Fragile PT-symmetry in a solvable model

Abstract

One of the simplest pseudo-Hermitian models with real spectrum (viz., square-well on a real interval I of coordinates) is re-examined. A PT-symmetric complex deformation C of I is introduced and shown tractable via an innovated approach to matching conditions. The result is surprising: an arbitrarily small deformation I -> C implies a sudden collapse (i.e., the spontaneous PT-symmetry breaking) of virtually all the spectrum (i.e., up to its low-energy part).

Prepared for presentation during VI-th Meeting of Math. Physicists (Czech. Phys. Soc.), March 19 - 20, 2004, Olomouc, Czech Republic

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• Conditional observability
• Solvability and PT-symmetry in a double-well model with point interactions
• Quantum catastrophes: a case study
• Coupled-channel version of PT-symmetric square well
• Complete Set of Inner Products for a Discrete PT-symmetric Square-well Hamiltonian
• Matching method and exact solvability of discrete PT-symmetric square wells
• On the Role of the Normalization Factors ${\kappa }_n$ and of the Pseudo-Metric $\mathcal{P}\ne {\mathcal{P}}^{†}$ in Crypto-Hermitian Quantum Models
• Decays of degeneracies in PT-symmetric ring-shaped lattices
• Exactly solvable models with PT-symmetry and with an asymmetric coupling of channels
• PT-symmetric supersymmetry in a solvable short-range model
• Strengthened PT-symmetry with P $\ne$ P${}^{†}$
• Solvable non-Hermitian discrete square well with closed-form physical inner product
• Exceptional points and domains of unitarity for a class of strongly non-Hermitian real-matrix Hamiltonians
• N-site-lattice analogues of $V(x)=i{x}^3$
• Conjecture on the analyticity of PT-symmetric potentials and the reality of their spectra
• CPT-symmetric discrete square well
• Quantum square well with logarithmic central spike
• Phase transitions in quasi-Hermitian quantum models at exceptional points of order four
• Coulomb potential and the paradoxes of PT-symmetrization
• Twin Hamiltonians, three types of the Dyson maps, and the probabilistic interpretation problem in quasi-Hermitian quantum mechanics