Investigation of non-Hermitian Hamiltonians in the Heisenberg picture

Abstract

The Heisenberg picture for non-Hermitian but $\eta$-pseudo-Hermitian Hamiltonian systems is suggested. If a non-Hermitian but $\eta$-pseudo-Hermitian Hamiltonian leads to real second order equations of motion, though their first order Heisenberg equations of motion are complex, we can construct a Hermitian counterpart that gives the same second order equations of motion. In terms of a similarity transformation we verify the iso-spectral property of the Hermitian and non-Hermitian Hamiltonians and obtain the related eigenfunctions. This feature can be used to determine real eigenvalues for such non-Hermitian Hamiltonian systems. As an application, two new non-Hermitian Hamiltonians are constructed and investigated, where one is non-Hermitian and non-PT-symmetric and the other is non-Hermitian but PT-symmetric. Moreover, the complementarity and compatibility between our treatment and the PT symmetry are discussed.

v1: 8 pages, no figures; v2: clarifications added; v3: 14 pages, title slightly changed, new content added, almost completely rewritten; v4: 18 pages, minor clarification added, conclusion unchanged; v5: 13 pages, title slightly changed, references and clarifications added, final version to appear in Physics Letters A

References in corpus (4)

• Making Sense of Non-Hermitian Hamiltonians
• No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model
• Interactions of Hermitian and non-Hermitian Hamiltonians
• Spontaneous breaking of permutation symmetry in pseudo-Hermitian quantum mechanics

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• Wheeler-DeWitt equation and the applicability of crypto-Hermitian interaction representation in quantum cosmology
• Hybrid form of quantum theory with non-Hermitian Hamiltonians
• Pseudo-invariant approach for a particle in a complex time-dependent linear potential
• Heisenberg and Heisenberg-Like Representations via Hilbert Space Bundle Geometry in the Non-Hermitian Regime
• Hermitian--to--quasi-Hermitian quantum phase transitions
• Analytic approximation for eigenvalues of a class of $\mathcal{P}\mathcal{T}$ symmetric Hamiltonians
• Non-stationary quantum mechanics in hybrid non-Hermitian interaction representation
• Consistent treatment of quantum systems with a time-dependent Hilbert space
• Universal quantum control over non-Hermitian continuous-variable systems
• Quantum mechanics using two auxiliary inner products
• On the construction of pseudo-Hermitian Hamiltonians by means of similarity transformations