Global Asymptotics for Functions of Parabolic Cylinder and Solutions of the Schrodinger Equation with a Potential in the Form of a Nonsmooth Double Well

In the paper, an approach is discussed that makes it possible to obtain global formulas in terms of Airy functions Ai and Bi of compound argument for the asymptotics of the functions of parabolic cylinder D-v(z) for real z and large.. The parabolic cylinder functions are determined from the Schrodinger equation, with potential in the form of a quadratic parabola, whose asymptotic solution can be constructed using the semiclassical approximation. In this case, the Bohr-Sommerfeld condition singles out the functions with an integer index whose asymptotics is determined only by the function Ai. For noninteger indices, the function Bi also contributes into the asymptotics. When choosing a potential in the form of a double well composed of two quadratic parabolas, an asymptotic solution can be constructed by gluing the asymptotic solutions for each of the wells separately. In this case, the condition of continuous differentiability of the resulting function gives the quantization condition for the levels of energy. In the vicinity of the top of the barrier (the transition level), this condition differs from the Bohr-Sommerfeld condition and, therefore, in the asymptotics, in addition to the function Ai, the function Bi also contributes. Similar expressions in the form of Airy functions and the quantization condition are also obtained for asymptotic solutions of the Schrodinger equation with a potential in the form of an arbitrary nonsmooth double well.