A system of three quantum particles with point-like interactions

Consider a quantum three-particle system consisting of two fermions of unit mass and another particle of mass m>0 interacting in a point-like manner with the fermions. Such systems are studied here using the theory of self-adjoint extensions of symmetric operators: the Hamiltonian of the system is constructed as an extension of the symmetric energy operator H₀=−(1/2)((1/m)Δ_y+Δ_x1+Δ_x2), which is defined on the functions in L₂(R³)⊗L₂^asym(R³×R³) that vanish whenever the position of the third particle coincides with the position of a fermion. To construct a natural family of extensions of H₀, one must solve the problem of self-adjoint extensions for an auxiliary sequence {T_l, l=0,1,2,…} of symmetric operators acting in L₂(R³). All the operators T_l with even l are self-adjoint, and for every odd l there are two numbers 0l⁽¹⁾l⁽²⁾<∞ such that T_l is self-adjoint and lower semibounded for m>m_l⁽²⁾, and has deficiency indices for m⩽m_l⁽²⁾. When m∈[m_l⁽¹⁾,m_l⁽²⁾], every self-adjoint extension of T_l which is invariant under rotations of R³ is lower semibounded, but if 0l⁽¹⁾, then it has an infinite sequence of eigenvalues {λ_n} of multiplicity 2l+1 such that λ_n→−∞ as n→∞ (the Thomas effect). It follows from the last fact that there is a sequence of bound states of H₀ with spectrum P²/(2(m+2))+z_n, where the numbers z_n<0 cluster at 0 (Efimov's effect). Bibliography: 19 titles