Instantons and transseries of the Mathieu potential deformed by a $\mathcal{P}\mathcal{T}$-symmetry parameter

Abstract

We investigate the non-perturbative effects of a deformation of the Mathieu differential equation consistent with $\mathcal{P}\mathcal{T}$ symmetry. First, we develop a connection between the non-Hermitian and Hermitian scenarios by a reparameterization in the complex plane, followed by a restriction of the $\mathcal{P}\mathcal{T}$ deformation parameter. The latter is responsible for preserving the information about $\mathcal{P}\mathcal{T}$ symmetry when we choose to work in the Hermitian scenario. We note that this factor is present in all non-perturbative results and in the transseries representation of the deformed Mathieu partition function that we have obtained. In quantum mechanics, we found that the deformation parameter of $\mathcal{P}\mathcal{T}$ symmetry has an effect on the real instanton solution for the deformed Mathieu potential in the Hermitian scenario. As its value increases, the non-Hermiticity factor makes it smoother for the instanton to pass from one minimum to another, that is, it modifies the instanton width. The explanation for this lies in the fact that the height of the potential barrier decreases as we increase the value of the deformation parameter. We present how this effect extends to the multi-instanton level and to the bounce limit of an instanton-anti-instanton pair. As an application of the obtained results, we show that the equation of motion under a tilted version of the potential in the Hermitian scenario compares to the resistively shunted junction (RSJ) model for the Josephson junction.

arXiv comment15 pages, 8 figures; new layout; added references; fixed typos in (old numbering) Eqs.(10), (12), (17)-(19), (29); added definition of instanton and instanton action; included identification of parameters; fixed the labels for axes in the Figs.1 and 4; added and rewrote some paragraphs in Introduction, Sec.II, V and Appendix A; changed titles of Sec.VII and Appendix A; corrected references

References in corpus (17)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• PT-Symmetric Quantum Mechanics
• Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• Lectures on non-perturbative effects in large N gauge theories, matrix models and strings
• A Primer on Resurgent Transseries and Their Asymptotics
• Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions
• Nonperturbative Ambiguities and the Reality of Resurgent Transseries
• Resurgence theory, ghost-instantons, and analytic continuation of path integrals
• Pseudo-Hermiticity for a Class of Nondiagonalizable Hamiltonians
• Pseudo-Unitary Operators and Pseudo-Unitary Quantum Dynamics
• Resurgence in sine-Gordon quantum mechanics: Exact agreement between multi-instantons and uniform WKB
• The Power of Perturbation Theory
• Resurgence and Holomorphy: From Weak to Strong Coupling
• Interaction of Flexural Phonons with Electrons in Graphene: A Generalized Dirac Equation in Corrugated Surfaces
• A New Boson realization of Fusion Polynomial Algebras in Non-Hermitian Quantum Mechanics : $\gamma$-deformed $su(2)$ generators, Partial $\mathcal{P}\mathcal{T}$-symmetry and Higgs algebra