Quartic anharmonic oscillator

Quartic anharmonic oscillator \(V(x)=\tfrac12 x^2+\lambda x^4\): the reference case of divergent perturbation theory (Bender–Wu). The perturbative series has zero radius of convergence but is Borel-summable; the exact spectrum is computed numerically by finite differences in the Python backend (via gw2py). For the quartic/sextic comparison see also the «quartic/sextic anharmonic oscillator» page.

The quartic is the historical example where perturbation theory diverges: the series in \(\lambda\) is asymptotic (zero radius of convergence) but Borel-summable, and the comparison with the exact numerical computation is the testing ground.

Quartic potential

\[ V(x)=\tfrac12 x^2 + \lambda\,x^4,\qquad \lambda\ge 0. \]

An apparently small perturbation of the harmonic oscillator; in fact it qualitatively changes the analytic structure (a single irregular singularity at infinity, triconfluent-Heun type).

Perturbative series and divergence

At first order \(\varepsilon_n\simeq(n+\tfrac12)+\tfrac{3\lambda}{4}(2n^2+2n+1)\). But the full series \(\varepsilon_0(\lambda)=\sum_k a_k\lambda^k\) has factorially growing coefficients,

\[ a_k \sim (-1)^{k+1}\,\frac{\sqrt6}{\pi^{3/2}}\,3^k\,\Gamma\!\big(k+\tfrac12\big), \]

The radius of convergence is zero: the series is asymptotic, not convergent. This is the Bender–Wu result (1969).

Borel summability

The series is nonetheless Borel-summable: the Borel transform converges and its resummation returns the exact energy. It is the prototype of the link between divergent series and finite physical results.

State to represent

State type Level \(n_1\)

Level \(n_2\)

Potential parameter

Quartic coupling \(\lambda\) = 0.1

Dimensionless units (\(\hbar=m=1\)). At \(\lambda=0\) the harmonic oscillator is recovered; increasing \(\lambda\) the levels rise more than linearly.

Why the series diverges

The factorial growth of the coefficients reflects the instantons of the problem (at \(\lambda<0\) the potential is unstable): the zero radius of convergence is the signature of the singularity at \(\lambda=0\) along the negative real axis.

Numerical vs perturbative

The finite-difference solver gives the exact energy for any \(\lambda\); the perturbative series approximates it well only when truncated at the optimal order (\(k\sim1/(3\lambda)\)), then worsens — typical behaviour of asymptotic series.

Computational evidence

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

References

C. M. Bender, T. T. Wu, «Anharmonic Oscillator», Phys. Rev. 184, 1231 (1969). doi.
C. M. Bender, T. T. Wu, Phys. Rev. D 7, 1620 (1973). doi.
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford Univ. Press.
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, Bender-Wu, asymptotic series, Borel summability, Heun equation, perturbation theory, quantum mechanics