Dona Ghosh

Accidental crossings of eigenvalues in the one-dimensional complex PT-symmetric Scarf-II potential

Abstract So far, the well known two branches of real discrete spectrum of complex PT-symmetric Scarf-II potential are kept isolated. Here, we suggest that these two need to be brought together as doublets: E ± n ( λ ) with n = 0 , 1 , 2 …  . Then if strength ( λ ) of the imaginary part of the potential is varied smoothly some pairs of real eigenvalue curves can intersect and cross each other at λ = λ ⁎ ; this is unlike one-dimensional Hermitian potentials. However, we show that the corresponding eigenstates at λ = λ ⁎ are identical or linearly dependent denying degeneracy in one dimension, once again. Other pairs of eigenvalue curves coalesce to complex-conjugate pairs completing the scenario of spontaneous breaking of PT-symmetry at λ = λ c . To re-emphasize, sharply at λ = λ ⁎ and λ c , two real eigenvalues coincide, nevertheless their corresponding eigenfunctions become identical or linearly dependent and the Hamiltonian looses diagonalizability.

A new solvable complex PT-symmetric potential

Abstract We propose a new solvable one-dimensional complex PT-symmetric potential as V ( x ) = i g sgn ( x ) | 1 − exp ⁡ ( 2 | x | / a ) | and study the spectrum of H = − d 2 / d x 2 + V ( x ) . For smaller values of a , g 1 , there is a finite number of real discrete eigenvalues. As a and g increase, there exist exceptional points (EPs), g n (for fixed values of a ), causing a scarcity of real discrete eigenvalues, but there exists at least one. We also show these real discrete eigenvalues as poles of reflection coefficient. We find that the energy-eigenstates ψ n ( x ) satisfy (1): PT ψ n ( x ) = 1 ψ n ( x ) and (2): PT ψ E n ( x ) = ψ E n ⁎ ( x ) , for real and complex energy eigenvalues, respectively.

Real Discrete Spectrum of Complex PT-Symmetric Scattering Potentials

We investigate the parametric evolution of the real discrete spectrum of several complex PT symmetric scattering potentials of the type

Divergence of

V(x)=-V_1 F_e(x) + i V_2 F_o(x), V_1>0, F_e(x)>0

\langle {p}^{6}\rangle

by varying

in discontinuous potential wells

V_2

Solvable models of an open well and a bottomless barrier: one dimensional exponential potentials

slowly. Here

Transparency of the complex PT-symmetric potentials for coherent injection

e,o

Expectation values of

stand for even and odd parity and

p^2

F_{e,o}(\pm \infty)=0