Quantum Graphs: Coulomb-Type Potentials and Exactly Solvable Models

Golovaty Yuriy

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Quantum Graphs: Coulomb-Type Potentials and Exactly Solvable Models

机译：Quantum Graphs: Coulomb-Type Potentials and Exactly Solvable Models

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Abstract We study the Schrödinger operators on a non-compact star graph with the Coulomb-type potentials having singularities at the vertex. The convergence of regularized Hamiltonians Hεdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$H_varepsilon $$end{document} with cutoff Coulomb potentials coupled with (αδ+βδ′)documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$(alpha delta +beta delta ')$$end{document}-like ones is investigated. The 1D Coulomb potential and the δ′documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$delta '$$end{document}-potential are very sensitive to their regularization method. The conditions of the norm resolvent convergence of Hεdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$H_varepsilon $$end{document} depending on the regularization are established. The limit Hamiltonians give the Schrödinger operators with the Coulomb-type potentials in a mathematically precise meaning, ensuring the correct choice of vertex conditions. We also describe all self-adjoint realizations of the formal Coulomb Hamiltonians on the star graph.

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《Annales Henri Poincare》 |2023年第8期|2557-2585|共29页
作者
Golovaty Yuriy;
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作者单位

Ivan Franko National University of Lviv;

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正文语种英语
中图分类理论物理学;
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Schrödinger operator; Coulomb potential; δ′documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$delta '$$end{document}-Potential; Quantum graph; Vertex coupling condition; Solvable model; Point interaction; Primary 34L40; 81Q35; Secondary 34E10; 81Q10;

Quantum Graphs: Coulomb-Type Potentials and Exactly Solvable Models

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