Quantum graphs: Coulomb-type potentials and exactly solvable models

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 $H_{\epsilon }$ with cut-off Coulomb potentials coupled with $(\alpha \delta +\beta {\delta }^{\prime })$-like ones is investigated.The 1D Coulomb potential and the ${\delta }^{\prime }$-potential are very sensitive to their regularization method. The conditions of the norm resolvent convergence of $H_{\epsilon }$ 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.

Accepted for publication in Annales Henri Poincaré, 25 pages, 8 figures

References in corpus (2)

• Coupling in the singular limit of thin quantum waveguides
• Self-adjoint extensions of Coulomb systems in 1,2 and 3 dimensions

Cited by in corpus (2)

• On negative eigenvalues of 1D Schrödinger operators with ${\delta }^{\prime }$-like potentials
• On eigenvibrations of branched structures with heterogeneous mass density