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arXiv · 2207.10403

Quantum graphs: Coulomb-type potentials and exactly solvable models

Yuriy Golovaty
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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 cut-off Coulomb potentials coupled with $(αδ+βδ')$-like ones is investigated.The 1D Coulomb potential and the $δ'$-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 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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Yuriy Golovaty. 2023-01-08. Quantum graphs: Coulomb-type potentials and exactly solvable models. https://doi.org/10.1007/s00023-023-01270-9

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