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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 \(H_\varepsilon \) with cutoff Coulomb potentials coupled with \((\alpha \delta +\beta \delta ')\)-like ones is investigated. The 1D Coulomb potential and the \(\delta '\)-potential are very sensitive to their regularization method. The conditions of the norm resolvent convergence of \(H_\varepsilon \) 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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  1. Department of Mechanics and Mathematics, Ivan Franko National University of Lviv, 1 Universytetska str., Lviv, 79000, Ukraine

    Yuriy Golovaty

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  1. Yuriy Golovaty
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Golovaty, Y. Quantum Graphs: Coulomb-Type Potentials and Exactly Solvable Models. Ann. Henri Poincaré 24, 2557–2585 (2023). https://doi.org/10.1007/s00023-023-01270-9

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