On point-like interaction of three particles: two fermions and another particle. II

Abstract

This work continues \cite{bib1} where the construction of Hamiltonian $H$ for the system of three quantum particles is considered. Namely the system consists of two fermions with mass $1$ and another particle with mass $m>0$. In the present paper, like in \cite{bib1}, we study the part $T_{l=1}$ of auxilliary operator $T={\oplus }_{l=0}^{\infty }{T}_l$ involving the construction of the resolvent for the operator $H$. In this work together with the previous one two constants $0<m_1<m_0<\infty$ were found such that: 1) for $m>m_0$ the operator $T_{l=1}$ is selfadjoint but for $m⩽m_0$ it has the deficiency indexes $(1,1)$; 2) for $m_1<m<m_0$ any selfadjoint extension of $T_{l=1}$ is semibounded below; 3) for $0<m<m_1$ any selfadjoint extension of $T_{l=1}$ has the sequence of eigenvalues $\{{\lambda }_n<0,n>n_0\}$ with the asymptotics \[ λ_n = λ_0 e^{δn} + O(1),\quad n\to\infty, \] where ${\lambda }_0<0$, $\delta >0$, $n_0>0$ and there is'nt other spectrum on the interval $\lambda <{\lambda }_{n_0}$.