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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Perturbation of translation invariant positivity preserving semigroups on $ L\sp{2}({\bf R}\sp{N})$


Authors: Ira W. Herbst and Alan D. Sloan
Journal: Trans. Amer. Math. Soc. 236 (1978), 325-360
MSC: Primary 47D05; Secondary 35P99, 81.47
MathSciNet review: 0470750
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Abstract: The theory of singular local perturbations of translation invariant positivity preserving semigroups on $ {L^2}({{\mathbf{R}}^N},{d^N}x)$ is developed. A powerful approximation theorem is proved which allows the treatment of a very general class of singular perturbations. Estimates on the local singularities of the kernels of the semigroups, $ {e^{ - tH}}$, are given. Eigenfunction expansions are derived. The local singularities of the eigenfunction and generalized eigenfunctions are discussed. Results are illustrated with examples involving singular perturbations of --$ \Delta$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0470750-2
PII: S 0002-9947(1978)0470750-2
Keywords: Positivity preserving, Levy-Khintchine formula, semigroups, singular perturbations, generalized eigenfunctions
Article copyright: © Copyright 1978 American Mathematical Society