Diffusion bound and reducibility for discrete Schrödinger equations with tangent potential
The lattice Schrödinger equations with a spatial localized real bounded (tangent) potential are considered. The diffusion for a class of such equations is studied in detail. This problem falls within the same general category of bounds on the higher Sobolev norms (\(H^1\) or beyond) for the continuum nonlinear Hamiltonian partial differential equation in a compact domain. It is proven that the growth of the \(H^1\) norm of the solution of the given lattice Schrödinger equations is at most logarithmic for some fixed initial data. Furthermore, it is supposed that the linear equation has a time quasi-periodic potential. Then it is proven that the linear equation can be reduced to an autonomous equation for the most values of the frequency. The Birkhoff normal form is used to study the dynamics of the respective Hamiltonian system described by the considered lattice Schrödinger equations.
scientific article; zbMATH DE number 6152752
scientific article