  Background: Phase-equivalent transformations (PETs) are well-known in quantum
scattering and inverse scattering theory. PETs do not affect scattering phase
shifts and bound state energies of two-body system but are conventionally
supposed to modify two-body bound state observables such as the rms radius and
electromagnetic moments. Purpose: In order to preserve all bound state
observables, we propose a new particular case of PETs, a deuteron-equivalent
transformation (DET-PET), which leaves unchanged not only scattering phase
shifts and bound state (deuteron) binding energy but also the bound state wave
function. Methods: The construction of DET-PET is discussed; equations defining
the simplest DET-PETs are derived. We apply these simplest DET-PETs to the
JISP16 $NN$ interaction and use the transformed $NN$ interactions in
calculations of $^3$H and $^4$He binding energies in the No-core Full
Configuration (NCFC) approach based on extrapolations of the No-core Shell
Model (NCSM) basis space results to the infinite basis space. Results: We
demonstrate the DET-PET modification of the $np$ scattering wave functions and
study the DET-PET manifestation in the binding energies of $^3$H and $^4$He
nuclei and their correlation (Tjon line). Conclusions: It is shown that some
DET-PETs generate modifications of the central component while the others
modify the tensor component of the $NN$ interaction. DET-PETs are able to
modify significantly the $np$ scattering wave functions and hence the off-shell
properties of the $NN$ interaction. DET-PETs give rise to significant changes
in the binding energies of $^3$H (in the range of approximately 1.5 MeV) and
$^4$He (in the range of more than 9 MeV) and are able to modify the correlation
patterns of binding energies of these nuclei.
