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Lamé band-edge eigenvalues
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Numbers
$n$
$m$
$j$ 
$h_{n,j}(m)$
1
1/2
0:
1/2
1
1/2
1:
1
1
1/2
2:
3/2
1
1/4
0:
1/4
1
1/4
1:
1
1
1/4
2:
5/4
1
3/4
0:
3/4
1
3/4
1:
1
1
3/4
2:
7/4
2
1/2
0:
1.267949192431122706472553658494127633057194746189619371944193020548066983091199962918853813242751424
2
1/2
1:
3/2
2
1/2
2:
3
2
1/2
3:
9/2
2
1/2
4:
4.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
2
1/4
0:
0.6972243622680053534403893662647520268743517130773768936447734718864165258534947773976904589907546412
2
1/4
1:
5/4
2
1/4
2:
2
2
1/4
3:
17/4
2
1/4
4:
4.302775637731994646559610633735247973125648286922623106355226528113583474146505222602309541009245359
2
3/4
0:
1.697224362268005353440389366264752026874351713077376893644773471886416525853494777397690458990754641
2
3/4
1:
7/4
2
3/4
2:
4
2
3/4
3:
19/4
2
3/4
4:
5.302775637731994646559610633735247973125648286922623106355226528113583474146505222602309541009245359
3
1/2
0:
2.050510257216821901802715925294108608034052519343329871567307432749039622542684973460140566895359765
3
1/2
1:
2.127016653792583114820734600217600389167078294708409173412426233886516908063020966480712623141326482
3
1/2
2:
5.050510257216821901802715925294108608034052519343329871567307432749039622542684973460140566895359765
3
1/2
3:
6
3
1/2
4:
6.949489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235
3
1/2
5:
9.872983346207416885179265399782399610832921705291590826587573766113483091936979033519287376858673518
3
1/2
6:
9.949489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235
3
1/4
0:
5/4
3
1/4
1:
1.594875162046672802935138632120449293215847431675718349911378119904607205531636472787283473866497552
3
1/4
2:
3.087722339831620668001106455567281466280444860674783173142495147207405561360761778655751891620699705
3
1/4
3:
5
3
1/4
4:
21/4
3
1/4
5:
9.405124837953327197064861367879550706784152568324281650088621880095392794468363527212716526133502448
3
1/4
6:
9.412277660168379331998893544432718533719555139325216826857504852792594438639238221344248108379300295
3
3/4
0:
2.587722339831620668001106455567281466280444860674783173142495147207405561360761778655751891620699705
3
3/4
1:
2.594875162046672802935138632120449293215847431675718349911378119904607205531636472787283473866497552
3
3/4
2:
27/4
3
3/4
3:
7
3
3/4
4:
8.912277660168379331998893544432718533719555139325216826857504852792594438639238221344248108379300295
3
3/4
5:
10.40512483795332719706486136787955070678415256832428165008862188009539279446836352721271652613350245
3
3/4
6:
43/4
4
1/2
0:
2.788897449072021413761557465059008107497406852309507574579093887545666103413979109590761835963018565
4
1/2
1:
2.809584240176570445434369886455533719411771646588262846394298108982975367246760278517884403938456865
4
1/2
2:
7.354248688935409409498384246360739574289740816917549819631665540798931176769716372239607113525456389
4
1/2
3:
7.809584240176570445434369886455533719411771646588262846394298108982975367246760278517884403938456865
4
1/2
4:
10
4
1/2
5:
12.19041575982342955456563011354446628058822835341173715360570189101702463275323972148211559606154314
4
1/2
6:
12.64575131106459059050161575363926042571025918308245018036833445920106882323028362776039288647454361
4
1/2
7:
17.19041575982342955456563011354446628058822835341173715360570189101702463275323972148211559606154314
4
1/2
8:
17.21110255092797858623844253494099189250259314769049242542090611245433389658602089040923816403698144
4
1/4
0:
1.821994859549012470118777608235647683481932022136251608641650775557244790606779458404240533610195588
4
1/4
1:
2.007359312880714853594933827370905764290984373869155780469960786027802564613678883448837397017075282
4
1/4
2:
4.458618734850890155500157877398966468957514952606794406790423476756833637340544880098466786021075668
4
1/4
3:
5.958497377870818818996768492721479148579481633835099639263331081597862353539432744479214227050912779
4
1/4
4:
6.635701974340424059959345585990795133169987088820053981432411079458075633134784296961950975726050184
4
1/4
5:
10.49264068711928514640506617262909423570901562613084421953003921397219743538632111655116260298292472
4
1/4
6:
10.54138126514910984449984212260103353104248504739320559320957652324316636265945511990153321397892433
4
1/4
7:
16.54150262212918118100323150727852085142051836616490036073666891840213764646056725552078577294908722
4
1/4
8:
16.54230316611056346992187680577355718334808088904369440992593814498467957625843624463380849066375423
4
3/4
0:
3.457696833889436530078123194226442816651919110956305590074061855015320423741563755366191509336245772
4
3/4
1:
3.458497377870818818996768492721479148579481633835099639263331081597862353539432744479214227050912779
4
3/4
2:
9.458618734850890155500157877398966468957514952606794406790423476756833637340544880098466786021075668
4
3/4
3:
9.507359312880714853594933827370905764290984373869155780469960786027802564613678883448837397017075282
4
3/4
4:
13.36429802565957594004065441400920486683001291117994601856758892054192436686521570303804902427394982
4
3/4
5:
14.04150262212918118100323150727852085142051836616490036073666891840213764646056725552078577294908722
4
3/4
6:
15.54138126514910984449984212260103353104248504739320559320957652324316636265945511990153321397892433
4
3/4
7:
17.99264068711928514640506617262909423570901562613084421953003921397219743538632111655116260298292472
4
3/4
8:
18.17800514045098752988122239176435231651806797786374839135834922444275520939322054159575946638980441
5
1/2
0:
3.505853361808499887010837597738809655858666135767279809329321792152206412366258530180776368581130106
5
1/2
1:
3.510874706923942680298777063562141363559471084034415264600245058868198557085190745945749268806423755
5
1/2
2:
9.637069481314314492489077386700990628092875795937539272124204033004463682730215260018786647838505584
5
1/2
3:
9.803847577293368119417660975482382899171584238568858115832579061644200949273599888756561439728254273
5
1/2
4:
13.63121611950581460547823978896218097223420966017025946279488224085225727036395672983801027925737548
5
1/2
5:
15
5
1/2
6:
16.36878388049418539452176021103781902776579033982974053720511775914774272963604327016198972074262452
5
1/2
7:
20.19615242270663188058233902451761710082841576143114188416742093835579905072640011124343856027174573
5
1/2
8:
20.36293051868568550751092261329900937190712420406246072787579596699553631726978473998121335216149442
5
1/2
9:
26.48912529307605731970122293643785863644052891596558473539975494113180144291480925405425073119357624
5
1/2
10:
26.49414663819150011298916240226119034414133386423272019067067820784779358763374146981922363141886989
5
1/4
0:
2.373327362805702578826479152877583953676955505839067870685645908298985552749704087408720094638897307
5
1/4
1:
2.462552160311898299720153583940617994134576759328518742628173471257283853528116032904554101529531410
5
1/4
2:
6.027974976831469307543407039614485207080028761428054096508265942218194576299549760630011677936988892
5
1/4
3:
7.091673086804016060321168098794256080623055139232130680934320415659249577560484332193071376972263923
5
1/4
4:
8.461266003692760563338483085781381011279767694684251961873483147096148267715716324648141476683344485
5
1/4
5:
11.86199380715357682352581729117029225978651498270908420459457002729376370908824150765147020970951074
5
1/4
6:
12.04649008684981984866670870774559428164176022741863284606913253006209660628910620248510965437006057
5
1/4
7:
17.90832691319598393967883190120574391937694486076786931906567958434075042243951566780692862302773608
5
1/4
8:
17.91540663350153685783503776134103503504327679947668016744087094460486617953457958794313842867775821
5
1/4
9:
25.67545403253452487675402912488908974607890825796239705277725650144895243738364245944397568876095785
5
1/4
10:
25.67553493631871084378988425263992051127821101115331305742260152771970881741134403688487866769295054
5
3/4
0:
4.324465063681289156210115747360079488721788988846686942577398472280291182588655963115121332307049460
5
3/4
1:
4.324545967465475123245970875110910253921091742037602947222743498551047562616357540556024311239042147
5
3/4
2:
12.08459336649846314216496223865896496495672320052331983255912905539513382046542041205686157132224179
5
3/4
3:
12.09167308680401606032116809879425608062305513923213068093432041565924957756048433219307137697226392
5
3/4
4:
17.95350991315018015133329129225440571835823977258136715393086746993790339371089379751489034562993943
5
3/4
5:
18.13800619284642317647418270882970774021348501729091579540542997270623629091175849234852979029048926
5
3/4
6:
21.53873399630723943666151691421861898872023230531574803812651685290385173228428367535185852331665551
5
3/4
7:
22.90832691319598393967883190120574391937694486076786931906567958434075042243951566780692862302773608
5
3/4
8:
23.97202502316853069245659296038551479291997123857194590349173405778180542370045023936998832206301111
5
3/4
9:
27.53744783968810170027984641605938200586542324067148125737182652874271614647188396709544589847046859
5
3/4
10:
27.62667263719429742117352084712241604632304449416093212931435409170101444725029591259127990536110269
6
1/2
0:
4.216089075431141043611411094166378323407315079569525305647081054988568379992705269666671569329744205
6
1/2
1:
4.217230099678924335470499026470206173504271960193483817218427428623756483955180834859337661308553543
6
1/2
2:
11.83484861008831998682390561254398302203108684646405619478551575218626314890445822679127688101310993
6
1/2
3:
11.88567458208462849732454465182532799671527478963147273093910476640728368533391610872371598140076010
6
1/2
4:
17.49281678891191797508047446092055593802650895453894856806418011557774557089546106630545047215847455
6
1/2
5:
18.16844448240570416185404562535512182321100282943798891372067733778352720137873527386437832009220656
6
1/2
6:
21
6
1/2
7:
23.83155551759429583814595437464487817678899717056201108627932266221647279862126472613562167990779344
6
1/2
8:
24.50718321108808202491952553907944406197349104546105143193581988442225442910453893369454952784152545
6
1/2
9:
30.11432541791537150267545534817467200328472521036852726906089523359271631466608389127628401859923990
6
1/2
10:
30.16515138991168001317609438745601697796891315353594380521448424781373685109554177320872311898689007
6
1/2
11:
37.78276990032107566452950097352979382649572803980651618278157257137624351604481916514066233869144646
6
1/2
12:
37.78391092456885895638858890583362167659268492043047469435291894501143162000729473033332843067025579
6
1/4
0:
2.901880864420670842580550387699631675224194986927292442416178594122399452746762780440172816865216746
6
1/4
1:
2.941776836049786232046805327335712932652571670995174719281207957439805975265657170870535000901598412
6
1/4
2:
7.689291564825708265736998105409112836843582737921216281570214887021930821094462883780657013923350283
6
1/4
3:
8.361261548680392815037497890115003379902840888803074530677582836675905858420139405274794656285725783
6
1/4
4:
10.67506164373821859096951841793358085325077570298904886528957939354338224848313473911555104580063889
6
1/4
5:
13.49839256989897436744149709497857219033744419815467275404243856037634665353466384258179089145661982
6
1/4
6:
14
6
1/4
7:
19.57950145973055592748441455083739944072047872338043339302061752246178129343776452362280588293514304
6
1/4
8:
19.61381282867559825905520118365640572253384234765121166785348970830324407799879514052170116887207810
6
1/4
9:
27.30983059405123940051169757768571487700998413085015252667635348218384737119967898654767410764178177
6
1/4
10:
27.31070843517429173426300189459088716315641726207878371842978511297806917890553711621934298607664972
6
1/4
11:
36.80923699158905125747808755904759717937668038781649207630179964086231284814209607110239946077913118
6
1/4
12:
36.80924466316551230739473001071038174899118696243244702444075230403097422077130733992257496846206626
6
3/4
0:
5.190755336834487692605269989289618251008813037567552975559247695969025779228692660077425031537933740
6
3/4
1:
5.190763008410948742521912440952402820623319612183507923698200359137687151857903928897600539220868819
6
3/4
2:
14.68929156482570826573699810540911283684358273792121628157021488702193082109446288378065701392335028
6
3/4
3:
14.69016940594876059948830242231428512299001586914984747332364651781615262880032101345232589235821823
6
3/4
4:
22.38618717132440174094479881634359427746615765234878833214651029169675592200120485947829883112792190
6
3/4
5:
22.42049854026944407251558544916260055927952127661956660697938247753821870656223547637719411706485696
6
3/4
6:
28
6
3/4
7:
28.50160743010102563255850290502142780966255580184532724595756143962365334646533615741820910854338018
6
3/4
8:
31.32493835626178140903048158206641914674922429701095113471042060645661775151686526088444895419936111
6
3/4
9:
33.63873845131960718496250210988499662009715911119692546932241716332409414157986059472520534371427422
6
3/4
10:
34.31070843517429173426300189459088716315641726207878371842978511297806917890553711621934298607664972
6
3/4
11:
39.05822316395021376795319467266428706734742832900482528071879204256019402473434282912946499909840159
6
3/4
12:
39.09811913557932915741944961230036832477580501307270755758382140587760054725323721955982718313478325
7
1/2
0:
4.924419415440558061670803807695080873583314781323210413506402494044898754581085595978620737717120156
7
1/2
1:
4.924667372255278328062720396473421409111725021210690133529141651234941825829737443543047082032316759
7
1/2
2:
13.98616199866105970393928745905423630987309291146694932519398661823535699630799895857872365361912207
7
1/2
3:
14
7
1/2
4:
21.29196034653585134843848287349582823483910148941503795274512520102160095159017031206640489543706509
7
1/2
5:
21.55569832185520055618912118792167161151233119001108767971847112927500637064726365620083880550030841
7
1/2
6:
26.23021776331534970616999922213667279854932335927129904105754107683114270986325694946630197953506318
7
1/2
7:
28
7
1/2
8:
29.76978223668465029383000077786332720145067664072870095894245892316885729013674305053369802046493682
7
1/2
9:
34.44430167814479944381087881207832838848766880998891232028152887072499362935273634379916119449969159
7
1/2
10:
34.70803965346414865156151712650417176516089851058496204725487479897839904840982968793359510456293491
7
1/2
11:
42
7
1/2
12:
42.01383800133894029606071254094576369012690708853305067480601338176464300369200104142127634638087793
7
1/2
13:
51.07533262774472167193727960352657859088827497878930986647085834876505817417026255645695291796768324
7
1/2
14:
51.07558058455944193832919619230491912641668521867678958649359750595510124541891440402137926228287984
8
1/2
0:
5.632172492987664645014810316276365664602105728495366024146830804458771054669301265858293723949367256
8
1/2
1:
5.632224662861870641115797728990629437908377484370614968299568020386574015716848733099278840929513411
8
1/2
2:
16.12004412237548593582258633584369015836129389106402040796466650118318052695451926961329324922260490
8
1/2
3:
16.12354214386461181383662701027701570064248497925758378792692639593945869335157925119862813454510227
8
1/2
4:
24.96392041962613568404495106031654644964777481146611776611752317394155151146089187992384648949048003
8
1/2
5:
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Definition
The Lamé band-edge eigenvalues $h_{n,0}(m)<\cdots<h_{n,2n}(m)$ are the $2n+1$ finite-gap edges of the real-line Schrödinger operator $H_{n,m}=-d^2/dx^2+n(n+1)m\operatorname{sn}^2(x\mid m)$ [1].
Parameters
$n$
—   Order ($n$ is a positive integer.)
$m$
—   Elliptic parameter ($0<m<1$.)
$j$
—   Band-edge index ($j$ is an integer with $0\leq j\leq 2n$.)
Formulas
(1)
$\operatorname{spec}(H_{n,m})=\bigcup_{r=0}^{n-1}[h_{n,2r}(m),h_{n,2r+1}(m)]\,\cup\,[h_{n,2n}(m),\infty)$ [1].
(2)
$h_{1,0}(m)=m,\quad h_{1,1}(m)=1,\quad h_{1,2}(m)=1+m$ [1].
(3)
$h_{n,j}(m)+h_{n,2n-j}(1-m)=n(n+1)$. In particular, $h_{n,n}(1/2)=n(n+1)/2$.
Comments
(4)
Here $\operatorname{sn}$ is the Jacobi elliptic sine, and $m=k^2$ is the elliptic parameter, not the modulus $k$. The energy $h$ has the normalization of [2]: the kinetic coefficient is $1$, the mean potential is not subtracted, and the potential has period $2K(m)$, where $K$ is the complete elliptic integral of the first kind.
(5)
The index $j$ starts at zero and orders the finite band edges by increasing energy. These edges admit Lamé polynomial eigenfunctions, the eight solution types of [3]. Periodic and antiperiodic boundary conditions mean that both the function and its derivative acquire multiplier $+1$ or $-1$, respectively, on translation by $2K(m)$. The higher closed-gap eigenvalues inside the infinite band are excluded [1].
(6)
At fixed $m$, these energies are the ordered roots of the Lamé spectral polynomial $P_n(h,m)$. Rational parameters give algebraic energies; rational roots are written exactly.
(7)
The parameter $m=1/2$ is the self-complementary lemniscatic case, with equal real and complementary quarter periods [5]. The complementary pair $m=1/4,3/4$ supplies bounded exploratory references for simple rational elliptic potentials on both sides of this symmetric case. The pair is a noncanonical reference choice, not a claim that either parameter is a distinguished Lamé application.
Programs
(P1)
Sage
# Download the attached generate.py into the current directory.
from generate import band_edges, LameBandEdges
band_edges(2, "1/2", digits=100)
# The default selection uses n<=12 at m=1/2 and n<=6 at m=1/4,3/4.
list(LameBandEdges().enumerate())
References
[1]
Robert S. Maier, Lamé polynomials, hyperelliptic reductions and Lamé band structure, Philosophical Transactions of the Royal Society A 366 (2008), 1115–1153. See §§1 and 7, Definition 7.1, and equations (54)–(56). (arXiv) (doi)
Links
Similar tables
Lamé spectral polynomials —   The exact spectral polynomials have these energies as their ordered roots at fixed elliptic parameter.
Complete elliptic integral of the first kind —   The integral $K(m)$ determines the period $2K(m)$ of the Lamé potential.
Characteristic values of the Mathieu equation —   The Mathieu equation supplies another spectral family for a second-order differential operator with a periodic potential.
Data properties
Entries are of type: real number
digits: 100 (Unknown key)
How well the digits are known: proven
Table is complete: no (it contains all $0\leq j\leq2n$ at $m=1/2$ for $1\leq n\leq12$, and at $m=1/4,3/4$ for $1\leq n\leq6$. The lemniscatic slice covers the symmetric family; the complementary pair supplies explicitly exploratory references for simple rational potentials with unequal period shapes. All edges are retained to cover complete spectra at each chosen order. The fractions and order cutoffs are noncanonical reference choices, independent of decimal notation; the shallower asymmetric slices give breadth before further depth, without a dense parameter mesh)
How they were obtained:

Exact rational arithmetic constructs the invariant polynomial sectors of [3], with the finite eigenvalue interpretation of [4]. Their characteristic polynomials are multiplied and factored over $\mathbb Q$. Linear factors yield the 38 exact rational entries. For the other 226 entries, exact Sturm counts isolate one real root, rational bisection refines the interval, and outward-rounded interval conversion preserves the enclosure. Every written 100-significant-digit decimal is checked to contain exactly one intended root; the ordered intervals are disjoint. The largest rational isolating width is less than $4.6\times10^{-125}$.

more

The checker compares the low-order polynomials with the independent formulas in [1], equations (54)–(56), and checks every selected specialization available in the Lamé spectral polynomial table. It checks the complementary-parameter identity and all twelve lemniscatic central roots. A separate Fourier Galerkin computation of the original periodic differential operator, with periodic and antiperiodic boundary conditions, compares all 264 energies using 48 and 64 modes on each side. Free-operator spectra and the known order-one energies are its controls. The maximum absolute discrepancy is less than $4.5\times10^{-12}$; this numerical comparison is a diagnostic and does not certify the 100 digits. Exact root certification supplies that guarantee. The attached checks are repeated on the database values.



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