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The distributional Borel summability and the large coupling \(\Phi ^ 4\) lattice fields. (English) Zbl 0648.40007

Following ’t Hooft we extend the Borel sum and the Watson-Nevanlinna criterion by allowing distributional transforms. This enables us to prove that the characteristic function of the measure of any \(g^{-2}\Phi^ 4\) finite lattice field is the sum of a power series expansion obtained by fixing exponentially small terms in the coefficients. The same result is obtained for the trace of the double well semigroup approximated by the n th order Trotter formula.

MSC:

40G10 Abel, Borel and power series methods
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[1] Beltrami, E.J., Wohlers, M.R.: Distributions and the boundary values of analytic functions. New York: London: Academic Press 1966 · Zbl 0186.19202
[2] Boas, R., Jr.: Entire functions. New York: Academic Press 1954 · Zbl 0058.30201
[3] Brézin, E., Parisi, G., Zinn-Justin, J.: Perturbation theory at large orders for a potential with degenerate minima. Phys. Rev.16 D, 408 (1977)
[4] Colombeau, J.F.: New generalized functions and multiplication of distributions. Amsterdam: North-Holland 1984 · Zbl 0532.46019
[5] Crutchfield, W.Y. II.: No horn of singularities for the double well anharmonic oscillator. Phys. Lett.77 B, 109-113 (1978)
[6] Crutchfield, W.Y. II.: Method for Borel-summing instanton singularities. Introduction. Phys. Rev.19 D, 2370-2384 (1979)
[7] Eckmann, J.-P., Wittwer, P.: Computer methods and Borel summability applied to Feigenbaum equation. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0598.58040
[8] Graffi, S., Grecchi, V.: The Borel sum of the double-well perturbation series and the Zinn-Justin conjecture. Phys. Lett.121 B, 410-414 (1983)
[9] Hardy, G.H.: Divergent series. Oxford: Clarendon Press 1949 · Zbl 0032.05801
[10] ’t Hooft, G.: The ways of subnuclear physics. Proceedings of the international school of subnuclear physics, Erice (1979), Zichichi, A. (ed.), pp. 943-971. New York: Plenum 1979
[11] ’t Hooft, G.: Private communication via B. Simon
[12] Khuri, N.N.: Zeros of the Gell-Mann-Low function and Borel summations in renormalizable theories. Phys. Lett.82 B, 83-88 (1979)
[13] Nevanlinna, F.: Ann. Acad. Sci. Fenn.12 A, No. 3 (1918-1919)
[14] Reed, M., Simon, B.: Methods of modern mathematical physics. Vol. IV. New York: Academic Press 1978 · Zbl 0401.47001
[15] Rosen, J.: Proc. Am. Math. Soc.66, 114-118 (1977)
[16] Simon, B.: Functional integration and quantum physics. New York: Academic Press 1979 · Zbl 0434.28013
[17] Simon, B.: Int. J. Quantum Chem.21, 3-25 (1982)
[18] Sokal, A.: An improvement of Watson’s theorem on Borel summability. J. Math. Phys.21, 261-263 (1980) · Zbl 0441.40012
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