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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Analyticity preserving properties of resolvents for degenerate diffusion operators in one dimension


Author: Masaaki Tsuchiya
Journal: Proc. Amer. Math. Soc. 90 (1984), 91-94
MSC: Primary 47D05; Secondary 26E05, 34A25, 35A99, 60J60
MathSciNet review: 722422
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Abstract: Let $ L = a(x)({d^2}/d{x^2}) + b(x)(d/dx) + c(x)$ be a diffusion operator on a compact interval $ I = [{r_0},{r_1}]$ investigated by S. N. Ethier. Here, assume that $ a$, $ b$ and $ c$ are real analytic functions on $ I$, $ a(x) > 0$ for $ x \in ({r_0},{r_1})$, $ a({r_i}) = 0 \leqslant {( - 1)^i}b({r_i})(i = 0,1)$, and both $ {r_i}(i = 0,1)$ are simple zeros of $ a(x)$. It is shown that the resolvent $ \left\{ {{G_\lambda }} \right\}$ for $ L$ has the analyticity preserving property for sufficiently large $ \lambda $, so that the equation $ (L - \lambda )u = f$ is solvable in the space of real analytic functions on $ I$. Some examples are given to show that the condition on $ L$ is best possible.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0722422-2
PII: S 0002-9939(1984)0722422-2
Article copyright: © Copyright 1984 American Mathematical Society