Rate of convergence for periodic homogenization of convex Hamilton–Jacobi equations in one dimension

Let u ε and u be viscosity solutions of the oscillatory Hamilton–Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence O ( ε ) of u ε → u as ε → 0 + for a large class of convex Hamiltonians H ( x , y , p ) in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension n = 1.

Calculation of Resonance S-Matrix Poles by Means of Analytic Continuation in the Coupling Constant

The method of analytic continuation in the coupling constant in combination with the use of statistical Padé approximation is applied to the determination of complex S-matrix poles, i.e. to the determination of resonance energy and widths. These parameters are of vital importance in many physical, chemical and biological processes. It is shown that an alternative to the method of analytic continuation in the coupling constant exists which in principle makes it possible to locate several resonances at once, in contrast to the original method which yields parameters of only one resonance. In addition the new approach appears to be less sensitive to the choice of the perturbation interaction used for the analytical continuation than the original method. In this paper both approaches are compared and tested for model analytic separable potential. It is shown that the new variant of the method of analytic continuation in the coupling constant is more robust and efficient than the original method and yields reasonable results even for data of limited accuracy.

Exact treatment of the relativistic double ring-shaped Kratzer potential using the quantum Hamilton–Jacobi formalism

We study the Klein–Gordon equation with noncentral and separable potential under the condition of equal scalar and vector potentials and we obtain the corresponding relativistic quantum Hamilton–Jacobi equation. The application of the quantum Hamilton–Jacobi formalism to the double ring-shaped Kratzer potential leads to its relativistic energy spectrum as well as the corresponding eigenfunctions.