Hyperelliptic action integral

Elsner, Bernhard. "Hyperelliptic action integral." Annales de l'institut Fourier 49.1 (1999): 303-331. <http://eudml.org/doc/75338>.

@article{Elsner1999,
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.},
author = {Elsner, Bernhard},
journal = {Annales de l'institut Fourier},
keywords = {complex WKB method; hyperelliptic curves; Stokes lines; non compact Riemann surfaces},
language = {eng},
number = {1},
pages = {303-331},
publisher = {Association des Annales de l'Institut Fourier},
title = {Hyperelliptic action integral},
url = {http://eudml.org/doc/75338},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Elsner, Bernhard
TI - Hyperelliptic action integral
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 1
SP - 303
EP - 331
AB - 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.
LA - eng
KW - complex WKB method; hyperelliptic curves; Stokes lines; non compact Riemann surfaces
UR - http://eudml.org/doc/75338
ER -