The Schrödinger equation with spatial white noise: The average wave function

The Schrödinger equation with spatial white noise: the average wave function

We prove a representation for the average wave function of the Schrödinger equation with a white noise potential in d = 1, 2, in terms of the renormalized self-intersection local time of a Brownian motion.

The Schrödinger equation with spatial white noise potential

We consider the linear and nonlinear Schrödinger equation with a spatial white noise as a potential in dimension 2. We prove existence and uniqueness of solutions thanks to a change of unknown originally used in [8] and conserved quantities. 2010 Mathematics Subject Classification AMS:

Scaled Schrödinger equation and the exact wave function.

We propose the scaled Schrödinger equation and the related principles, and construct a general method of calculating the exact wave functions of atoms and molecules in analytical forms. The nuclear and electron singularity problems no longer occur. Test applications to hydrogen atom, helium atom, and hydrogen molecule are satisfactory, implying a high potentiality of the proposed method.

Collapse of solitary excitations in the nonlinear Schrödinger equation with nonlinear damping and white noise.

Effective noise theory for the nonlinear Schrödinger equation with disorder.

For the nonlinear Shrödinger equation with disorder it was found numerically that in some regime of the parameters Anderson localization is destroyed and subdiffusion takes place for a long time interval. It was argued that the nonlinear term acts as random noise. In the present work, the properties of this effective noise are studied numerically. Some assumptions made in earlier work were veri...