$S$-matrix pole symmetries for non-Hermitian scattering Hamiltonians

The complex eigenvalues of some non-Hermitian Hamiltonians, e.g., parity-time-symmetric Hamiltonians, come in complex-conjugate pairs. We show that for non-Hermitian scattering Hamiltonians (of a structureless particle in one dimension) possessing one of four certain symmetries, the poles of the $S$-matrix eigenvalues in the complex momentum plane are symmetric about the imaginary axis, i.e., they are complex-conjugate pairs on the complex-energy plane. This applies even to states which are not bounded eigenstates of the system, i.e., antibound or virtual states, resonances, and antiresonances. The four Hamiltonian symmetries are formulated as the commutation of the Hamiltonian with specific antilinear operators. Example potentials with such symmetries are constructed and their pole structures and scattering properties are calculated.

Figure 1

Example of configuration of poles (filled circles) and zeros (open circles) of the $S$-matrix eigenvalues in the complex momentum plane for Hermitian Hamiltonians. Poles in the upper half-plane $[Im(p)>0]$ correspond to bound eigenstates of the Hamiltonian, i.e., localized states with negative energy. Poles in the lower half-plane correspond to virtual states $[Re(p)=0]$, resonances $[Re(p)>0]$, and antiresonances $[Re(p)<0]$. Singularities with a negative imaginary part correspond to states that do not belong to the Hilbert space since they are not normalizable. However, they can produce observable effects in the scattering amplitudes, in particular, when they approach the real axis. The pole structures of symmetries IV, V, and VII (see Table 1) are similar, but pole pairs are also possible in the upper half-plane.

Figure 2

Poles and pole trajectories of the time-reversal symmetric potential (24) for (a) varying $V_0$ with $a=2b$; (b) varying $a$ with $b=0.5p_0, V_0>0$; and (c) varying $b$ with $a=p_0, V_0>0$. At pole collisions we connect each of the incoming trajectories with a different emerging trajectory but the choice of outgoing branch is arbitrary since the two colliding poles lose their identity.

Figure 3

Transmission and reflection coefficients of the time-reversal symmetric potential (24) with $a=p_0, b=0.5p_0$, and $V_0>0$.

Figure 4

Poles and pole trajectories for the parity pseudo-Hermitian potential (28) (a) varying $V_0$ with $a=b$; (b) varying $a$ with $b=p_0, V_0>0$; and (c) varying $b$ with $a=0.5p_0, V_0>0$.

Figure 5

Transmission and reflection coefficients for $a=b=0.5p_0$ and $V_0>0$.