A characterization of singular Schrödinger operators on the half-line
Copyright policy )A characterization of singular Schrödinger operators on the half-line
AbstractWe 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.
- University Federico II of Naples Italy
- University of Milan Italy
- National Institute for Nuclear Physics Italy
- University of Vienna Austria
- Gran Sasso Science Institute Italy
34L40, 34E15, 35J10, 03H05, 26E35, 47S20, math-ph, 35J10, FOS: Physical sciences, math.FA, 34L40, Nonstandard analysis, Mathematics - Spectral Theory, math.MP, Schrödinger operators; singular perturbations; point interactions; nonstandard analysis;, 101032 Funktionalanalysis, Singular perturbations for ordinary differential equations, FOS: Mathematics, 103019 Mathematische Physik, Schrödinger operators, Spectral Theory (math.SP), Mathematical Physics, 47S20, 103019 Mathematical physics, QUANTUM-MECHANICS, point interactions, Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.), math.SP, 101032 Functional analysis, 34E15, Mathematical Physics (math-ph), 03H05, nonstandard analysis, PERTURBATIONS, Functional Analysis (math.FA), Mathematics - Functional Analysis, 26E35, Nonstandard operator theory, AMS subject classification 34L40 34E15 35J10 03H05 26E35 47S20, Schrodinger operators, singular perturbations, PARTICLE, POINT INTERACTIONS
34L40, 34E15, 35J10, 03H05, 26E35, 47S20, math-ph, 35J10, FOS: Physical sciences, math.FA, 34L40, Nonstandard analysis, Mathematics - Spectral Theory, math.MP, Schrödinger operators; singular perturbations; point interactions; nonstandard analysis;, 101032 Funktionalanalysis, Singular perturbations for ordinary differential equations, FOS: Mathematics, 103019 Mathematische Physik, Schrödinger operators, Spectral Theory (math.SP), Mathematical Physics, 47S20, 103019 Mathematical physics, QUANTUM-MECHANICS, point interactions, Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.), math.SP, 101032 Functional analysis, 34E15, Mathematical Physics (math-ph), 03H05, nonstandard analysis, PERTURBATIONS, Functional Analysis (math.FA), Mathematics - Functional Analysis, 26E35, Nonstandard operator theory, AMS subject classification 34L40 34E15 35J10 03H05 26E35 47S20, Schrodinger operators, singular perturbations, PARTICLE, POINT INTERACTIONS
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