Free Schrodinger equation: Difference between revisions

From DispersiveWiki
Jump to navigationJump to search
mNo edit summary
mNo edit summary
Line 13: Line 13:
In the analysis of [[Schrodinger equations|nonlinear Schrodinger equations]] it is of importance to obtain several [[Schrodinger estimates|linear and nonlinear estimates]] for the free and inhomogeneous Schrodinger equation.
In the analysis of [[Schrodinger equations|nonlinear Schrodinger equations]] it is of importance to obtain several [[Schrodinger estimates|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 <math>Vu</math> to the right-hand side, yielding the [[Schrodinger equation with potential]].  Or one replace the Laplacian <math>\Delta = \partial_k \partial_k</math> with a covariant Laplacian <math>(\partial_k + i A_k)(\partial_k + i A_k)</math>, leading to the [[magnetic Schrodinger equation]].  Finally, one can replace the underlying spatial domain <math>R^d</math> with a Riemannian manifold <math>(M,g)</math>, and the Laplacian with the Laplace-Beltrami operator <math>\Delta_g</math>, 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]].
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 <math>Vu</math> to the right-hand side, yielding the [[Schrodinger equation with potential]].  Or one replace the Laplacian <math>\Delta = \partial_k \partial_k</math> with a covariant Laplacian <math>(\partial_k + i A_k)(\partial_k + i A_k)</math>, leading to the [[magnetic Schrodinger equation]].  Finally, one can replace the underlying spatial domain <math>{\mathbb R}^d</math> with a Riemannian manifold <math>\, (M,g)\!</math>, and the Laplacian with the Laplace-Beltrami operator <math>\, \Delta_g\!</math>, 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]].


[[Category:Equations]]
[[Category:Equations]]
[[Category:Schrodinger]]
[[Category:Schrodinger]]

Revision as of 06:16, 15 June 2009

The free Schrodinger equation

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

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

being 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 to the right-hand side, yielding the Schrodinger equation with potential. Or one replace the Laplacian with a covariant Laplacian , leading to the magnetic Schrodinger equation. Finally, one can replace the underlying spatial domain with a Riemannian manifold , and the Laplacian with the Laplace-Beltrami operator , 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.