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Bernhard Elsner
Hyperelliptic action integral
Annales de l'institut Fourier, 49 no. 1 (1999), p. 303-331, doi: 10.5802/aif.1675
Article PDF | Reviews MR 2000f:14069 | Zbl 0935.32012

Résumé - Abstract

Applying the ``exact WKB method" (cf. Delabaere-Dillinger-Pham) to the stationary one-dimensional Schrödinger equation with polynomial potential, one is led to a multivalued complex action-integral function. This function is a (hyper)elliptic integral; the sheet structure of its Riemann surface above the plane of its values has interesting properties: the projection of its branch-points is in general a dense subset of the plane, and there is a group of symmetries acting on the surface. The distribution of the branch points on the surface is of crucial importance, because it gives the position for the obstacles to Borel-Laplace summation of the WKB-symbols. In ``Approche de la résurgence" by B. Candelpergher, J.-C. Nosmas et F. Pham, p. 103-105, an attempt has been made towards giving an explicit construction of the surface with paper, scissors and glue; here we give the correct construction and in addition we prove that each surface constructed in this way comes from a polynomial potential. Along the way we are lead to an elementary conjecture in the theory of holomorphic functions.

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