# The inverse*s*-wave scattering problem for a class of potentials depending on energy

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## Abstract

The inverse scattering problem is considered for the radial*s*-wave Schrödinger equation with the energy-dependent potential*V*^{+}(*E*,*x*)=*U*(*x*)+2\(\sqrt E \)*Q*(*x*). (Note that this problem is closely related to the inverse problem for the radial*s*-wave Klein-Gordon equation of zero mass with a static potential.) Some authors have already studied it by extending the method given by Gel'fand and Levitan in the case*Q*=0. Here, a more direct approach generalizing the Marchenko method is used. First, the Jost solution*f*^{+}(*E*,*x*) is shown to be generated by two functions*F*^{+}(*x*) and*A*^{+}(*x*,*t*). After introducing the potential*V*^{−}(*E*,*x*)=*U*(*x*)−2\(\sqrt E \)*Q*(*x*) and the corresponding functions*F*^{−}(*x*) and*A*^{−}(*x*,*t*), fundamental integral equations are derived connecting*F*^{+}(*x*),*F*^{−}(*x*),*A*^{+}(*x*,*t*) and*A*^{−}(*x*,*t*) with two functions*z*^{+}(*x*) and*z*^{−}(*x*);*z*^{+}(*x*) and*z*^{−}(*x*) are themselves easily connected with the binding energies*E* _{ n } ^{ + } and the scattering “matrix”*S*^{+}(*E*),*E*>0 (the input data of the inverse problem). The inverse problem is then reduced to the solution of these fundamental integral equations. Some specific examples are given. Derivation of more elaborate results in the case of real potentials, and applications of this work to other inverse problems in physics will be the object of further studies.

## Keywords

Neural Network Statistical Physic Integral Equation Input Data Complex System## Preview

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## References

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