Semiclassical analysis of a complex quartic Hamiltonian

Abstract

It is necessary to calculate the C operator for the non-Hermitian PT-symmetric Hamiltonian H=\half p^2+\halfμ^2x^2-λx^4 in order to demonstrate that H defines a consistent unitary theory of quantum mechanics. However, the C operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian H=\half p^2+\half μ^2x^2+igx^3-λx^4, whose C operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for C having a nontrivial dependence on the dynamical variables x and p and on the parameter λ.

4 pages

References in corpus (4)

• Extension of PT-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction
• Scalar Quantum Field Theory with Cubic Interaction
• Solvable PT-symmetric model with a tunable interspersion of non-merging levels
• A Perturbative Treatment Of A Generalized PT-Symmetric Quartic Anharmonic Oscillator

Cited by in corpus (5)

• Making Sense of Non-Hermitian Hamiltonians
• Non-perturbative tests for the Asymptotic Freedom in the $\mathcal{P}\mathcal{T}$-symmetric $(-{\varphi }^4{)}_{3+1}$ theory
• PT-symmetric quartic anharmonic oscillator and position-dependent mass in a perturbative approach
• On some meaningful inner product for real Klein-Gordon fields with positive semi-definite norm
• Vacuum Stability of the $\mathcal{P}\mathcal{T}$-Symmetric $\left(-{\varphi }^4\right)$ Scalar Field Theory