TY - JOUR

T1 - Path integrals, asymptotics, and singular perturbations

AU - McLaughlin, David W.

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1972

Y1 - 1972

N2 - In this paper we evaluate several Feynman path integrals asymptotically with respect to various parameters in order to gain mathematical insight into the asymptotic evaluation of function space integrals with oscillatory integrands, a type of integral which is beginning to appear in areas of physics other than quantum mechanics. In each integral studied, the integrand factors into a product of two functionals, one of which is dominant in the limit under consideration. By systematically exploiting this feature, we obtain the asymptotic behavior of path integrals for the physical situations of (1) weakly complex potentials, (2) high energy and complex potentials, (3) weak real potentials, and (4) strong real potentials. In the complex cases, the techniques indicate a means to handle (complex valued) turning points. In the sections treating strong and weak potentials, we relate the relative ease with which one may exploit the factorization of the integrand to the theories of regular and singular perturbations. In the singular case, several examples are presented, one of which is a high energy evaluation of the path integral associated with the "Langer transformed" radial equation. Finally, using more conventional techniques, we construct the complete asymptotic series for each case, thus formally establishing that we have obtained the leading term in an asymptotic expansion of the path integral.

AB - In this paper we evaluate several Feynman path integrals asymptotically with respect to various parameters in order to gain mathematical insight into the asymptotic evaluation of function space integrals with oscillatory integrands, a type of integral which is beginning to appear in areas of physics other than quantum mechanics. In each integral studied, the integrand factors into a product of two functionals, one of which is dominant in the limit under consideration. By systematically exploiting this feature, we obtain the asymptotic behavior of path integrals for the physical situations of (1) weakly complex potentials, (2) high energy and complex potentials, (3) weak real potentials, and (4) strong real potentials. In the complex cases, the techniques indicate a means to handle (complex valued) turning points. In the sections treating strong and weak potentials, we relate the relative ease with which one may exploit the factorization of the integrand to the theories of regular and singular perturbations. In the singular case, several examples are presented, one of which is a high energy evaluation of the path integral associated with the "Langer transformed" radial equation. Finally, using more conventional techniques, we construct the complete asymptotic series for each case, thus formally establishing that we have obtained the leading term in an asymptotic expansion of the path integral.

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U2 - 10.1063/1.1666052

DO - 10.1063/1.1666052

M3 - Article

AN - SCOPUS:1842751420

VL - 13

SP - 784

EP - 796

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 5

ER -