Feasibility of perturbative generation of bound-states from resonances or virtual states

I investigate whether it is possible to generate bound-states from resonances or virtual states through first-order perturbation theory. Using contact-type potentials as those appeared in pionless effective field theory, I show that it is possible to obtain negative-energy states by sandwiching a next-to-leading order (NLO) interaction with the leading-order (LO) wavefunctions, under the presence of LO resonances or virtual states. However, at least under the framework of time-independent Schr\"odinger equation and Hermitian Hamiltonian, there is an inability to create bound-states with structure similar to those formed by the non-perturbative treatments.