Affiliations: [a] Department of Mathematics, University of Rome La Sapienza, Piazza Aldo Moro 5, 00185 Rome, Italy. E-mail: gianfa@sissa.it | [b] International School for Advanced Studies – SISSA, via Bonomea 265, 34136 Trieste, Italy | [c] Institute of Mathematics, Ludwig Maximilian University of Munich, Theresienstr. 39, 80333 Munich, Germany. E-mails: michel@math.lmu.de, alemiche@sissa.it
Correspondence:
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Corresponding author: Alessandro Michelangeli, Institute of Mathematics, Ludwig Maximilian University of Munich, Theresienstr. 39, 80333 Munich, Germany. E-mail: michel@math.lmu.de.
Abstract: We discuss the general model of a Schrödinger quantum particle constrained on a straight half-line with given self-adjoint boundary condition at the origin and an interaction potential supported around the origin. We study the limit when the range of the potential scales to zero and its magnitude blows up. We show that in the limit the dynamics is generated by a self-adjoint negative Laplacian on the half-line, with a possible preservation or modification of the boundary condition at the origin, depending on the magnitude of the scaling and of the strength of the potential.
Keywords: point interaction, self-adjoint boundary conditions, singular scaling limits in Schrödinger operators, resolvent convergence, Konno–Kuroda resolvent formula
