Eigenvalues of the pure quartic, sextic, octic and decic oscillators $-y''+x^{2m}y$

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Numbers

$m$

$n$

$E^{(m)}_n$

1.0603620904841828996470460166926635455152087285290

3.7996730298013941687830941885125689577660654673274

7.4556979379867383921565913471857674881378195367491

1.1448024537970527637654575341495490765378125289799

4.3385987115139811916473368880601867793285474529433

9.0730845609214338560162490966647121795524567424038

1.2258201138004921915910860266459400337268063956724

4.7558744139607598693122409990423304081959100921951

10.244946977236854744232174212997988062170873570381

1.2988437006785213755123001647804857534795851841702

5.0978765292033805010497788940182055164673483248438

11.154318202156246849971760449916844817044273518128

Definition

For $m\geq2$, let $E^{(m)}_0<E^{(m)}_1<\cdots$ be the eigenvalues of $-y''+x^{2m}y=Ey$ on $L^2(\mathbb R)$, with no factor of $\tfrac12$ on the kinetic term. The index $n$ starts at $0$ for the ground state.

Parameters

$m$

— half the degree of the potential ($m\geq2$)

$n$

— energy level ($n\geq0$)

Formulas

(1)

If $E^{(m)}_n(\beta)$ is the eigenvalue of $-y''+\beta x^{2m}$ with $\beta>0$, then $E^{(m)}_n(\beta)=\beta^{1/(m+1)}E^{(m)}_n$.

(2)

The operator $-\tfrac12 y''+\lambda x^{2m}$ has eigenvalues $2^{-m/(m+1)}\lambda^{1/(m+1)}E^{(m)}_n$, with $E^{(m)}_n$ as defined in this table.

Comments

(3)

The rows $m=2,3,4,5$ are the pure quartic, sextic, octic and decic anharmonic oscillators, respectively [3].

(4)

With the same kinetic-term normalisation, the harmonic oscillator $-y''+x^2y=Ey$ has eigenvalues $2n+1$.

(5)

The eigenfunction for $E^{(m)}_n$ has $n$ nodes. The potential is even, so the eigenfunctions alternate parity: even $n$ gives an even eigenfunction and odd $n$ gives an odd eigenfunction.

References

[1]

M. H. Macfarlane, A high-precision study of anharmonic-oscillator spectra, Annals of Physics 271 (1999), 159-202. (doi)

[2]

A. Mushtaq, A. Noreen and K. Olaussen, Numerical solutions of quantum mechanical eigenvalue problems, Frontiers in Physics 8 (2020), article 390. (doi)

Links

[3]

Wikipedia: Anharmonic oscillator

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Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds the first three levels $0\leq n\leq2$ for $m=2,3,4,5$, the pure quartic, sextic, octic and decic oscillators, a conservative range chosen because these are the levels checked by four Galerkin truncations for each $m$ to at least 50 common significant digits)

How they were obtained:

The generator forms the Galerkin matrix of $-y''+x^{2m}$ in the harmonic-oscillator basis and finds finite-matrix eigenvalues by bisection using an $LDL^T$ Sturm count, without a dense high-precision eigensolver. For $m=2,3$ it compares truncations $N=700,900$ at basis frequencies $\omega=2,5/2$; for $m=4$ it compares $N=800,1000$ at $\omega=3,7/2$; for $m=5$ it compares $N=1600,2000$ at $\omega=4,5$. The stored rows are the digits common to all four computations for that $m$, and the worst stored row retained 51 matching significant digits.

As controls, the same code with $m=1$ gives the harmonic-oscillator eigenvalues $2n+1$, and the three quartic rows agree with the 30-decimal values in [2]. The high-precision spectral study [1] gives an outside reference for the $x^{2m}$ oscillator family and its scaling conventions.