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Prolate spheroidal differential eigenvalues
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Numbers
$m$
$S$
$n$ 
$\chi_n(c)$
0
1
0:
0.7396372387275077423045640021435829542173842383561180260297228963015707895011862751831177565869868671
0
1
1:
3.437881837578678487317165577884247677697864814620435928105359034748907521607921476984843420318391035
0
1
2:
7.346774054804104963542093658254609381168807845165356157477586313102772692677281458479279255550176169
0
1
3:
13.28196050216820134764174912151149343112901986746031582516742663251728973447305078157574946014760373
0
1
4:
21.26062653470516922696478517429088431766998951652736695646014183543933999963895577241043976276445452
0
1
5:
31.25114365826224825318938246513992106987491116214234534566805640493038168215763454072445558301666302
0
1
6:
43.24597758649363113198807645019422572573060079887607688192953725316471120580164335830949165715963299
0
1
7:
57.24282794757827674623910901437976508238312777889663420386936718289300658666578411446462856479500313
0
1
8:
73.24075942004733541910141450289063220889025150414920493487222602346800011552955778797564851213020733
0
1
9:
91.23932563707088185647444527547147809714261240405633010283004139922634218950804299258681752802304185
0
1
10:
111.2382900034477535197012183303338058790104455588945741381174619719852207182494070758484179853237430
0
1
11:
133.2375171293101639826683249014669354933167120503943654657774741756502040822192619054763883996301931
0
2
0:
2.283575944805409772036930450333231255207632593234122663047724423819502100002326750514198150539886795
0
2
1:
7.225471797613234455472177992941503734330688443125203739447269463777967584169453875278069637920920074
0
2
2:
11.71257167608532124641355834854354961929237024697383013439105302711742767112492537721706223843057724
0
2
3:
17.39166672775308856820301860000660659195257749269919822486713794078919526164029838529726215604684226
0
2
4:
25.18030205244861705740627768669181600328380693180725949917984263002641849891718341388764195201053606
0
2
5:
35.08881732831798768185280185679619226706125338112271677507313164257692375577421450705376877172174281
0
2
6:
47.04191962495861689985916637199203651824609083797610155917292862053376859002526242779704386082567060
0
2
7:
61.01400292828784004969413992578060598734066796343061203042338328984138934800656914328740362694026918
0
2
8:
76.99586444298258452852052818199505193239199177134915616237121545091216585192198245843944480669184412
0
2
9:
94.98336662204057088585290524135883617689528577328648486010663501543420746012827198805157618162653561
0
2
10:
114.9743729998213782692678087299730107902625749154776593538889317604446189359887199249286434762339975
0
2
11:
136.9676781384101850862800809304435554871518937086671185990838858332834617053931292875190345317308509
0
4
0:
5.494094019053958488451597490956331713879222897893001293968301646738504795937852265441960824087562827
0
4
1:
16.86876957021940362018148564477001931359718128760388459191420692520369656536270737623922446354007878
0
4
2:
26.83416413313985398131104604695588651624376454108809495355291854728584526794157449323449474294751287
0
4
3:
35.20641058697378336869737815157160592922589252584695640157285465479734721464423842580290799425003276
0
4
4:
42.91603375679911364728050647899942669719875728192594388682397293473315385833147342737691954571194195
0
4
5:
51.87416869079950838773546975336724704083727158229352790511853358550536145503058961742369270139365286
0
4
6:
63.15858874295850318035902624678768366853503376971157899489929666007642234370518617550644786115852969
0
4
7:
76.75995855247881225179783225164096995702541238633180731439577832475120658123599087223526747810682054
0
4
8:
92.51701320838409718073104787607046666606707291305338465532832032096701789385497460381843512452225858
0
4
9:
110.3543780322108906817273387909813143104627830435663076225241751424816839255280918867092925266427590
0
4
10:
130.2389120982647919563441722194870710144062320210487680592416230636518354219696951304674852833140960
0
4
11:
152.1535879359296457740242630756965940781986705543655953617032469172734520767157448200006375092785692
0
8
0:
11.79968500025679876394815450170848318690717163175749726678443945963645854832286537568288928584548887
0
8
1:
35.86126143142712724843372292850357033251097469095594357144045682532229898838338375808204031503080260
0
8
2:
58.80260020345197559907012487047866257159917171041438600081302186420460500758177681137413722059167886
0
8
3:
80.51805967946424024588463754305547727754597625410105480709986871667316764395475652503731498465709775
0
8
4:
100.8541310172002993006740707163277544823077936110144801869330918191971898055888010361125557686218704
0
8
5:
119.5580806566878474793134236278814495597419139200486897600442951198587277870546212997977896276840819
0
8
6:
136.2292340756357880943531803521623507488553460634579568334057111360973921603018238443350520085540165
0
8
7:
150.7130562784683553122010045471915997052207753765339156733599424673348768859868179931410806181776171
0
8
8:
164.2550191130415159765464215888973817692482104882605099957460914508326617782628215743622376169873370
0
8
9:
179.2087545848027664341644923981338076520052199845013965582936124280554284858742040000875981944276958
0
8
10:
196.9277853995966937285599428801532619478723169140777099075277664381686002371709325595629846567979919
0
8
11:
217.3924972385462533675907506789094241876539920858958557488051994172978375541557821366841817405193602
Definition
The order-zero prolate spheroidal differential eigenvalue $\chi_n(c)$ is the $n$th eigenvalue, in increasing order starting at $n=0$, for which $((1-x^2)\psi'(x))'+(\chi_n(c)-c^2x^2)\psi(x)=0$ has a nonzero solution bounded at both endpoints of $-1<x<1$ [1]. The bandwidth is $c=\pi S/2$.
Parameters
$m$
—   angular order ($m=0$)
$S$
—   Shannon number ($S>0$)
$n$
—   mode index ($n\geq 0$)
Formulas
(1)
$\chi_n(c)=\lambda_n^0(c^2)+c^2$, where $\lambda_n^0$ is the separation constant in the DLMF equation [2].
(2)
$\chi_n(0)=n(n+1)$ and $n(n+1)<\chi_n(c)<n(n+1)+c^2$ for $c>0$ [1].
Comments
(3)
The angular order $m$ and the mode index $n$ are distinct. This table concerns only $m=0$. The real eigenfunction may be chosen with $\int_{-1}^1\psi_n(x;c)^2\,dx=1$ and $\psi_n(1;c)>0$; its normalization does not change the eigenvalue.
(4)
The Shannon number $S=2c/\pi$ is the trace of the integral operator on $[-1,1]$ with kernel $\sin(c(x-y))/(\pi(x-y))$, with its diagonal defined by continuity [1]. The selected integral Shannon numbers represent small effective dimensions.
(5)
At zero bandwidth the differential problem reduces to that for $P_n(x)$. The zero-bandwidth values are a limit of the positive-bandwidth family.
Programs
(P1)
Python
# SciPy provides a double-precision approximation.
from math import pi
from scipy.special import pro_cv

def chi(S, n):
    return pro_cv(0, n, pi * S / 2)

chi(1, 0)  # approximately 0.7396372387275103
References
[1]
A. Osipov and V. Rokhlin, On the evaluation of prolate spheroidal wave functions and associated quadrature rules (2013), §§3.1–3.2. (arXiv)
Links
Data properties
Entries are of type: real number
Table is complete: no (it holds all modes $0\leq n\leq11$ at angular order $m=0$ for $S\in\{1,2,4,8\}$, a finite benchmark selection of small integral effective dimensions that reaches both sides of the concentration transition)
How well the digits are known: heuristic (agreement-checked)
How they were obtained:

The attached generator uses the symmetric tridiagonal representation in the orthonormal Legendre basis, separately for even and odd degrees [1]. Sturm counts select the ordered eigenvalues. Finite truncation is the method described in [3].

more

Values are compared at 160 and 220 decimal working digits and with 80, 120 and 180 basis terms in each parity block. At least 110 significant digits of agreement are required before 100 significant digits are retained. An independent endpoint Taylor-series solution of the differential equation imposes the even or odd condition at the origin; it also requires at least 110 significant digits of agreement. Zero-bandwidth Legendre eigenvalues provide a known-answer control. SciPy pro_cv provides an external double-precision comparison on every entry [4].

These checks support an agreement-based accuracy claim. They do not supply a rigorous bound for the infinite-operator truncation or for the endpoint-series remainder. The attached checks repeat the bounds, ordering and independent boundary-condition test on the values read back from the database.



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