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Journal of Physics A Mathematical and General

Generalized PT symmetry and real spectra

Carl M. Bender, Michael Berry, Aikaterini Mandilara

Published Jul 25, 2002• 255 citations
Publisher

Abstract

The fact that eigenvalues of PT -symmetric Hamiltonians H can be real for some values of a parameter and complex for others is explained by showing that the matrix elements of H , and hence the secular equation, are real, not only for PT but also for any antiunitary operator A satisfying A 2 k = 1 with k odd. The argument is illustrated by a 2 × 2 matrix Hamiltonian, and two examples of the generalization are given.

DOI
10.1088/0305-4470/35/31/101

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The fact that eigenvalues of PT -symmetric Hamiltonians H can be real for some values of a parameter and complex for others is explained by showing that the matrix elements of H , and hence the secular equation, are real, not only for PT but also for any antiunitary operator A satisfying A 2 k = 1 with k odd. The argument is illustrated by a 2 × 2 matrix Hamiltonian, and two examples of the generalization are given.
Two formulations for constructing a non-Hermitian matrix with all real eigenvalues are studied. They are called PT symmetry and pseudo-Hermiticity in the literature. Explicit 2×2 matrices of both forms are provided. They are characterized by six real parameters and are hence more general than Hermitian matrices. The equivalence of the two formulations is established. A 2×2 matrix with all real eigenvalues is PT-symmetric and pseudo-Hermitian at the same time. The application in time-dependent problems is discussed and a new geometry phase is obtained.
This paper proposes to broaden the canonical formulation of quantum mechanics.Ordinarily, one imposes the condition H † = H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition.This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum.However, replacing this mathematical condition by the weaker and more physical requirement H ‡ = H, where ‡ represents combined parity reflection and time reversal PT , one obtains new classes of complex Hamiltonians whose spectra are still real and positive.This generalization of Hermiticity is investigated using a complex deformation H = p 2 + x 2 (ix) ǫ of the harmonic oscillator Hamiltonian, where ǫ is a real parameter.The system exhibits two phases: When ǫ ≥ 0, the energy spectrum of H is real and positive as a consequence of PT symmetry.However, when -1 < ǫ < 0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because PT symmetry is spontaneously broken.The phase transition that occurs at ǫ = 0 manifests itself in both the quantum-mechanical system and the underlying classical system.Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians H = p 2 + x 2N (ix) ǫ with N integer and ǫ > -N ; each of these complex Hamiltonians exhibits a phase transition at ǫ = 0.These PT -symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.