Gel'fand-Levitan theory of the inverse problem for singular potentials
The Gel'fand-Levitan theory with two potentials is generalized to the case where the first potential, assumed to be known, is singular and repulsive at the origin. It is shown that the only modifications required are redefinitions of the regular solution and the Jost function. Some of the proofs are only sketched here, and will be given in more detail in a separate paper.
KeywordsInverse Problem Regular Solution Singular Part Regular Case Singular Potential
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- 1).FADDEEV, L.D. (1963), J. Math. Phys. 4, 72–104Google Scholar
- 2).CHADAN, K., and SABATIER, P.C.(1977) Inverse Problems in Quantum Scattering Theory, Springer-Verlag.Google Scholar
- 3).BAETEMAN, M.L., and CHADAN, K. (1976), Nucl. Phys. A255, 34–49; (1977), Ann. Inst. Henri Poincaré, AXXIV 1–16.Google Scholar
- 4).PEARSON, D.B. (1975), Commun. Math. Phys. 40, 125–146.Google Scholar
- 5).SIMON, B. (1971): Quantum Mechanics for Hamiltonians defined as Quadratic Forms, Princeton University Press. See also the recent treatise by REED, M., and SIMON, B., Vol. II, III, IV.Google Scholar
- 6).AMREIN, W.O., JAUCH, J.M., and SINHA, K.B. (1978). Scattering Theory in Quantum Mechanics, W.A. Benjamin, Inc.Google Scholar
- 7).See reference 4 where a good list of papers treating scattering by singular potentials is given.Google Scholar
- 8).FRANK, W.M., LAND, D.J., and SPECTOR, R.M. (1971), Rev. Mod. Phys., 43, 36–98.Google Scholar
- 9).NEWTON, R.G. (1966): Scattering Theory of Waves and Particles, McGRAW-HILL.Google Scholar
- 10).TITCHMARSH, E.C. (1948): Theory of Fourier Integrals, pp. 172 ff.Google Scholar