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On Pointlike Interaction between Three Particles: Two Fermions and Another Particle

ISRN Mathematical Physics
Publication Date
  • Physics


The problem of construction of self-adjoint Hamiltonian for quantum system consisting of three pointlike interacting particles (two fermions with mass 1 plus a particle of another nature with mass 𝑚 > 0 ) was studied in many works. In most of these works, a family of one-parametric symmetrical operators { 𝐻 𝜀 , 𝜀 ∈ ℝ 1 } is considered as such Hamiltonians. In addition, the question about the self-adjointness of 𝐻 𝜀 is equivalent to the one concerning the self-adjointness of some auxiliary operators { 𝒯 𝑙 , 𝑙 = 0 , 1 , … } acting in the space 𝐿 2 ( ℝ 1 + , 𝑟 2 𝑑 𝑟 ) . In this work, we establish a simple general criterion of self-adjointness for operators 𝒯 𝑙 and apply it to the cases 𝑙 = 0 and 𝑙 = 1 . It turns out that the operator 𝒯 𝑙 = 0 is self-adjoint for any 𝑚 , while the operator 𝒯 𝑙 = 1 is self-adjoint for 𝑚 > 𝑚 0 , where the value of 𝑚 0 is given explicitly in the paper.

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