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arXiv:1105.2742 · Journal of Physics A Mathematical and Theoretical

Two-parametric PT-symmetric quartic family

Alexandre Eremenko, Andrei Gabrielov

Published Apr 12, 2012• 0 citations
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Abstract

We describe a parametrization of the real spectral locus of the two-parametric family of PT-symmetric quartic oscillators. For this family, we find a parameter region where all eigenvalues are real, extending the results of Dorey et al (2007 J. Phys. A: Math Theor. 40 R205–83) and Shin (2005 J. Phys. A: Math. Gen. 38 6147–66; 2002 Commun. Math. Phys. 229 543–64).

DOI
10.1088/1751-8113/45/17/175206

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We describe a parametrization of the real spectral locus of the two-parametric family of PT-symmetric quartic oscillators. For this family, we find a parameter region where all eigenvalues are real, extending the results of Dorey et al (2007 J. Phys. A: Math Theor. 40 R205–83) and Shin (2005 J. Phys. A: Math. Gen. 38 6147–66; 2002 Commun. Math. Phys. 229 543–64).
We introduce a Hamiltonian H α, β depending on the two complex parameters α and β and develop a through analysis of the spectral properties of this Hamiltonian and its adjoint. We determine the regions in the space of the complex parameters α and β, where H α, β is (i) Hermitian, (ii) non-Hermitian with real spectrum and (iii) non-Hermitian but PT -symmetric. In the case when H α, β is non-Hermitian having real spectrum, we derive a closed formula for a family of the metric operators, depending on two arbitrary real positive parameters, which render the Hamiltonian H α, β Hermitian. In a particular case we calculate the Hermitian counterpart of H α, β .
A major mathematical problem in PT-symmetric quantum mechanics is to determine whether or not the spectrum of any given non self-adjoint but PT-symmetric Schroedinger operator is real. In this case the PT-symmetry is called "proper". Clearly in this connection an equally important issue is the spontaneous violation of the PT-symmetry which might occur in a PT-symmetric operator family. The spontaneous violation of the PT-symmetry is defined as the transition from real values of the spectrum to complex ones at the variation of the parameter labeling the family. Its occurrence is referred to also as the PTPTPT-symmetric phase transition. \nThis chapter intends to present a review of recent results concerning these two mathematical points, within the standard notions of spectral theory for Hilbert space operators. The main technical instrument is represented by perturbation theory.
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.