On pointlike interaction between three particles: two fermions and another particle (Q454476)

Summary: The problem of construction of self-adjoint Hamiltonian for quantum system consisting of three pointlike interacting particles (two fermions with mass 1 plus a particle of another nature with mass \(m > 0\)) was studied in many works. In most of these works, a family of one-parametric symmetrical operators \(\{H_\epsilon, \epsilon \in \mathbb R^1\}\) is considered as such Hamiltonians. In addition, the question about the self-adjointness of \(H_\epsilon\) is equivalent to the one concerning the self-adjointness of some auxiliary operators \(\{\mathcal T_l, l = 0, 1, \dots\}\) acting in the space \(L_2(\mathbb R^1_+, r^2 dr)\). In this work, we establish a simple general criterion of self-adjointness for operators \(\mathcal T_l\) and apply it to the cases \(l = 0\) and \(l = 1\). It turns out that the operator \(\mathcal T_{l = 0}\) is self-adjoint for any \(m\), while the operator \(\mathcal T_{l = 1}\) is self-adjoint for \(m > m_0\), where the value of \(m_0\) is given explicitly in the paper.