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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 230245, 18 pages
http://dx.doi.org/10.5402/2012/230245
Research Article

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

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetnyi 19, Moscow, Russia

Received 3 April 2012; Accepted 29 April 2012

Academic Editors: J. Banasiak and M. Martins

Copyright © 2012 Robert Adol'fovich Minlos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 π‘š>0) was studied in many works. In most of these works, a family of one-parametric symmetrical operators {π»πœ€,πœ€βˆˆβ„1} is considered as such Hamiltonians. In addition, the question about the self-adjointness of π»πœ€ is equivalent to the one concerning the self-adjointness of some auxiliary operators {𝒯𝑙,𝑙=0,1,…} acting in the space 𝐿2(ℝ1+,π‘Ÿ2π‘‘π‘Ÿ). In this work, we establish a simple general criterion of self-adjointness for operators 𝒯𝑙 and apply it to the cases 𝑙=0 and 𝑙=1. It turns out that the operator 𝒯𝑙=0 is self-adjoint for any π‘š, while the operator 𝒯𝑙=1 is self-adjoint for π‘š>π‘š0, where the value of π‘š0 is given explicitly in the paper.