Applied Mathematics Seminar

Speaker: Jose Vega-Guzman, Howard University

Title: On the solution of some nonautonomous evolution equations<p>

Abstract: Solution methods for certain linear and nonlinear evolution equations will be presented. Emphasis will be placed mainly on the analytical treatment of nonautonomous differential equations, which are challenging to solve despite the existent numerical and symbolic computational software programs available. Ideas from the transformation theory are adopted allowing one to solve the problems under consideration from a non-traditional perspective. The Cauchy initial value problem is considered for a class of nonautonomous and inhomogeneous linear diffusion-type equation on the entire real line. Explicit transformations are used to reduce the equations under study to their corresponding standard forms emphasizing on natural relations with certain Riccati(and/or Ermakov)-type systems. This results will serve as a base to create a strong commutative relation, and to construct explicitly the minimum-uncertainty squeezed states for quantum harmonic oscillators. For the last, it is shown that the product of the variances attains the required minimum value only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the
variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied.