The (𝜙²ⁿ)₂ field Hamiltonian for complex coupling constant

Rosen, Lon; Simon, Barry

doi:10.1090/S0002-9947-1972-0292436-4

The $(\phi ^{2n})_{2}$ field Hamiltonian for complex coupling constant
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by Lon Rosen and Barry Simon PDF

Trans. Amer. Math. Soc. 165 (1972), 365-379 Request permission

Erratum: Trans. Amer. Math. Soc. 172 (1972), 508.

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 $\left \| {{H_0}{N^r}R_\beta ^s} \right \| < \infty$, 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 $|\beta | \to 0$ with $|\arg \beta | \leqq \Theta$. 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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