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Eigenvalues of the quartic anharmonic oscillator $-y''+(x^2+\lambda x^4)y$
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Numbers
$\lambda$
$n$ 
$E_n(\lambda)$
1/8
0:
1.079410395210256991287619792328582726650058425343618771820032159625357130497234713975202836191141381
1/8
1:
3.369789878265709308685016189216812292640552699665136573088923030036576563926792183432631958383305893
1/8
2:
5.893512863792018390427315164856170039974944096950050228430845176652804117996300484088927933544123106
1/8
3:
8.604262908158421967737343009182058604275761894350760042840379806588405929655791768026257033340849998
1/8
4:
11.47454675624305170117676509961470465309514499565092281909821088478306743940435992986617337353539901
1/8
5:
14.48549785335369321656493363236095837015338365725019381613846078870715046619073651764019042393993933
1/8
6:
17.62314948912817378971743250107655805739220102886661042950897665655221809092828254975380104463280145
1/8
7:
20.87664165266752545971877785173946474864234685390957088109381331468132136454080907066473119480848157
1/8
8:
24.23723083527248354083776301912986377372788847931576856550659069958350443260609556878804653294606038
1/8
9:
27.69769134400999536482535403503554862005514475617772712871322564231899083752745722478538130099614042
1/8
10:
31.25192896538599561915323993557620911894715290314710874592528099928704283585138529725308007336556181
1/6
0:
1.101589436370785394595005309829551735787751265119885392341278644002598865471728794059593165517282597
1/6
1:
3.466949604519811667839360250829978788257957085872508596261062276533104255759207826521202498443852357
1/6
2:
6.115060887126414549654810103673462938756975028979557617448115461266342109902766708585230780386333548
1/6
3:
8.983064361428597413877513178426274680290190067509531291625669815519082709473828661243857764215279762
1/6
4:
12.03611064351919694434937562809446478339643903876199135677839640834768648593583220604053761804615952
1/6
5:
15.25096407458534122900769690152022110371825464793503892370613394309865724372983376033512424929596510
1/6
6:
18.61072819021518993453293405031118853289922147313948829982603319401077180952844429403094693262798199
1/6
7:
22.10242583999139058585414363242874911252368397357476598293105344590351797843671372205044960334738475
1/6
8:
25.71570364282975512847650551016139908512347892752999021505825726249577022473220148702616029913171799
1/6
9:
29.44206548128963170278178967482543444936929611939775177128583415851449250849908056318156313822070349
1/6
10:
33.27438583289195863317948130673225848809613551120722201282558332478524289356528818329042771032379776
1/4
0:
1.141901839539148990454163377984541051903941698076066469463383581179227495096026700832957630592531457
1/4
1:
3.639482049866188290668871816347292992908706202835329242241159070578965787316605797224387754024745522
1/4
2:
6.500905725742838461153551159756822022514380815637990419892660316556634367814881292485245787868281412
1/4
3:
9.633627909179422731461231888520148944505266390852205384990338810215848965173999875926009954872162989
1/4
4:
12.99102590486625489842329290675294966786606409660001062053519053494979786996792001784488453141880185
1/4
5:
16.54300099024994889633746310221126443851701520383697962566236077078514092796909544520088504000745263
1/4
6:
20.26810197643100392203296538164833804485494954033799981418809492684163156202127089571616494473776437
1/4
7:
24.15007729136102486555037495222763225184666953296002965126935010893523757166644502126962190300355426
1/4
8:
28.17609045368534169618674109130142319888818984236999218386173754963310125209998193865181089495933969
1/4
9:
32.33568855412641224258551167119799544837858053158837598494746536646851137140831489295749622623173540
1/4
10:
36.62015859492688821029465161806089815147056776965265549963957765619728503146094701521338330532692380
1/3
0:
1.178113465814610339784667464346328999926396785302595380585073740048697646966696468211031772362733255
1/3
1:
3.791014425707524322202748533653617042056145114158728041396883266307287974702146962395361420182164681
1/3
2:
6.833676443122706953940146171909893195300538678061261814126234976099162138172398559053926934887397345
1/3
3:
10.18773932766081675186722377021342685753527798963656568892035814045347081744732964422229934146766144
1/3
4:
13.79737252703827832422383350696339905919259837883024673036401370135050850068487134599865502027047573
1/3
5:
17.62712812657714898946010945539189253059575493890885746306614823624607464008855688674472219223317571
1/3
6:
21.65202146224877734820688446220494670628931399340407712553584874299402346934852439629901946462613011
1/3
7:
25.85326480375607374731943249091415947804732708170356644796663264162376853199793109232231685780369266
1/3
8:
30.21609924360655004370783253755845923118054440449504871859404176394752969150111350571738329510190946
1/3
9:
34.72855667778798708953789094960418165271318406729382504474470590535590023813037637387112578508714399
1/3
10:
39.38069433060648433674302061255839143238397836787161657109026561367414151132759523594437001270415667
1/2
0:
1.241854059651497321716071465974241396400034507238277965084734650125925496377537767958782702606958912
1/2
1:
4.051932328333313994170140685592195145441730075812377933455887117541122947154813800708012459057417142
1/2
2:
7.396900638756165707144934064427198956981284891380829907138152453286744068977280672435934219015942237
1/2
3:
11.11515427711363800867133817392675386559734147983780830086319085266911682122600077442022687712092563
1/2
4:
15.13684574711999049660804720154005955917486578444262290758334463175706611758017704181588600445276235
1/2
5:
19.41829575322669831705668419587687155823987673339073374758892252579850236178030563005605064327218489
1/2
6:
23.92908724126157873869532741450481351346728667846189720092567544462402450466167497971783916676338605
1/2
7:
28.64653039794302604259101682559400328098543302872318472186916031046780330349534086330267109114498022
1/2
8:
33.55290557249545989635905402974332208086712257161814679478835508093598830235607736853718824033267479
1/2
9:
38.63390860797234760312060824489212602869606047384918780559897503995958264493408341272392832401052012
1/2
10:
43.87769872474737662368252296785496640973543791875338454928347639797978513935145929686049518224366941
2/3
0:
1.297396125479060103126064339439101677092833369229363495862410954258718086877258730798564682597227607
2/3
1:
4.274775654669183821802811836897073013366050363004765997175659022138990479532047250664337223693626311
2/3
2:
7.870646283530196406650286028521750093821039489352743581031885729023400036953233737250376707478893833
2/3
3:
11.88782543932922110367647626506863310668606670614875749846834335024639919035376736754105299761229454
2/3
4:
16.24582391942989200873927709283706874698484209204647067582140673804138060706535766454376554089742475
2/3
5:
20.89459264079110472063546685239969031574412059777013592371944169455493267201066304375313552686363798
2/3
6:
25.79950783026004234930213387793623813787892962789193410734672337927071236451805170183204435284484834
2/3
7:
30.93485255698329377146077861409582972595504971292232894942223550869956780661943449749747842273070574
2/3
8:
36.28060679168636195045755979106014191250754353989213377827641094030815245062931962500337235658163823
2/3
9:
41.82064888020607647714793366369035053488140638824072459833594886989346808658831024731458096773812930
2/3
10:
47.54165956983471559579951997499297179561569786374663332137834548113524696221885059920964349020508715
3/4
0:
1.322872581463487858215080139416958333224272512512781257090212285371418716489820176137653989510044279
3/4
1:
4.375889248196178255536502702038539296929427251210982049672580482762445261828459095281581396653153663
3/4
2:
8.083870917219202894009391891409835059893017339930094483360264094988574035288847705845228243773584056
3/4
3:
12.23389340873043661253125011863211091995457868522103438903630740320257609164782296130011316512030771
3/4
4:
16.74094059360428150598468831786726767442832538468200738853089652997448074592516702287106287013490864
3/4
5:
21.55221736509632214713077530155342514900675169345529143067994730478773564597364920599990367866244246
3/4
6:
26.63128512024648497176771422019436586732684243293333613737036436537436618638450814780770574736088357
3/4
7:
31.95111933552058847816630225199941976329430888499455224882025706321339263376498893680888367935067339
3/4
8:
37.49070533202322011100810652583499901359549904916180256794763550119117366868519989419260840993844010
3/4
9:
43.23313494530637216060566954374933468514353798254956204159200297623860146643344598487398462633111704
3/4
10:
49.16444867044472665039728705058503818449124767522424323022717096471042708863781710866516103149027454
1
0:
1.392351641530291855657507876609934184600066711220834088906349323877567431875646528590973563467791759
1
1:
4.648812704212077536377032917260584488898860447882825934823424910341008252816076588279145639554119683
1
2:
8.655049957759309688116539457377308026273939986602632848055562015955620493646006200429249974659616150
1
3:
13.15680389804987507920977204038231467465014830179100966377306029593556519940963708997106840530942701
1
4:
18.05755743630325289477123964652543485324473508246805290779494284390576321398588139341692940903865046
1
5:
23.29744145122318908486448199209812382812079006195109084026047468848511282625204622501485893217014564
1
6:
28.83533845950424884013363571549983816085706021423392346498250467941291354754715167406529650906679845
1
7:
34.64084832111133254288452761815634203376768401553268680354733698252993920685721165927338588600756590
1
8:
40.69038608210644472527893148158246478601010434102850872320718187687097313475783470458913368552763122
1
9:
46.96500950567552798409644332417511425254429089045798162780669056671982648326504379227326643517558125
1
10:
53.44910213966526460083150645975947696224384977945168556607244540121011684523277387914707984685236106
4/3
0:
1.472956697647174192433160623577833332537318850775265567913430051782892275709833968527488491065935035
4/3
1:
4.961286470042171842863809537055777127459733301549617322509330060047491993252068136824358897200475593
4/3
2:
9.302747476358323548116626517317949038598043845151942497902082971577897520170065709457098546615233228
4/3
3:
14.19759606534150639381014477538066750938604983740030433110256616230832421193762130019065502606677162
4/3
4:
19.53711894174475991782559397821139513665733108048266717404385939982714186216474490501016825278886277
4/3
5:
25.25380411966435743340108354174872729698193250199286930116089692936101940251950452013351100927196207
4/3
6:
31.30147348206589507247022508033167617962998530601953007181028563422630140377278428490416564723615665
4/3
7:
37.64606928086838688713860506102644214158567111589264389278086212364352261015253405044851802002789912
4/3
8:
44.26121791750088434951437401439627876259647943921143362778946227874847136154619443075535589110791192
4/3
9:
51.12576792122586644688919470508929952085829822588593725861885240893108853825573808775012485487151449
4/3
10:
58.22230161917658415838206560856754146702420184762061658684519562374168065086540469935170395509872361
3/2
0:
1.509415693287616840634198585142778744847157956569142635228738101155421083455936727362062028272583131
3/2
1:
5.101449442145987866036948826270011064916487689139606257613978452228458257894197442006534086623667949
3/2
2:
9.591537272056562672399942251943413875951300309954144881762781350605709147228141695558113122712801289
3/2
3:
14.66009194987800054536696229826597267359245372925841938803643797980345028106646847749523583654122766
3/2
4:
20.19318255405882946745104843369693895141560290728909471886911133013608873431794412421053000287983418
3/2
5:
26.11998864009803440309198117791529392499138828058755422569382893349601540872035969591288546009400785
3/2
6:
32.39213985968262711063953312251452015663491742535628085497002111843806105824133245380185697233510220
3/2
7:
38.97399215477609400077761692303816140891504532678539912377443881082597902238099819915218316274013893
3/2
8:
45.83796146797537781918428119380418865417821964502240066433712483429504489937224610806057882794964593
3/2
9:
52.96193672335204209911822114029304686404760524312049983875816438067971398473701209506881195384720871
3/2
10:
60.32771743176602387117012633434763820519516916802006221385081226483197277191723267582159743827743917
2
0:
1.607541302468547538708171929473248382081047203719075902642037259869777154502771545753422159859515040
2
1:
5.475784536016868521912474563739999415667991366875796146162011142062198250851185786463628719588191530
2
2:
10.35858337527878191804572572842158141819373877431093232386016306612454618464800021670412010002245148
2
3:
15.88480796878193009343001544119895305892912248624258608310820358426875176504827861942009685164291080
2
4:
21.92716618825494585159810342703014865589041759988740310163406847025744896465850702834910729120933595
2
5:
28.40627820905766052060430346210998460197686712401942480558637474608146808602747262885373155353185079
2
6:
35.26809823227617922626053015082494721718558345792708609512558688095518306264854771143857198855651237
2
7:
42.47287097351653958640940604905628001991425877867659132336479246251309362433707902997484001163527811
2
8:
49.98987281900384719430301175554200270204255196946139571419167957852726781620848263789403865077116173
2
9:
57.79450223844589817497338753687643722556745115396902786112628902543578456904813232488047643983791120
2
10:
65.86652608041113079066883118365232640923813391900450730204990103835659178026678543002340887344244532
3
0:
1.769588844280393518255295938842426818646411219399536666623546556413018995054694478795822629162267765
3
1:
6.086896442439898381574009900153910913828539093793725372181498798974714358433069464360149030721807590
3
2:
11.60065817774456947672262078011437557060749802108460533160123119677522688040729965758224586109696566
3
3:
17.85931650214142740938515453446260864035444009433518977495023765266114323276067891695953827340489546
3
4:
24.71503545873142297023541681571175053123926928924811744111263448075744311492139985702069307676457123
3
5:
32.07509317238299987846474779882474601579015250917368532920728360606980063302221394085387434509012519
3
6:
39.87658741889396635558505291192411738529130021414382875318129151315813701511725150007084905905558047
3
7:
48.07333712851443402120618139174511314681106340553373465627420171269096359075386949967676571664813203
3
8:
56.62970599281528471158752545113044005500518353853313608180427190996834307646667858639148879545310850
3
9:
65.51719193284409705143697527954067795042376373614269855870947044633958845668459857071359758395859339
3
10:
74.71237587279875263732244303729436839236529961717018638771017466390718692302521072268655062413544282
4
0:
1.903136945459000022293850722201023931817313964690420813149118263904470277503395813675858423264848190
4
1:
6.585735642868931845383992870543423816264488910074475135186072170283041054904963024875620248920050660
4
2:
12.60776113489305219979329091755229072449119368642933326900378690420260521267137856492225605292982502
4
3:
19.45464634540741003107821931981263348168752049912317848075332385301123666881630554909032553583943921
4
4:
26.96255167207717876765418157211501432388724829549318495513743105885630321818102624231522354535708146
4
5:
35.02826479850614185197933641752413289303670541316929573842794336149397842417110327762755364728220038
4
6:
43.58191278359303179488635048225869586977266854837470829991685651431126852055923739684663072816452311
4
7:
52.57225031211362098205170977851631778990946104217206881901807746221561008880776799179635054332764204
4
8:
61.95976567587673915051884898675557535418738505117669727973488475877974029831034789380335270699076828
4
9:
71.71287753333194229306035712219872701112827556695091760991253222270943788030666012823544461042416369
4
10:
81.80565017090208991922919442453373763622844045700572767772290829662307835442258560325191583040521536
6
0:
2.120532929394275013736510953174065013012524819610719852384593932148902749391147003364910402936827530
6
1:
7.391326014966009075793628264934138064731771625285118815322952253788641019449351567387172146128613047
6
2:
14.22518143910658255173814662649686153010665730998714752309575952288585325768665021374902045767870214
6
3:
22.00946709912227100963254305109877517724111177060305921167611660496408273599234432703021481108435813
6
4:
30.55540672518218717971615313202952068605360036317678440439686695338711532472505559218519212661673553
6
5:
39.74335324484715110551352470283060649897053617370607071266793696626996282208129439085210933318344217
6
6:
49.49250225918509709578382674965376580132807856460025808032863459709330118715903987356306720489497616
6
7:
59.74365835957382559791010616179642224431819949003491667752436219278644310453265078477847001192498318
6
8:
70.45122140177500365332725132155138245293728816577605852590036858614621081676168760788461136543768362
6
9:
81.57876944520599983437780501556757727550456292938569838787334577177579699170166414845065750306359648
6
10:
93.09640867173952419402937566634074232780286075040886401813769854075689417175707359431961883119909382
8
0:
2.297577828252073167582987472793968264673741540078165244637591473167615203105101649002000308193087608
8
1:
8.043131331694987562245243619968208460745733279751811924707197874111022221008037216432948659329899492
8
2:
15.52796036764321999893840057684599354943964199152793973035453262353481796959527506637839005846586085
8
3:
24.06259482140560899966863313079337290564240151272348620419974623535656480053462137956915393488783523
8
4:
33.43862332044408427078035378429454299732699961419487388788279897817995090304433864900614400914598127
8
5:
43.52342264175461976999377229614477656384536170767995000264411092925499946938100699993337846975131582
8
6:
54.22754935801666107343353196128888407150042138965040554106301385986772992441818835026710673581658981
8
7:
65.48551992816509255922879902433210166663955049460297989728661598808063826823937023585632321135953109
8
8:
77.24691558829575009975921798783419952104429251265204786209548146175494806903554545180453642865728870
8
9:
89.47148149414350385809282802576861817026475017742727490414345371049494412397278121094895838241691271
8
10:
102.1261884600841392395919011584897186429019786972145261345449580531892279778708523305719433604233412
Definition
For $\lambda>0$, this table gives the eigenvalues $E_n(\lambda)$ of the Schrödinger operator $-y''(x)+(x^2+\lambda x^4)y(x)=E y(x)$ on $L^2(\mathbb{R})$ [1]. The eigenvalues are listed in increasing order, and the level index $n$ starts at $0$ for the ground state.
Parameters
$\lambda$
—   quartic coupling ($\lambda>0$)
$n$
—   energy level (integer with $n\geq0$)
Formulas
(1)
If $\mathcal{E}_n(\alpha,\beta)$ denotes the $n$th eigenvalue of $-y''+(\alpha x^2+\beta x^4)y=E y$, then $\mathcal{E}_n(\alpha,\beta)=\alpha^{1/2}E_n(\beta\alpha^{-3/2})$ for $\alpha>0$ and $\beta>0$. Thus $E_n(\lambda)=\mathcal{E}_n(1,\lambda)$.
(2)
With $p=-i\,\frac{d}{dx}$, if $e_n(g)$ is the $n$th eigenvalue of the common $\frac12p^2+\frac12x^2+g x^4$ normalisation, then $e_n(g)=\frac12 E_n(2g)$ [1].
(3)
With $p=-i\,\frac{d}{dx}$, if $f_n(g)$ is the $n$th eigenvalue of the $p^2+\frac14x^2+g x^4$ normalisation used by Bender and Wu, then $f_n(g)=\frac12 E_n(8g)$ [1].
Comments
(4)
The index $n$ is the number of nodes of the square-integrable eigenfunction. The rows with even $n$ have even parity, and the rows with odd $n$ have odd parity.
(5)
The harmonic limit $\lambda=0$ is not included. In this convention it has the exact eigenvalues $E_n(0)=2n+1$.
References
[1]
C. M. Bender and T. T. Wu, Anharmonic oscillator, Physical Review 184 (1969), 1231-1260. (doi)
[2]
H. Taşeli and M. Demiralp, Studies on algebraic methods to solve linear eigenvalue problems: generalised anharmonic oscillators, Journal of Physics A: Mathematical and General 21 (1988), 3903-3919. (doi)
[3]
F. M. Fernández, On the Rayleigh-Ritz variational method, 2022. (arXiv)
Links
Similar tables
Dirichlet eigenvalues of the classical planar domains —   stores eigenvalues of differential operators with boundary conditions rather than one-dimensional confining Schrödinger operators
Hermite polynomials in physicist's convention —   in the harmonic limit $\lambda=0$, the eigenfunctions are the Hermite functions $H_n(x)e^{-x^2/2}$, up to normalisation
Hermite polynomials in probabilist's convention —   in the harmonic limit $\lambda=0$, the eigenfunctions are $He_n(z)e^{-z^2/4}$ with $z=\sqrt{2}x$, up to normalisation
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds all $0\leq n\leq10$ for $\lambda\in\{1/8,1/6,1/4,1/3,1/2,2/3,3/4,1,4/3,3/2,2,3,4,6,8\}$ (165 entries). The selection includes the small positive ratios $a/b$ with $1\leq a,b\leq4$, together with $1/6,1/8,6,8$ to extend the weak- and stronger-coupling cases. These are selected couplings, not all positive $\lambda$; the harmonic limit $\lambda=0$ remains outside this table.)
How they were obtained:

The generator computes Rayleigh-Ritz eigenvalues in a harmonic-oscillator basis, with even and odd parity separated. For each coupling it uses the two truncations $N=360$ and $N=480$, and the two basis frequencies $\omega_1=\max(1.3,(1+3\lambda)^{1/3})$ and $\omega_2=1.18\omega_1$.

more

Eigenvalues of the finite Galerkin matrices are isolated by Sturm counts from an $LDL^T$ factorisation, rather than by a dense eigensolver. Each entry is the decimal whose last-place interval contains the union of the four computed values; the generator refuses to write a row unless that still leaves 100 significant digits. Rayleigh-Ritz eigenvalues are upper bounds for the true eigenvalues by the variational principle [3], so the agreement between truncations is not a rigorous error bound. Before entries are written, the generator checks the $\lambda=0$ harmonic-oscillator control for both parities, checks the even levels against the values tabulated by Taşeli and Demiralp [2], and checks that all stored rows retain at least 100 digits across the four runs.



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