Stokes multipliers for a class of ordinary differential equations

Abstract

A new method is presented for calculating the Stokes multipliers for a class of linear second-order ordinary differential equations. The Stokes multipliers allow the asymptotic solutions of these equations to be continued across the Stokes lines on which they are dominant. The differential equations, of the class considered here, have an irregular singular point at infinity and a singular point at the origin, which may be either regular or irregular. The Stokes multipliers, as functions of the coefficients in the differential equation, are obtained in the form of convergent infinite series, whose terms must be obtained from the solution of recursion relations, which are derived. In the case of Whittaker’s equation (when the origin is a regular singular point), the known results are obtained analytically. When the origin is an irregular singular point, numerical evaluation of the series is necessary, but the method seems to be quite efficient for use with digital computers. In the special case of an equation, with two irregular singular points, which can be transformed to Mathieu’s equation, the numerical results for the Stokes multiplier show good agreement with available known results for the characteristic exponents of Mathieu’s equation.

On the calculation of Stokes multipliers for linear differential equations of the second order

“…Other analytical methods also exist [5,18]. The novelty of our approach is that we employ inverse factorial series in place of the more usual inverse power series for the asymptotic expansions of as,i and a S) 2-The advantage of the inverse factorial expansions is that the coefficients are available explicitly.…”

Multi-channel scattering problems: an analytically solvable model

“…Then, physical quantities such as eigenvalues, scattering matrices can still be solved in an exact analytical form, providing that the connection problem of the asymptotic solution is known. The Stokes phenomenon [27] of the asymptotic solution of the ordinary differential equation provides a powerful tool to deal with these kinds of problems [28][29][30]. Generalizing the real variable to the complex variable and tracing the asymptotic solution around the complex plane, the connection matrix which connects the asymptotic solution in the complex plane can be expressed in terms of Stokes constants.…”

Curve crossing problem with Gaussian type coupling: analytically solvable model

“…Then, physical quantities such as eigenvalues, scattering matrices can still be solved in an exact analytical form, providing that the connection problem of the asymptotic solution is known. The Stokes phenomenon [27] of asymptotic solution of the ordinary differential equation provides a powerful tool to deal with these kinds of problems [28][29][30]. Generalizing the real variable to the complex variable and tracing the asymptotic solution around the complex plane, the connection matrix which connects the asymptotic solution in the complex plane can be expressed in terms of Stokes constants.…”