## Instantons in quantum mechanics and resurgent expansions.(English)Zbl 1247.81135

Summary: Certain quantum mechanical potentials give rise to a vanishing perturbation series for at least one energy level (which as we here assume is the ground state), but the true ground-state energy is positive. We show here that in a typical case, the eigenvalue may be expressed in terms of a generalized perturbative expansion (resurgent expansion). Modified Bohr-Sommerfeld quantization conditions lead to generalized perturbative expansions which may be expressed in terms of nonanalytic factors of the form $$\exp(-a/g)$$, where $$a>0$$ is the instanton action, and power series in the coupling $$g$$, as well as logarithmic factors. The ground-state energy, for the specific Hamiltonians, is shown to be dominated by instanton effects, and we provide numerical evidence for the validity of the related conjectures.

### MSC:

 81Q15 Perturbation theories for operators and differential equations in quantum theory 81S40 Path integrals in quantum mechanics 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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### References:

 [1] LeGuillou, J.C.; Zinn-Justin, J., Large-order behaviour of perturbation theory, (1990), North-Holland Amsterdam [2] Brézin, E.; Parisi, G.; Zinn-Justin, J., Phys. rev. D, 16, 408, (1977) [3] Bogomolny, E.B., Phys. lett. B, 91, 431, (1980) [4] Zinn-Justin, J., Quantum field theory and critical phenomena, (2002), Clarendon Press Oxford, Chapter 43 [5] Forkel, H. [6] Zinn-Justin, J., J. math. phys., 22, 511, (1981) [7] Zinn-Justin, J., Nucl. phys. B, 192, 125, (1981) [8] Zinn-Justin, J., Nucl. phys. B, 218, 333, (1983) [9] Zinn-Justin, J., J. math. phys., 25, 549, (1984) [10] Zinn-Justin, J., (), Several results have also been reported in [11] J. Écalle, Les Fonctions Résurgentes, Tomes I-III, Publications Mathématiques d’Orsay, France, 1981-1985 [12] Stingl, M. [13] Jentschura, U.D.; Zinn-Justin, J., J. phys. A, 34, L253, (2001) [14] () [15] J. Zinn-Justin, U.D. Jentschura, Multi-instantons and exact results I: Conjectures, WKB expansions, and instanton interactions, Ann. Phys. (N.Y.), in press (2004) · Zbl 1054.81020 [16] J. Zinn-Justin, U.D. Jentschura, Multi-instantons and exact results II: Specific cases, higher-order effects, and numerical calculations, Ann. Phys. (N.Y.), in press (2004) · Zbl 1054.81022 [17] Herbst, I.W.; Simon, B., Phys. lett. B, 78, 304, (1978) [18] Witten, E., Nucl. phys. B, 185, 513, (1981) [19] Aoyama, H.; Kukuchi, H.; Okuochi, I.; Sato, M.; Wada, S., Nucl. phys. B, 533, 644, (1999) [20] Herbst, I.W.; Simon, B., Phys. rev. lett., 41, 67, (1978) [21] Franceschini, V.; Grecchi, V.; Silverstone, H.J., Phys. rev. A, 32, 1338, (1985) [22] Caliceti, E.; Grecchi, V.; Maioli, M., Commun. math. phys., 157, 347, (1993) [23] Caliceti, E., J. phys. A, 33, 3753, (2000) [24] Jentschura, U.D., Phys. rev. D, 62, 076001, (2000) [26] Itzykson, C.; Zuber, J.B., Quantum field theory, (1980), McGraw-Hill New York · Zbl 0453.05035
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