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

TL;DR

This paper analyzes solutions to the Schrödinger equation with a time-dependent quadratic Hamiltonian, providing explicit formulas for the evolution operator and establishing well-posedness results for both linear and nonlinear cases.

Contribution

It derives explicit integral kernel formulas for the evolution operator and proves local and global well-posedness for the nonlinear Schrödinger equation with time-dependent quadratic Hamiltonians.

Findings

Explicit formula for the evolution operator kernel.

Conditions for local and global Strichartz estimates.

Well-posedness results for nonlinear cases, including damped harmonic Schrödinger equation.

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.