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 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 which is defined on the functions in that vanish whenever the position of the third particle coincides with the position of a?fermion. To construct a?natural family of extensions of?, one must solve the problem of self-adjoint extensions for an auxiliary sequence of symmetric operators acting in?. All the operators? with even? are self-adjoint, and for every odd? there are two numbers such that is self-adjoint and lower semibounded for , and has deficiency indices for . When , every self-adjoint extension of? which is invariant under rotations of? is lower semibounded, but if , then it has an infinite sequence of eigenvalues? of multiplicity such that as (the Thomas effect). It follows from the last fact that there is a?sequence of bound states of? with spectrum , where the numbers cluster at?0 (Efimov's effect). Bibliography: 19 titles.

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