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Transactions of the American Mathematical Society

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

 

 

The $ (\phi\sp{2n})\sb{2}$ field Hamiltonian for complex coupling constant


Authors: Lon Rosen and Barry Simon
Journal: Trans. Amer. Math. Soc. 165 (1972), 365-379
MSC: Primary 81.47
DOI: https://doi.org/10.1090/S0002-9947-1972-0292436-4
Erratum: Trans. Amer. Math. Soc. 172 (1972), 508.
MathSciNet review: 0292436
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Abstract: We consider hamiltonians $ {H_\beta } = {H_0} + \beta {H_I}(g)$, where $ {H_0}$ is the hamiltonian of a free Bose field $ \phi (x)$ of mass $ m > 0$ in two-dimensional space-time, $ {H_I}(g) = \smallint g(x):P(\phi (x)):dx$ where $ g \geqq 0$ is a spatial cutoff and P is an arbitrary polynomial which is bounded below, and the coupling constant $ \beta $ is in the cut plane, i.e. $ \beta \ne $ negative real. We show that $ {H_\beta }$ generates a semigroup with hypercontractive properties and satisfies higher order estimates of the form $ \left\Vert {{H_0}{N^r}R_\beta ^s} \right\Vert < \infty $, where N is the number operator, $ {R_\beta } = {({H_\beta } + b)^{ - 1}}$, r a positive integer, and $ \beta $, s, and b are suitably chosen. For any $ 0 \leqq \Theta < \pi $, $ {R_\beta }$ converges in norm to $ {R_0}$ as $ \vert\beta \vert \to 0$ with $ \vert\arg \beta \vert \leqq \Theta $. Finally we discuss applications of these results and establish asymptotic series and Borel summability for various objects in the real $ \beta $ theory.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0292436-4
Keywords: Boson fields, polynomial interaction, hamiltonian, vacuum expectation values, coupling constant analyticity, asymptotic series, Borel summability
Article copyright: © Copyright 1972 American Mathematical Society