"Striped" Rectangular Rigid Box with Hermitian and non-Hermitian $\mathcal{P}\mathcal{T}$ Symmetric Potentials

Eigenspectra of a spinless quantum particle trapped inside a rigid, rectangular, two-dimensional (2D) box subject to diverse inner potential distributions are investigated under hermitian, as well as non-hermitian antiunitary $\mathcal{P}\mathcal{T}$ (composite parity and time-reversal) symmetric regimes. Four sectors or “stripes” inscribed in the rigid box comprising contiguously conjoined parallel rectangular segments with one side equaling the entire width of the box are studied. The stripes encompass piecewise constant potentials whose exact, complete energy eigenspectrum is obtained employing matrix mechanics. Various striped potential compositions, viz. real valued ones in the hermitian regime as well as complex, non-hermitian but $\mathcal{P}\mathcal{T}$ symmetric ones are considered separately and in conjunction, unraveling among typical lowest lying eigenvalues, retention and breakdown scenarios engendered by the $\mathcal{P}\mathcal{T}$ symmetry, bearing upon the strength of non-hermitian sectors. Some states exhibit a remarkable crossover of symmetry making' and breaking’: while a broken $\mathcal{P}\mathcal{T}$ gets reinstated for an energy level, higher levels may couple to continue with symmetry breaking. Further, for a charged quantum particle a $\mathcal{P}\mathcal{T}$ symmetric electric field, furnished with a striped potential backdrop, also reveals peculiar retention and breakdown $\mathcal{P}\mathcal{T}$ scenarios. Depictions of prominent probability redistributions relating to various potential distributions both under norm-conserving unitary regime for hermitian Hamiltonians and non-conserving ones post $\mathcal{P}\mathcal{T}$ breakdown are presented.