| 講演要旨： |
Asymptotic expansions have been used for over 200 years. Stirling's approximation for n! is a prime example. In physics, there is a large class of problems that can be solved by an asymptotic expansion technique called the JWKB method. It was invented in 1817 by the Italian astronomer Carlini to study planetary orbits, and Carlini obtained what we would now call an asymptotic expansion for the Bessel function Jp(px) for large p. The Carlini method was rediscovered for quantum mechanics by Jeffreys, Wentzel, Kramers, and Brillouin, for whom the JWKB method is named, and has been used for over 80 years. Its unique appeal is an explicit, tractable procedure to generate the wave function as a series in powers of ￣h. Its disadvantages are its divergence, its singularities at classical turning points with consequent “Stokes lines” and “connection formulas,” and long-standing confusion about connection-formula “directionality.” Practical calculations typically have used low-order partial sums with error estimates to justify results. Related to the JWKB expansion are a number of perturbation theory problems, for instance, the hydrogen atom in an electric field, in which there is a small physical parameter. Expansions of the wave function and energy with respect to this parameter are asymptotic, not convergent. For 20 years it has been known that the JWKB expansion is Borel summable to the exact function the series represents, leading to the exact JWKB method. Although there have been many mathematical applications, there have been few practical physical calculations using the exact JWKB method to go beyond lowest orders. This lecture will focus on the JWKB method and its Borel summability, and then illustrate it by applying it to a prototypical problem involving tunneling. Both practical and theoretical complications are encountered and overcome. These involve coalescing Stokes lines, numerical cancellations, and how to take approximate Borel sums. For the inverted parabola potential, the Borel sum of the JWKB series for the solution of the Schr¨odinger equation is proportional to the Weber function. The proportionality constant in the JWKB theory is an infinite series, conjectured by Sato to sum to a particular gamma function. A short proof of Sato’s formula will be given. |
