A quantum-mechanical anharmonic oscillator with a most interesting spectrum

Abstract: We revisit the problem posed by an anharmonic oscillator with a potential given by a polynomial function of the coordinate of degree six that depends on a parameter $λ$. The ground state can be obtained exactly and its energy $E_{0}=1$ is independent of $λ$. This solution is valid only for $λ>0$ because the eigenfunction is not square integrable otherwise. Here we show that the perturbation series for the expectation values are Padé and Borel-Padé summable for $λ>0$. When $% λ<0$ the spectrum exhibits an infinite number of avoided crossings at each of which the eigenfunctions undergo dramatic changes in their spatial distribution that we analyze by means of the expectation values $\langle x^{2}\rangle $.

• The paper investigates the spectral properties and perturbative expansions of a sextic anharmonic oscillator, highlighting an exact ground state for λ ≥ 0.
• The paper demonstrates that while Padé and Borel-Padé techniques successfully sum the divergent series for λ > 0, they fail for λ < 0 due to non-analytic behavior.
• The paper reveals intricate avoided crossings and eigenfunction localization transitions as λ varies, emphasizing nonperturbative effects in multi-well potentials.

Spectral Properties of a Quantum-Mechanical Sextic Anharmonic Oscillator

Introduction

This paper investigates the spectral structure and perturbative properties of a quantum anharmonic oscillator characterized by a sextic polynomial potential dependent on a tuning parameter $\lambda$. This model, originally discussed by Herbst and Simon, displays unique analytic, algebraic, and numerical features, particularly in how the ground state can be solved exactly and how the spectrum evolves as the parameter $\lambda$ transitions between positive and negative values. The authors extend and complement earlier works by providing a thorough high-precision analysis of perturbation series, their summability, and the role of avoided crossings and localization transitions in the eigenfunction behavior.

The Model and Exact Ground State

The Hamiltonian under study is:

$$H(\lambda) = p^2 + x^2 - 12\lambda x^2 + 8\lambda x^4 + 16\lambda^2 x^6 \, .$$

An exact ground-state eigenfunction exists for $\lambda \geq 0$:

$$\varphi(x) = \exp \left( -\frac{1}{2}x^2 - \lambda x^4 \right)$$

with eigenvalue $E_0 = 1$, independent of $\lambda$. The solution ceases to be physical for $\lambda < 0$ due to the lack of square integrability.

Perturbative Expansions and (Non)Summability

For $\lambda \geq 0$, the perturbative expansion of the ground-state energy is trivial ($E_0^{(j)} = 0$ for $\lambda$0), while the expansion for the ground-state wavefunction and derived expectation values such as $\lambda$1 exhibits factorial divergence. Asymptotic analysis of the series coefficients, enabled by the hypervirial perturbation method, reveals a dominant behavior $\lambda$2 for the $\lambda$3-order correction. Despite divergence, Padé and Borel-Padé techniques can successfully sum these series for $\lambda$4, yielding quantitatively accurate results for physical observables.

In contrast, when $\lambda$5, both Padé and Borel-Padé summations fail due to obstruction from non-analytic behavior associated with the spectrum’s nontrivial structure and the proliferation of multiple wells in the potential. For negative $\lambda$6, the lowest-lying eigenvalue approaches its limiting value in a non-perturbative manner:

$\lambda$7

where numerical fitting indicates $\lambda$8, $\lambda$9, and $$H(\lambda) = p^2 + x^2 - 12\lambda x^2 + 8\lambda x^4 + 16\lambda^2 x^6 \, .$$0.

Spectrum Structure, Avoided Crossings, and Eigenfunction Localization

As $$H(\lambda) = p^2 + x^2 - 12\lambda x^2 + 8\lambda x^4 + 16\lambda^2 x^6 \, .$$1 crosses zero from the positive into the negative regime, the potential undergoes topological transitions. For $$H(\lambda) = p^2 + x^2 - 12\lambda x^2 + 8\lambda x^4 + 16\lambda^2 x^6 \, .$$2, the system features three well-separated minima (triple-well regime). The spectral analysis reveals intricate patterns: the ground and first excited states ($$H(\lambda) = p^2 + x^2 - 12\lambda x^2 + 8\lambda x^4 + 16\lambda^2 x^6 \, .$$3, $$H(\lambda) = p^2 + x^2 - 12\lambda x^2 + 8\lambda x^4 + 16\lambda^2 x^6 \, .$$4) remain isolated, while higher excited states cluster and become quasi-degenerate, undergoing sequences of avoided crossings as $$H(\lambda) = p^2 + x^2 - 12\lambda x^2 + 8\lambda x^4 + 16\lambda^2 x^6 \, .$$5 is tuned. These transitions are characterized by abrupt changes in $$H(\lambda) = p^2 + x^2 - 12\lambda x^2 + 8\lambda x^4 + 16\lambda^2 x^6 \, .$$6, signifying dramatic spatial reorganization of the eigenfunctions.

Notably, for $$H(\lambda) = p^2 + x^2 - 12\lambda x^2 + 8\lambda x^4 + 16\lambda^2 x^6 \, .$$7, the spectrum approaches that of the pure harmonic oscillator, but higher-lying states exhibit traces of their prior quasi-degenerate structure and avoided crossings, evidencing intricate interplay between potential topology and eigenfunction localization. The pattern in which higher excited states participate in avoided crossings depends on parity and index, shaping the density and structure of states in the continuum.

Numerical Techniques and Asymptotic Estimates

The analytical investigation is supported by extensive perturbative calculations and high-precision numerical diagonalization (Rayleigh-Ritz with large oscillator bases) in the multi-well regime. The hypervirial approach enables systematic computation and asymptotic estimation of high-order corrections, illuminating the scaling and summability properties of the perturbation expansions.

Implications and Outlook

The analysis deepens the understanding of quasi-exactly solvable systems, nonperturbative effects, and summability of divergent series in quantum mechanics. The dramatic change in eigenstate localization revealed at avoided crossings underscores the physical consequences of the intersections between analytic structure in parameter space and potential topology. The robust summability for $$H(\lambda) = p^2 + x^2 - 12\lambda x^2 + 8\lambda x^4 + 16\lambda^2 x^6 \, .$$8 stands in contrast to the essential nonperturbativity for $$H(\lambda) = p^2 + x^2 - 12\lambda x^2 + 8\lambda x^4 + 16\lambda^2 x^6 \, .$$9, highlighting subtle limitations of resummation techniques in presence of degeneracy and tunneling.

Future avenues include the exploration of more general multi-well and parameter-dependent models, rigorous characterization of Borel and non-Borel summability across quantum phase transitions, and potential application of advanced resummation or numerical analytic continuation techniques to further clarify the structure of the spectrum for negative parameter regimes.

Conclusion

The paper provides an authoritative and high-resolution study of a quantum anharmonic oscillator with sextic potential, revealing the profound interplay between perturbative summability, avoided crossings, and eigenfunction localization transitions as a function of the tuning parameter. The results contribute significant nuance to the understanding of spectral transitions and non-analyticity in one-dimensional quantum systems, and offer a firm reference for further studies on quasi-exact and nonperturbative phenomena in quantum mechanics.