Spectral properties of a pseudorelativistic system of two particles with finite masses (Q5930860)

The authors consider the Hamiltonian \[ H=\sqrt{-\Delta_1+m_1^2}+\sqrt{-\Delta_2+m_2^2}+V(r_{12}) \] where \(m_i\) and \(r_i=(x_{1i},x_{2i},x_{3i})\) are the mass and the coordinates of the \(i\)th particle, \(i=1,2\); \(r_{12}=r_1-r_2\). The interaction potential \(V(r_{12})\) has the form \(Z|r_{12}|^{- \gamma }\), \(Z<0\), \(0<\gamma \leq 2\). \(H\) is restricted to the invariant subspace corresponding to a fixed value of the momentum \(p\), and the problem is to study spectral properties of the resulting operator \(H_p\). If \(\gamma <2\), the discrete spectrum \(\sigma_d(H_p)\) is infinite; its asymptotics is found. For \(\gamma =2\), conditions are found for the finiteness and infiniteness of \(\sigma_d(H_p)\). Depending on the value of \(Z\), \(\sigma_d(H_p)\) can be infinite for any \(p\), or only for \(|p|\) large enough.