scholarly journals Generalized Separable Potential Theory and Bateman's Method on the Scattering Problem

1974 ◽  
Vol 52 (2) ◽  
pp. 550-566 ◽  
Author(s):  
S. Oryu
Keyword(s):  
Potential Theory ◽  
Scattering Problem ◽  
Separable Potential

NONLOCAL SEPARABLE SOLUTIONS OF THE INVERSE SCATTERING PROBLEM

1993 ◽  
Vol 08 (18) ◽  
pp. 3163-3184 ◽  
Author(s):  
TONY GHERGHETTA ◽  
YOICHIRO NAMBU
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Bound State ◽  
Separable Potential ◽  
Nonlocal Potential ◽  
S Wave ◽  
Weakly Bound ◽  
Square Well ◽  
Nonlocal Separable Potential

We extend the nonlocal separable potential solutions of Gourdin and Martin for the inverse scattering problem to the case where sin δ0 has more than N zeroes, δ0 being the s-wave scattering phase shift and δ0(0) − δ0(∞) = Nπ. As an example we construct the solution for the particular case of 4 He and show how to incorporate a weakly bound state. Using a local square well potential chosen to mimic the real 4 He potential, we compare the off-shell extension of the nonlocal potential solution with the exactly solvable square well. We then discuss how a nonlocal potential might be used to simplify the many-body problem of liquid 4 He .

Two discrete forms of the inverse scattering problem

1977 ◽  
Vol 55 (16) ◽  
pp. 1434-1441 ◽  
Author(s):  
M. Hron ◽  
M. Razavy
Keyword(s):  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Transcendental Equation ◽  
Separable Potential ◽  
Large Sphere ◽  
Approximate Form ◽  
The Matrix ◽  
Reaction Theory ◽  
Quantum Mechanical Systems ◽  
Discrete Spectra

The inverse problem for a class of quantum mechanical systems with discrete spectra is studied. First, for the case of a particle moving in a large sphere and interacting with a separable potential, it is shown that the potential can be found from the analytic properties of the eigenvalues, if they are given as the roots of a transcendental equation. The same technique is also applied for the construction of the form factor from the eigenfunctions of a two channel Wigner–Weisskopf model, when the system is enclosed in a large sphere. Then a completely different method of inversion is developed for the determination of the matrix elements of a tri–diagonal shell-model Hamiltonian. In this case the input data are the energy eigenvalues and the reduced widths. This method utilizes the continued.J-fraction expansion of a quantity which is analogous to the R-matrix of the nuclear reaction theory. The same technique is also used to investigate the exact inversion of the scattering problem, when the phase shifts are calculated by solving the finite-difference approximate form of the Schrödinger equation.

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