Exponentially growing solutions and the cauchy problem

Abstract

We establish a representation formula for solutions to the Schrödinger equation in a domain in terms of the cauchy data on a part of ∂W The formula and the idea of the derivation are different front that of Carleman type even in the case when V(x) is a constant We make use of a family of special solutions to the equation

where D is a triangle, tetrahedron for n=2,3, respectively and is the characteristic function of D.

Keywords:

• Exponentially growing solution
• Cauchy problem
• stationary schrödinger equation
• representation of solutions of partial differential equations
• Formula of the Carleman type
• the Carleman function
• the Greem-Faddeev function
• inverse boundary value problem
• inverse scattering problem

AMS:

• 35A35
• 35C15
• 35505
• 35R05
• 35R25
• 35R30