Free Schrodinger equation

The free Schrodinger equation

$i\partial _{t}u+\Delta u=0$

where u is a complex-valued function in ${\mathbb {R} }\times {\mathbb {R} }^{d}$, describes the evolution of a free non-relativistic quantum particle in d spatial dimensions. One can also consider the inhomogeneous (forced) Schrodinger equation

$i\partial _{t}u+\Delta u=F$

where F is a given forcing term. A general solution for this equation can be written down as

$u=u_{0}+\int dt'd^{d}x'{\frac {1}{[4\pi i(t-t')]^{d \over 2}}}e^{\frac {i|x-x'|^{2}}{4(t-t')}}F[u,x',t']$

being $u_{0}$ a solution of the homogeneous equation. This is generally the starting point for applying perturbation theory.

In the analysis of nonlinear Schrodinger equations it is of importance to obtain several linear and nonlinear estimates for the free and inhomogeneous Schrodinger equation.

Some linear perturbations of the free Schrodinger equation are also of interest in the nonlinear theory (in part because one can view nonlinear equations as linear equations in which certain coefficients themselves depend on the solution). For instance, one can add a potential term $\,Vu\!$ to the right-hand side, yielding the Schrodinger equation with potential. Or one replace the Laplacian $\Delta =\partial _{k}\partial _{k}$ with a covariant Laplacian $(\partial _{k}+iA_{k})(\partial _{k}+iA_{k})$, leading to the magnetic Schrodinger equation. Finally, one can replace the underlying spatial domain ${\mathbb {R} }^{d}$ with a Riemannian manifold $\,(M,g)\!$, and the Laplacian with the Laplace-Beltrami operator $\,\Delta _{g}\!$, yielding the Schrodinger equation on manifolds. One can also allow the manifolds to have boundaries (and assume appropriate boundary conditions), leading to the Schrodinger equation with obstacles.