S -matrix pole symmetries for non-Hermitian scattering Hamiltonians

Abstract

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) possesing 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 in 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.

On beautiful analytic structure of the S-matrix

“…Our results for the attractive case [20] can be readily applied for the analysis of the s-wave Klein-Gordon equation with exponential scalar and vector potentials [27, section 4]. At the same time a proper understanding of analytic structure of the S-matrix of essentially a textbook model will do no harm when attempting to generalize the presented results in the direction of non-Hermitian scattering Hamiltonians [16]. Last but not the least, we hope to stimulate search for further exactly solvable S-matrix models.…”

“…i i where the first factor including the Jost function,   , is only a function of k, and only the second factor, r  ( ) I x i , depends on both k and r. In virtue of (11), the first factor is finite for any a = r  ( ) I 0 i . Let us ignore for a while the first k-dependent prefactors in (16). Then -( ) f k r , , which is typically exponentially increasing on the physical sheet as  ¥ r , would become suddenly exponentially decreasing for  ¥ r for any  r Îi , i.e.…”

“…Ma's finding inspired and motivated many authors, such as in now classical [3,[8][9][10][11][12][13]. There is a noticeable recent revival of interest in the complex analytic structure of the S-matrix related either to (i) the so-called modal expansion of the scattered field [14,15], or to (ii) non-Hermitian scattering Hamiltonians [16]. The differences with potentials of finite range highlighted above bear important consequences when attempting to generalize modal expansion of the scattered field for a finite range potential [14,15] to potentials of infinite range.…”

On the Heisenberg condition in the presence of redundant poles of the S-matrix

“…This is because essential conclusions of our analysis will not change if the exact equalities involving r-dependence were replaced by asymptotic ones. Whether redundant poles and zeros and the Heisenberg condition for other model cases, including non-Hermitian scattering Hamiltonians [12], show similar behaviour is the subject of future study. * * * The work of AEM was supported by the Australian Research Council and UNSW Scientia Fellowship.…”

“…At the same time, especially in connection with non-Hermitian scattering Hamiltonians [12], one witnesses a recent revival of interest in the analytic structure of the S-matrix leading to a number of surprising real applications [13]. In this regard, the potential (1) has an immense pedagogical value.…”