Imaginary Cubic Perturbation: Numerical and Analytic Study

Abstract

The analytic properties of the ground state resonance energy E(g) of the cubic potential are investigated as a function of the complex coupling parameter g. We explicitly show that it is possible to analytically continue E(g) by means of a resummed strong coupling expansion, to the second sheet of the Riemann surface, and we observe a merging of resonance and antiresonance eigenvalues at a critical point along the line arg(g) = 5 pi/4. In addition, we investigate the convergence of the resummed weak-coupling expansion in the strong coupling regime, by means of various modifications of order-dependent mappings (ODM), that take special properties of the cubic potential into account. The various ODM are adapted to different regimes of the coupling constant. We also determine a large number of terms of the strong coupling expansion by resumming the weak-coupling expansion using the ODM, demonstrating the interpolation between the two regimes made possible by this summation method.

18 pages; 4 figures; typographical errors corrected

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Cited by in corpus (14)

• Generalized Householder Transformations for the Complex Symmetric Eigenvalue Problem
• Precise Critical Exponents of the O(N)-Symmetric Quantum field Model using Hypergeometric-Meijer Resummation
• Weak-Coupling, Strong-Coupling and Large-Order Parametrization of the Hypergeometric-Meijer Approximants
• Interlace properties for the real and imaginary parts of the wave functions of complex-valued potentials with real spectrum
• Extrapolating the precision of the Hypergeometric Resummation to Strong couplings with application to the $\mathcal{P}\mathcal{T}-$Symmetric $i{\varphi }^3$ Field Theory
• N-site-lattice analogues of $V(x)=i{x}^3$
• A resurgence analysis for cubic and quartic anharmonic potentials
• Universal Large-order asymptotic behavior of the Strong-coupling and High-Temperature series expansions
• High-Order Parametrization of the Hypergeometric-Meijer Approximants
• Global solution of the cubic oscillator
• Soft quantum vibrations of PT-symmetric nonlinear ion chain
• An investigation of $\mathcal{P}\mathcal{T}$-symmetry breaking in tight-binding chains
• Critical Exponents of the O(N)-symmetric ${\varphi }^4$ Model from the \boldmath{\large{ ${\epsilon }^7$}} Hypergeometric-Meijer Resummation
• Two-parametric PT-symmetric quartic family