Quantum graphs: Coulomb-type potentials and exactly solvable models

TL;DR

This paper investigates the mathematical properties of Schrödinger operators with Coulomb-type potentials on star graphs, focusing on regularization, convergence, and self-adjoint realizations to ensure well-defined quantum models.

Contribution

It establishes conditions for norm resolvent convergence of regularized Hamiltonians and characterizes all self-adjoint realizations of Coulomb-type operators on star graphs.

Findings

Conditions for convergence of regularized Hamiltonians are derived.

All self-adjoint realizations of Coulomb Hamiltonians on star graphs are described.

Mathematically rigorous framework for Coulomb potentials on quantum graphs is provided.

Abstract

We study the Schr\"{o}dinger 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 $(\alpha \delta+\beta\delta')$-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 Schr\"{o}dinger 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.