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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Singular perturbations and nonstandard analysis


Authors: S. Albeverio, J. E. Fenstad and R. Høegh-Krohn
Journal: Trans. Amer. Math. Soc. 252 (1979), 275-295
MSC: Primary 34B25; Secondary 03H05, 26E35, 35P99
DOI: https://doi.org/10.1090/S0002-9947-1979-0534122-5
MathSciNet review: 534122
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Abstract: We study by methods of nonstandard analysis second order differential operators with zero order coefficients which are too singular to be defined by standard functions. In particular we study perturbations of the Laplacian in $ {R^3}$ given by potentials of the form $ \lambda {\Sigma _j}\delta \left( {x\, - \,{x_j}} \right)$. We also study Sturm-Liouville problems with zero order coefficients given by measures and prove that they satisfy the same oscillation theorems as the regular Sturm-Liouville problems.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0534122-5
Keywords: Singular perturbations, Schrödinger operators, Sturm-Liouville problems, nonstandard analysis
Article copyright: © Copyright 1979 American Mathematical Society