On Schroedinger Equation with Time-Dependent Quadratic Hamiltonian in $R^d$

Abstract

We study solutions to the Cauchy problem for the linear and nonlinear Schroedinger equation with a quadratic Hamiltonian depending on time. For the linear case the evolution operator can be expressed as an integral operator with the explicit formula for the kernel. As a consequence, conditions for local and global in time Strichartz estimates can be established. For the nonlinear case we show local well-posedness. As a particular case we obtain well-posedness for the damped harmonic nonlinear Schroedinger equation.

16 pages

References in corpus (5)

• Quantum Integrals of Motion for Variable Quadratic Hamiltonians
• The time-dependent Schroedinger equation, Riccati equation and Airy functions
• The Riccati Differential Equation and a Diffusion-Type Equation
• An Integral Form of the Nonlinear Schroedinger Equation with Variable Coefficients
• Hydrodynamic approach to constructing solutions of Hydrodynamic approach to constructing solutions of nonlinear Schrödinger equation in the critical case