Calculation of the Hidden Symmetry Operator for a $\cP\cT$-Symmetric Square Well

Abstract

It has been shown that a Hamiltonian with an unbroken $\cP\cT$ symmetry also possesses a hidden symmetry that is represented by the linear operator $\cC$. This symmetry operator $\cC$ guarantees that the Hamiltonian acts on a Hilbert space with an inner product that is both positive definite and conserved in time, thereby ensuring that the Hamiltonian can be used to define a unitary theory of quantum mechanics. In this paper it is shown how to construct the operator $\cC$ for the $\cP\cT$-symmetric square well using perturbative techniques.

arXiv comment10 pages, 2 figures

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• Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly
• Conceptual Problems in Scattering from Localized non-Hermitian Potentials
• Giving up the ghost
• Generalized Swanson models and their solutions
• Nonunique C operator in PT Quantum Mechanics
• $J$-self-adjoint operators with $\mathcal{C}$-symmetries: extension theory approach
• Differential Realization of Pseudo-Hermiticity: A quantum mechanical analog of Einstein's field equation
• Pseudo supersymmetric partners for the generalized Swanson model
• Exact solution of a non-Hermitian $\mathcal{PT}$-symmetric Heisenberg spin chain
• Overcritical PT-symmetric square well potential in the Dirac equation
• Unbounded C-symmetries and their nonuniqueness
• On the construction of pseudo-hermitian quantum system with a pre-determined metric in the Hilbert space
• p-Adic Schrödinger-Type Operator with Point Interactions
• Entangled Quantum State Discrimination using Pseudo-Hermitian System
• Small-$\epsilon$ behavior of the Non-Hermitian PT-Symmetric Hamiltonian $H=p^2+x^2(ix{)}^{\epsilon }$
• Conjecture on the analyticity of PT-symmetric potentials and the reality of their spectra
• A spin chain with non-Hermitian $\mathcal{PT}-$symmetric boundary couplings: exact solution, dissipative Kondo effect, and phase transitions on the edge
• Generation of families of spectra in PT-symmetric quantum mechanics and scalar bosonic field theory
• Nonuniqueness of the C operator in PT-symmetric quantum mechanics
• A Generalization of Gauge Symmetry, Fourth-Order Gauge Field Equations and Accelerated Cosmic-Expansion
• On pseudo-Hermitian operators with generalized $\mathcal{C}$-symmetries