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

 

Analytic domination with quadratic form type estimates and nondegeneracy of ground states in quantum field theory


Author: Alan D. Sloan
Journal: Trans. Amer. Math. Soc. 194 (1974), 325-336
MSC: Primary 81.47
MathSciNet review: 0345564
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Abstract: We present a theorem concerning the analytic domination by a semi-bounded selfadjoint operator H of another linear operator A which requires only the quadratic form type estimates

$\displaystyle \left\Vert {{H^{ - 1/2}}({{({\text{ad}}\;A)}^n}H){H^{ - 1/2}}u} \right\Vert \leq {c_n}\left\Vert u \right\Vert$

instead of the norm estimates

$\displaystyle \left\Vert {{{({\text{ad}}\;A)}^n}Hu} \right\Vert \leq {c_n}\left\Vert {Hu} \right\Vert$

usually required for this type of theorem. We call the new estimates ``quadratic form type", since they are sometimes equivalent to

$\displaystyle \vert({({\text{ad}}\;A)^n}Hu,u)\vert \leq {c_n}\vert(Hu,u)\vert.$

The theorem is then applied with H the Hamiltonian for the spatially cutoff boson field model with real, bounded below, even ordered polynomial self-interaction in one space dimension and $ A = \pi (g)$, the conjugate momentum to the free field. When the underlying Hilbert space of this model is represented as $ {L^2}(Q,dq)$ where dq is a probability measure on Q, the spectrum of the von Neumann algebra generated by bounded functions of certain field operators, then $ {e^{ - tH}}$ maximizes support in the sense that $ {e^{ - tH}}f$ is nonzero almost everywhere whenever f is not identically zero.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0345564-0
PII: S 0002-9947(1974)0345564-0
Keywords: Analytic vector, analytic domination, quantum fields, bosons, semigroup, positivity preserving
Article copyright: © Copyright 1974 American Mathematical Society