A characterization of singular Schrödinger operators on the half-line. Issue 4 (7th December 2021)

• Record Type: Journal Article
• Authors: Scandone, Raffaele; Luperi Baglini, Lorenzo; Simonov, Kyrylo
• Is Part Of: Canadian mathematical bulletin =. Volume 64:Issue 4(2021)
• Page Start: 923
• Page End: 941
• Publication Date: 2021-12-07
• Subjects: 34L40 -- 34E15 -- 35J10 -- 03H05 -- 26E35 -- 47S20; Schrödinger operators -- singular perturbations -- point interactions -- nonstandard analysis; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439520000958
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: We study a class of delta-like perturbations of the Laplacian on the half-line, characterized by Robin boundary conditions at the origin. Using the formalism of nonstandard analysis, we derive a simple connection with a suitable family of Schrödinger operators with potentials of very large (infinite) magnitude and very short (infinitesimal) range. As a consequence, we also derive a similar result for point interactions in the Euclidean space $\mathbb {R}^3$, in the case of radial potentials. Moreover, we discuss explicitly our results in the case of potentials that are linear in a neighborhood of the origin.