Application of complex canonical point transformations to linear second-order differential equations

Abstract

We present a canonical method to solve one-dimensional linear differential equations making use of pseudodifferential calculus. We apply two successive canonical point transformations on the cartesian momentum and position spaces to obtain a nonlinear complex-valued canonical transformation which maps a very simple linear differential equation into the desired differential equation. This method yields a closed contour integral representation for the exact solution in terms of arbitrary functions, which, may be determined from the mapping equations in a similar way to that followed in classical mechanics. This method does not require the completeness condition on the intermediary states and avoids calculation of the kernel of the generator. We explicitly develop the case of second-order differential equations and give some standard examples to show how this method works.