Quartic / sextic anharmonic oscillator

Anharmonic oscillator — quartic / sextic

Anharmonic oscillator with a quartic \(x^4\) or sextic \(x^6\) term: both fall in the Heun class, with no general closed form — the spectrum is numerical. The sextic is the paradigmatic case of quasi-exactly-solvable models (\(sl(2)\) algebra, Turbiner). Finite-difference computation in the Python backend (via gw2py).

In the variable \(x\) the equation has a single irregular singularity at infinity, of rank increasing with the potential's degree: the quartic is of triconfluent-Heun type, the sextic of higher rank. There is no general termination condition.

Potential

\[ V(x)=\tfrac12 x^2 + \lambda\,x^4 \quad(\text{quartic}),\qquad V(x)=\tfrac12 x^2 + \lambda\,x^6 \quad(\text{sextic}). \]

In the Heun hierarchy

In the variable \(x\) the equation has a single irregular singularity at infinity, of rank increasing with the potential's degree: the quartic is of triconfluent-Heun type, the sextic of even higher rank. There is no general termination condition: the spectrum is numerical.

First-order perturbative estimate (quartic): \[ \varepsilon_n \simeq \big(n+\tfrac12\big) + \tfrac{3\lambda}{4}\big(2n^2+2n+1\big). \]

Quasi-exact solvability (sextic)

The sextic is the paradigmatic case of quasi-exactly-solvable models: for special relations between the coefficients, a finite number of eigenstates has a closed form (\(sl(2)\) algebra, Turbiner).

State to represent

State type Level \(n_1\)

Level \(n_2\)

Potential parameters

Anharmonic term Coupling \(\lambda\) = 0.1

Tri/bi-confluent Heun

The quartic on \(\mathbb{R}\) is of triconfluent-Heun type; with parity \(z=x^2\) it becomes biconfluent. The sextic pushes even higher in the hierarchy. In no case does the series termination that quantizes \(_1F_1\) and \(_2F_1\) hold.

Bender–Wu divergence (quartic)

The quartic's perturbative series has zero radius of convergence (coefficients \(\sim(-1)^{k+1}3^k\,\Gamma(k+\tfrac12)\)): asymptotic but Borel-summable.

Quasi-exact solvability (sextic)

The sextic is Turbiner's historical example: the existence of a finite-dimensional invariant subspace of the \(sl(2)\) algebra generates a finite number of exact polynomial eigenstates, embedded in an otherwise only-numerical spectrum.

Computational evidence

Quartic \(\lambda=0.1\): the engine gives \(\varepsilon_0=0.5591\), consistent with the known converged value (\(0.5591463\)) and with \(\texttt{numpy.eigvalsh}\).

References

C. M. Bender, T. T. Wu, Phys. Rev. 184, 1231 (1969). doi.
A. V. Turbiner, «Quasi-exactly-solvable problems and \(sl(2)\) algebra», Commun. Math. Phys. 118, 467 (1988). doi.
A. G. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mechanics, IOP, 1994.
NIST DLMF, ch. 31 (Heun functions). dlmf.nist.gov/31.

WebNIR · CNR-IFAC | demo interface — numerical work is provided by the Python backend (gw2py).

Keywords: anharmonic oscillator, quartic, sextic, Heun equation, Bender-Wu, quasi-exact solvability, Turbiner, sl(2), quantum mechanics