Threshold Analysis of the Schrödinger Operator of the System of Three Particles with Masses \(m_1=m_2=\infty \) and \(m_3<\infty \)

We consider a family of parameter-dependent discrete Schrödinger operators corresponding to the Hamiltonian of a system of three arbitrary particles (either fermions or bosons) with masses \(m_1=m_2=\infty \) $m_1=m_2=\infty$ and \(m_3<\infty \) $m_3<\infty$ , on the integer lattice, \(\mathbb {Z}^3\) ${\mathbb{Z}}^3$ . The interactions of particles are described via zero-range attractive forces. We show that such operators might have an infinite discrete spectrum depending on the interaction energies, \(\mu _1\) ${\mu }_1$ and \(\mu _2\) ${\mu }_2$ , between the finite-mass particle and the infinite-mass ones. Infinitely many eigenvalues arise from threshold eigenvalues and threshold resonances when \(\mu _1=\mu _2\) ${\mu }_1={\mu }_2$ and only from threshold eigenvalues otherwise. Moreover, the condition for the existence of eigenvalues, threshold eigenvalues and threshold resonances as well as their dependence on the parameters of the operators are explicitly derived.