Scattering theory with localized non-Hermiticities

Abstract

In the context of the recent interest in solvable models of scattering mediated by non-Hermitian Hamiltonians (cf. H. F. Jones, Phys. Rev. D 76, 125003 (2007)) we show that the well known variability of the ad hoc choice of the metric Θ which defines the physical Hilbert space of states can help us to clarify several apparent paradoxes. We argue that with a suitable Θ a fully plausible physical picture of the scattering can be recovered. Quantitatively, our new recipe is illustrated on an exactly solvable toy model.

Hermitian–Non-Hermitian Interfaces in Quantum Theory

“…In the present context this means that once the metric (25) remains diagonal, in an interval of small = > 0 at least, we may conclude that the strongnon-locality effects as caused by the metric Θ and observed, say, in [21,31] are absent here. In this sense, our present model shares the weak-nonlocality merits of its predecessors in [22,23,28].…”

“…His interest was predominantly paid to the scattering [20] and his answers were discouraging [21]. In papers [22,23,28] we opposed his scepticism. We argued that the difficulties with the HSP-NHSP interface may be attributed to the ultralocal, point-interaction toy model background of his methodical analysis.…”

“…In our first paper [28] on the subject, we pointed out that Jones' discouraging "no-interface" conclusions remain strongly model-dependent. For another, weakly nonlocal choice of (non-Hermitian) (weakly nonlocal) , we encountered a much less drastic effect of the interchange of the mathematically equivalent Schrödinger equations (2) and (4) upon the predicted physical outcome of the scattering (see also the related footnote added in [21]).…”

“…One may conclude that the local-interaction nature and constructive tractability of the alternative models (10) contained in class F (NH) would render their isospectral partners (3) nonlocal. Thus, some of the weaker forms of the nonlocalities as sampled, for example, in [22,23,28] may be expected necessary for the constructive search for the nonempty interfaces F (퐻) ∩ F (NH) .…”

“…In [22,23,28], we proposed that one of the most efficient resolutions of at least some of the latter problems might be sought and found in the discretization of the coordinates. Thus, one replaces the real line of ∈ (−∞, ∞) by a discrete lattice of grid points 푗 such that 푗 = 0 + ℎ , with = .…”