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We realize the treatment of bound and continuum nuclear systems in theproximity of a three-body breakup threshold within the ab initio framework ofthe no-core shell model with continuum. Many-body eigenstates obtained from thediagonalization of the Hamiltonian within the harmonic-oscillator expansion ofthe no-core shell model are coupled with continuous microscopic three-clusterstates to correctly describe the nuclear wave function both in the interior andasymptotic regions. We discuss the formalism in detail and give algebraicexpressions for the case of core+$n$+$n$ systems. Usingsimilarity-renormalization-group evolved nucleon-nucleon interactions, weanalyze the role of $^4$He+$n$+$n$ clustering and many-body correlations in theground and low-lying continuum states of the Borromean $^6$He nucleus, andstudy the dependence of the energy spectrum on the resolution scale of theinteraction. We show that $^6$He small binding energy and extended radiicompatible with experiment can be obtained simultaneously, without recurring toextrapolations. We also find that a significant portion of the ground-stateenergy and the narrow width of the first $2^+$ resonance stem from many-bodycorrelations that can be interpreted as core-excitation effects.

Three-cluster dynamics within the ab initio no-core shell model with continuum: How many-body correlations and $\alpha$-clustering shape $^6$He