Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity
The smoothing properties of time-dependent linear Schrödinger operators with potentials growing super-quadratically are studied. Also, Strichartz type inequalities for these operators are proved which, as usual, can be applied to nonlinear Schrödinger equations. If the potential grows like a power of order \(m\) in the spatial variable the corresponding solution of the initial value problem (with initial data from state space) gains derivatives of order \(1/m\), and \(L(p,q)\)-norms of solutions can be estimated by derivatives of the initial data up to a specific order (Strichartz estimates). The proofs use a variety of methods, e.g. estimates of fundamental solutions of the corresponding equations, Calderon-Vaillancourt inequalities, scaled Hamiltonians, techniques of semi-classical analysis and pseudo-differential calculus, and results of Keel-Tao.
time-dependent linear Schrödinger operators
scientific article; zbMATH DE number 2092985
scientific article