, 1 and 2 are displayed as dashed lines. However, the numerical solutions only indicate anomalous sequences for $\widehat{L}=0$ and 1.

Figure 19

The real part of the 3 consecutive eigenvector solutions ${{}_{2}S}_{64,2}(x;c)$, ${}_2{S}_{32}(x;c)$, and ${{}_{2}S}_{65,2}(x;c)$ at $c=-100i$. When eigenvalues are sorted by the real part of ${{}_{s}A}_{\ell m}(c)$, these three eigenvectors correspond to ${62}^{\mathrm{nd}}$, ${63}^{\mathrm{rd}}$, and ${64}^{\mathrm{th}}$ eigenvalues. ${}_2{S}_{32}(x;c)={{}_2\widehat{S}}_{22}(x;c)$ corresponds to an anomalous sequence with $L=1(\widehat{L}=0)$, while ${{}_{2}S}_{64,2}(x;c)={}_2{\overline{S}}_{63,2}(x;c)$ and ${{}_{2}S}_{65,2}(x;c)={}_2{\overline{S}}_{64,2}(x;c)$ correspond to normal sequences with $L=62(\overline{L}=61)$ and $63(\overline{L}=62)$. At this large value of $L$, the normal sequences are not yet in the asymptotic regime.

Figure 20

The real part of the three eigenvector solutions ${{}_{2}S}_{64,2}(x;c)$, ${{}_{2}S}_{65,2}(x;c)$, and ${}_2{S}_{32}(x;c)$ at $c=-300i$. These are eigenvectors along the same three sequences as in Fig. 19 but further into the asymptotic regime. ${}_2{S}_{32}(x;c)={{}_2\widehat{S}}_{22}(x;c)$ corresponds to an anomalous sequence with $L=1(\widehat{L}=0)$, but it now corresponds to the ${190}^{\mathrm{th}}$ eigenvalue. The normal sequences ${{}_{2}S}_{64,2}(x;c)={{}_2\overline{S}}_{63,2}(x;c)$ and ${{}_{2}S}_{65,2}(x;c)={{}_2\overline{S}}_{64,2}(x;c)$ correspond to the ${62}^{\mathrm{nd}}$ and ${63}^{\mathrm{rd}}$ eigenvalues and are now in the asymptotic regime.

Figure 21

The real part of the $\widehat{L}=0$ eigenvector solutions ${{}_s\widehat{S}}_{\ell m}(x;c)$ with $s=m=2\to 7$ and $c=-200i$. The anomalous type of each increases by 2 from type-1 for $m=s=2$ to type-11 for $m=s=7$.

Figure 22

The real part of the $\widehat{L}=1$ eigenvector solutions ${{}_s\widehat{S}}_{\ell m}(x;c)$ with $s=m=4\to 7$ and $c=-200i$. The anomalous type of each increases by 2 from type-1 for $m=s=4$ to type-7 for $m=s=7$.

Figure 23

The real part of the $\widehat{L}=2$ eigenvector solution ${}_7{\widehat{S}}_{97}(x;c)$ at $c=-200i$. This solution is type-3 anomalous. A type-1 anomalous solution might exist as ${}_6{\widehat{S}}_{86}(x;c)$ but was not found after searching as far out as $L=378$.

Figure 24

The real part of the $\widehat{L}=0\to 3$ eigenvector solution ${}_{10}{\widehat{S}}_{\ell 9}(x;c)$ with $s\ne m$ and $c=-200i$. The anomalous type of each decreases by 4 from type-15 for $\widehat{L}=0$ to type-3 for $\widehat{L}=3$.

Figure 25

Eigenvalue sequences for ${{}_{4}A}_{\ell 4}(-i|c|)$. Note the unusual behavior of the $L=0$ and $L=28$ sequences which have quadratic leading-order behavior for the real components and linear leading-order behavior for the imaginary components. The left plot includes the sequences with $0\le L\le 28$, so the uppermost sequence is anomalous. The right plot includes 13 additional sequences to illustrate the behavior of these sequences following the transition of the $L=28$ sequence to anomalous prolate asymptotic behavior.

Figure 26

Eigenvalue sequences for ${{}_{7}A}_{\ell 5}(-i|c|)$. Note the unusual behavior of the $L=0$ and $L=7$ sequences which have quadratic leading-order behavior for the real components and linear leading-order behavior for the imaginary components. Like Fig. 25, this set of sequences has two anomalous sequences with the first at $L=0$. In this case, however, the second anomalous sequence appears sooner, and we can zoom in on a smaller region on the plots.

Figure 27

Eigenvalue sequences for ${{}_{9}A}_{\ell 10}(-i|c|)$. Note the unusual behavior of the $L=0$, $L=1$, and $L=10$ sequences which have quadratic leading-order behavior for the real components and linear leading-order behavior for the imaginary components. This case is similar to Fig. 26 but include three anomalous sequences.

Figure 28

Eigenvalue sequences for the real part of ${{}_{10}A}_{\ell 9}(-i|c|)$. The behavior of the imaginary part can be seen in Fig. 14. Note the unusual behavior of the $L=0$, $L=1$, $L=3$, and $L=78$ sequences which have quadratic leading-order behavior for the real components. The left plot includes the sequences with $0\le L\le 78$, so the uppermost sequence is anomalous. The upper-right plot focuses on the region of the plot were the $L=28$ anomalous sequence transitions to anomalous prolate asymptotic behavior. The lower-right plot focuses on the behavior of the first three anomalous sequences.

Figure 29

Eigenvalue sequences for the real part of ${{}_{s}A}_{\ell m}(-i|c|)$ showing only the real parts of anomalous sequences grouped by anomalous type. Each plot includes all sequences of a given type found with $0\le m\le 10$ and $0\le s\le 10$. The upper-left plot displays the type-1 anomalous sequences. Note that several additional sequences may exist, but may occur for such large values of $L$ that we have not found them. The upper-right plot displays the type-3 anomalous sequences. The lower-left and -right plots show, respectively the plots for type-5 and type-7 anomalous sequences.

Figure 30

Eigenvalue sequences for the imaginary part of ${{}_{s}A}_{\ell m}(-i|c|)$ showing only the imaginary parts of anomalous sequences grouped by anomalous type. Each plot includes all sequences of a given type found with $0\le m\le 10$ and $0\le s\le 10$. The upper-left plot displays the type-1 anomalous sequences. Note that several additional sequences may exist, but may occur for such large values of $L$ that we have not found them. The upper-right plot displays the type-3 anomalous sequences. The lower-left and -right plots show, respectively the plots for type-5 and type-7 anomalous sequences.