Competing PT potentials and re-entrant PT symmetric phase for a particle in a box

Abstract

We investigate the effects of competition between two complex, $\mathcal{P}\mathcal{T}$-symmetric potentials on the $\mathcal{P}\mathcal{T}$-symmetric phase of a "particle in a box". These potentials, given by $V_Z(x)=iZ\mathrm{sign}(x)$ and $V_{\xi }(x)=i\xi [\delta (x-a)-\delta (x+a)]$, represent long-range and localized gain/loss regions respectively. We obtain the $\mathcal{P}\mathcal{T}$-symmetric phase in the $(Z,\xi )$ plane, and find that for locations $\pm a$ near the edge of the box, the $\mathcal{P}\mathcal{T}$-symmetric phase is strengthened by additional losses to the loss region. We also predict that a broken $\mathcal{P}\mathcal{T}$-symmetry will be restored by increasing the strength $\xi$ of the localized potential. By comparing the results for this problem and its lattice counterpart, we show that a robust $\mathcal{P}\mathcal{T}$-symmetric phase in the continuum is consistent with the fragile phase on the lattice. Our results demonstrate that systems with multiple, $\mathcal{P}\mathcal{T}$-symmetric potentials show unique, unexpected properties.

7 pages, 3 figures

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• Exponentially Fragile PT-Symmetry in Lattices with Localized Eigenmodes
• Tridiagonal PT-symmetric N by N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime
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• Bound states emerging from below the continuum in a solvable PT-symmetric discrete Schroedinger equation
• Time invariant $\mathcal{P}\mathcal{T}$-product and phase locking in $\mathcal{P}\mathcal{T}$-symmetric lattice models
• Metric Operator For The Non-Hermitian Hamiltonian Model and Pseudo-Supersymmetry
• PT-symmetry enabled stable modes in multi-core fiber