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Eigenvalues of the quartic double well $-y''+(x^2-a^2)^2y$
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Numbers
$a$
$n$ 
$E_n(a)$
1/2
0:
0.93251751837161214795679128077017420476113257103299
1/2
1:
3.3962793298870063641567670946690868402310766754160
1/2
2:
6.8840966004392994577699048427791660146551278717424
1/2
3:
10.916371833189373003318852744032487694729981967799
1/2
4:
15.391675083688140944142345392107421542254436356031
1/2
5:
20.236551883175342556551997109431175481715413438255
1/2
6:
25.402743184670170554579939082907792515590859381940
1/2
7:
30.855167942808961999118238897059783282565860511220
1/2
8:
36.566993608583701828490497294327436652086719446610
1/2
9:
42.516907510077379579605839583604690881816293736840
1
0:
1.1377858481882225086396342454552475007884655889426
1
1:
2.7130278977676756391961261124876165656757588410548
1
2:
5.7824297093038696420457389259084269529613230322018
1
3:
9.3328681947663418353657173214148725502338579171750
1
4:
13.384724625277890940428229086449317339143225053291
1
5:
17.835182939856327657559032748504419144713957076983
1
6:
22.630196784297817625019048559500895571094669210501
1
7:
27.729948306342806305087228166841835806024096591781
1
8:
33.104395594723694891085061580762248403769704296984
1
9:
38.729876190385427289280338246217203628631653044523
3/2
0:
2.5830258609557839224440770838565406922013041410816
3/2
1:
2.8656607527180684736439146626749708717830310035440
3/2
2:
6.3428412376205258088854295126961550984536708751391
3/2
3:
8.6767193452183451151275915531779567292253669483894
3/2
4:
12.069692090756503087233922578723844257595050010183
3/2
5:
15.841709579930929733280775664155247002869679606227
3/2
6:
20.015491958730793261071900742472820063695783455244
3/2
7:
24.526027913081015627760325832509918696939767448690
3/2
8:
29.337983012592089372199308144901005081076735035934
3/2
9:
34.423052850765237422424412777615847758563859378575
2
0:
3.8636692793065243905881741614524626561820682663663
2
1:
3.8651856547271381625466861693193162020527866355036
2
2:
10.873449795773882903644857054351070448817190850536
2
3:
10.989086688497552637631413318781755364093889603515
2
4:
15.867979899735835776028318528704955886196754147480
2
5:
17.419662580123436419982273562099897628387872137160
2
6:
20.911822173411348368718748532178927079419330726435
2
7:
24.398938103825890278448915764568062400797174867274
2
8:
28.379385410143526744561513833164352009507348758905
2
9:
32.669939319108547061645811054395288468194796837302
5/2
0:
4.9168620795077917824929910453149316180083625439500
5/2
1:
4.9168621869909699235472977260635910923539873512862
5/2
2:
14.390822212006226611757167563353067217720143408602
5/2
3:
14.390843929836701910275516660272174154729636268765
5/2
4:
23.239885307289131703229007410803072085613594128114
5/2
5:
23.241663892834377227988959457895769827269433788424
5/2
6:
31.240625054062224796654565995682025994809521395878
5/2
7:
31.311849031231375578585652246125098246690004266850
5/2
8:
37.527707015434081269353566887640938061702447096798
5/2
9:
38.595243759326681130065150287989908498632849514908
3
0:
5.9432302823521999427306188047388732537233848195132
3
1:
5.9432302823522448149792211710063420258496722913204
3
2:
17.592881453822910926563388869865621353601372500635
3
3:
17.592881453840186509741140342401055412024427995501
3
4:
28.861334174618965967961989391247066102592256855248
3
5:
28.861334177634090962763291538636528022345274298417
3
6:
39.705632836075655432110396005639415194913224753175
3
7:
39.705633149347185476011064129959461736591954966404
3
8:
50.067243450360845653728118521817434727329631242426
3
9:
50.067264782081289963476363294634406718482077923591
Definition
For $a>0$, this table gives the eigenvalues $E_n(a)$ of the Schrödinger operator $-y''(x)+(x^2-a^2)^2y(x)=E y(x)$ on $L^2(\mathbb{R})$, with no factor of $\tfrac12$ on the kinetic term. The eigenvalues are listed in increasing order, and the level index $n$ starts at $0$ for the ground state.
Parameters
$a$
—   well parameter ($a>0$)
$n$
—   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 $E_n(a)=a^4+\mathcal{E}_n(-2a^2,1)$.
(2)
For $\beta>0$, $\mathcal{E}_n(\alpha,\beta)=\beta^{1/3} \mathcal{E}_n(\alpha\beta^{-2/3},1)$, with $\mathcal{E}_n$ as in (1).
Comments
(3)
The potential vanishes at its two minima $x=\pm a$ and rises to $a^4$ at $x=0$.
(4)
The index $n$ is the number of nodes of the square-integrable eigenfunction. Since the potential is even, even $n$ gives an even eigenfunction and odd $n$ gives an odd eigenfunction.
(5)
The limiting case $a=0$ is not included. It is the pure quartic oscillator stored in the table of pure quartic, sextic, octic and decic oscillator eigenvalues.
(6)
For levels with $E_n(a)<a^4$, the even and odd states form nearly degenerate pairs $(E_0,E_1)$, $(E_2,E_3)$, and so on. Their differences are tunneling splittings; in this table the first splitting decreases from $0.2826\ldots$ at $a=3/2$ to $4.49\cdot10^{-14}$ at $a=3$. See [1] and [2].
(7)
For the convention $-\tfrac12 y''+(x^2-a^2)^2y=\tilde E y$, the eigenvalues satisfy $\tilde E_n(a)=2^{-2/3}E_n(2^{1/6}a)$. The argument $2^{1/6}a$ is off the half-step grid used here for the rational $a$'s in this table.
References
[1]
Miklos Ronto and Eli Pollak, Upper and lower bounds for tunneling splittings in a symmetric double-well potential, RSC Advances 10 (2020), 34681-34689. (doi)
Links
Similar tables
Eigenvalues of the quartic anharmonic oscillator —   stores the positive-quadratic quartic oscillator, while this table stores the negative-quadratic double-well case with the constant shift chosen so that the minima are $0$
Eigenvalues of the pure quartic, sextic, octic and decic oscillators —   contains the limiting case $a=0$ for the quartic potential
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds $0\leq n\leq9$ for $a\in\{1/2,1,3/2,2,5/2,3\}$, a half-step grid from $a=1/2$, where no stored level has energy less than the barrier $a^4$, through $a=3$, where all ten stored levels have energy less than the barrier and form five near-degenerate pairs)
How they were obtained:

The generator computes Rayleigh-Ritz eigenvalues in the harmonic-oscillator basis for $-d^2/dx^2+\omega^2x^2$, with even and odd parity separated. For each $a$ it uses the two truncations $N=320$ and $N=460$, and the two basis frequencies $\omega=1$ and $\omega=13/10$. Eigenvalues of the finite Galerkin matrices are isolated by Sturm counts from a banded $LDL^T$ factorisation and by bisection, rather than by a dense high-precision eigensolver. Each entry is the real interval spanning those four computations; the widest interval in the dry run had relative diameter less than $4\cdot10^{-61}$.

more

As controls, the generator checks the $a=0$ pure-quartic limit against the pure oscillator table. It also checks $a=(6\cdot2^{-2/3})^{1/2}$, for which $(E_n(a)-a^4)/2^{2/3}$ gives the eigenvalues of $-\tfrac12 d^2/dx^2+x^4-6x^2$ tabulated by Ronto and Pollak [1]. Rayleigh-Ritz eigenvalues are upper bounds for the true eigenvalues by the variational principle, so the agreement between truncations is not a rigorous error bound.



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