The distributional Borel summability and the large coupling ?4 lattice fields

Caliceti, Emanuela; Grecchi, V.; Maioli, Marco

doi:10.1007/bf01221404

Cited by 10 publications

(7 citation statements)

References 17 publications

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“…2 . There is p 0 > 0 and there are positive constants a ί9 a 2 , c, c u c 2 such that r 3 )~7 /3 ] + η 0 r 2 (l + r 3 y Ί/6 }\pu\ 2 dr (9) uniformly for 0 < p ^ p o ,ueD(h p ). A similar estimate holds with h p replaced by K (ro= +co,ε = 0).…”

Section: Lemma 2 Let ε = ^R~1pmentioning

confidence: 99%

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Stark resonances: Asymptotics and distributional Borel Sum

1993

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We prove that the Stark effect perturbation theory of a class of bound states uniquely determines the position and the width of the resonances by Distributional Borel Sum. In particular the small field asymptotics of the width is uniquely related to the large order asymptotics of the perturbation coefficients. Similar results apply to all the "resonances" of the anharmonic and double well oscillators.

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Section: Lemma 2 Let ε = ^R~1pmentioning

confidence: 99%

“…Distributional Borel (DB) summability was defined in [7], following a suggestion of't Hooft [23] for double well problems. In particular a criterion for summability was proved [7] and some applications were performed to lattice field theories and to double well problems ( [7,8]) and to the justification of the semiclassical method [9].…”

Section: Introductionmentioning

confidence: 99%

Stark resonances: Asymptotics and distributional Borel Sum

1993

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“…It is known how to obtain this asymptotic behaviour of Q, from the asymptotic behaviour of the moments v(n) as n + cc [4]. Thus through hypotheses (H3)-(H5) the growth rate of the expansion coefficients a, and the allowed geometrical form of the domain G near zo are linked.…”

Section: An Integral Representationmentioning

confidence: 98%

“…. which grows fast enough to ensure that the coefficients an/v(n) define an entire function instead of requiring that these coefficients produce a power series with finite radius of convergence (as in the Watson-Nevanlinna reconstruction by Borel-summability [4,7,111). (c) The uniqueness part of the reconstruction is proved by controlling the interchange of two limits instead of relying on Carleman's results on quasi-analytic functions [lo].…”

Section: Introductionmentioning

confidence: 99%

How to reconstruct an analytic function from its asymptotic expansion?

Brüning

1996

Complex Variables, Theory and Application: An International Jou

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Asymptotic expansions of analytic functions at a boundary point of their domain of analyticity arise in a variety of problems in Applied Mathematics and Mathematical Physics. Relying on a concept of an asymptotic expansion which is slightly more restrictive than that of Poincart, but still considerably weaker than that of a 'strong asymptotic expansion', it is shown when and how the analytic function can be reconstructed from the asymptotic expansion. As in the famous Watson-Nevanlinna reconstruction by Bore1 summability moment summation methods are used. We find an interesting new link between the growth rate of the expansion coefficients with n and the allowed geometrical form of the domain of analyticity near the expansion point. The general reconstruction result is illustrated by a class of special reconstructions, the Mittag-Leffler reconstructions.

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“…where ψ(x) is restricted to the dense set of L 2 functions analytic on an angular sector The two DB complex conjugate sums [30][31][32][33][34] apply to the energy levels of these operators defined by the subdominant behaviors on the two pairs of sectors (S −2 , S 1 ) and (S −1 , S 2 ), respectively. The application of the perturbation theory leading to the DB sums has been thoroughly studied in [27] by numerical methods, specializing, in particular, to the calculation of the pair production rate obtained from the imaginary part of the resonance.…”

Section: )mentioning

confidence: 99%

PT-symmetric operators and metastable states of the 1D relativistic oscillators

Giachetti¹,

Grecchi²

2011

J. Phys. A: Math. Theor.

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We consider the one-dimensional Dirac equation for the harmonic oscillator and the associated second order separated operators giving the resonances of the problem by complex dilation. The same operators have unique extensions as closed PT -symmetric operators defining infinite positive energy levels converging to the Schrödinger ones as c → ∞. Such energy levels and their eigenfunctions give directly a definite choice of metastable states of the problem. Precise numerical computations shows that these levels coincide with the positions of the resonances up to the order of the width. Similar results are found for the Klein-Gordon oscillators, and in this case there is an infinite number of dynamics and the eigenvalues and eigenvectors of the PT -symmetric operators give metastable states for each dynamics.

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The distributional Borel summability and the large coupling ?4 lattice fields

Abstract: Abstract. We complete and correct some proofs of an earlier paper on distributional Borel summability and we add an application which can be useful in the discussion of semiclassical problems.

Cited by 10 publications

References 17 publications

Stark resonances: Asymptotics and distributional Borel Sum

Stark resonances: Asymptotics and distributional Borel Sum

How to reconstruct an analytic function from its asymptotic expansion?

PT-symmetric operators and metastable states of the 1D relativistic oscillators

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