Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries

Abstract

We extend the definition of generalized parity $P$, charge-conjugation $C$ and time-reversal $T$ operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and we use these generalized operators to describe the full set of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold classification. In particular we show that $TP$ and $CTP$ are the generators of the antiunitary symmetries; moreover, a necessary and sufficient condition is provided for a pseudo-Hermitian Hamiltonian $H$ to admit a $P$-reflecting symmetry which generates the $P$-pseudounitary and the $P$-pseudoantiunitary symmetries. Finally, a physical example is considered and some hints on the $P$-unitary evolution of a physical system are also given.

20 pages

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• The Pauli equation with complex boundary conditions
• Conjecture on the analyticity of PT-symmetric potentials and the reality of their spectra
• Infinite-dimensional representations of the rotation group and Dirac's monopole problem
• $\mathcal{P}\mathcal{T}$-symmetric mapping of three states and its implementation on a cloud quantum processor
• Quasistationary quaternionic Hamiltonians and complex stochastic maps