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Abstract <p> The paper deals with a Hamiltonian, namely, with a semi-bounded self-adjoint operator that is attributed to the problem of scattering of three one-dimensional particles with point interaction in pairs, in other words, with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3254_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\( \delta \)</EquationSource> </InlineEquation>-functional singular potential of interaction. The support of the potential in the Hamiltonian coincides with a symmetric star-graph having six leads on the two-dimensional plane. Due to the symmetry, we find that such a model is exactly solvable, which means that the eigenfunctions of the discrete spectrum and the generalized eigenfunctions of the essential (absolutely continuous) spectrum are determined explicitly, i.e., by quadrature. In this (first part) of our work we describe the discrete spectrum and the eigenfunctions. </p> <p> <b> DOI</b> 10.1134/S1061920825600850 </p>

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Three Identical One-Dimensional Quantum Particles with Point Interaction as a Solvable Model: I. Discrete Spectrum

  • 期刊论文

  • 发表日期：

    2025年9月29日

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  • M.A. Lyalinov

摘要

Abstract

The paper deals with a Hamiltonian, namely, with a semi-bounded self-adjoint operator that is attributed to the problem of scattering of three one-dimensional particles with point interaction in pairs, in other words, with \( \delta \) -functional singular potential of interaction. The support of the potential in the Hamiltonian coincides with a symmetric star-graph having six leads on the two-dimensional plane. Due to the symmetry, we find that such a model is exactly solvable, which means that the eigenfunctions of the discrete spectrum and the generalized eigenfunctions of the essential (absolutely continuous) spectrum are determined explicitly, i.e., by quadrature. In this (first part) of our work we describe the discrete spectrum and the eigenfunctions.

DOI 10.1134/S1061920825600850