|
The second-order ordinary differential equation (sODE) has been involved in a lot of branches of theoretical physics, and the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) method is one of the most powerful approximation to this type of equation with great success. However, the validity of the JWKB method has to be restricted to the region where the JWKB condition (or adiabatic condition) is fulfilled. Recently, we developed a powerful and effective method, the uniform asymptotic approximation, to accurately construct analytical solutions of sODE. The most remarkable feature of this method is that it provides a systematic and error controlled treatment to regions where the adiabatic condition is violated, and goes over to the JWKB approximations when the adiabatic condition is restored. In this talk we are going to provide a brief introduction about the uniform asymptotic approximation and its recent applications to study the non-adiabatic effects on primordial perturbations. Applications to several other problems in quantum mechanics, black hole physics, and cosmology have also been discussed.
|
