The inverse problem in the case of bound states

Yekken, Rabia; Ighezou, F.-Z.; Lombard, R.J.

doi:10.1016/j.aop.2007.09.005

Cited by 12 publications

(24 citation statements)

References 15 publications

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“…These starting points are particularly convenient to answer the question of the existence of the local equivalent potential, or at least to show different types of situations. For this purpose, use is made of the method for the inverse problem in the case of discrete states developed in a preceding paper [20]. In particular, conditions for a unique answer have been given in [20,21].…”

Section: The Equivalent Local Potentialmentioning

confidence: 99%

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Energy-dependent potentials and the problem of the equivalent local potential

Yekken¹,

Lombard²

2010

J. Phys. A: Math. Theor.

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The properties of the wave equation are studied in the case of energy-dependent potentials for bound sates. The nonlinearity induced by the energy dependence requires modification of the standard rules of quantum mechanics. These modifications are briefly recalled. Analytical and numerical solutions are given in the three-dimensional space for power-law radial shape potentials with a linear energy dependence. This last is chosen since it allows the construction of a coherent theory. Among the results, we stress the saturation of the spectrum observed for confining potentials: as the quantum numbers increase, the eigenvalues reach an upper limit. Finally, the problem of the equivalent local potential is discussed. The existence of analytical solutions presents a good opportunity to tackle this problem in detail.

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Section: The Equivalent Local Potentialmentioning

confidence: 99%

“…For this purpose, use is made of the method for the inverse problem in the case of discrete states developed in a preceding paper [20]. In particular, conditions for a unique answer have been given in [20,21].…”

Section: The Equivalent Local Potentialmentioning

confidence: 99%

Energy-dependent potentials and the problem of the equivalent local potential

Yekken¹,

Lombard²

2010

J. Phys. A: Math. Theor.

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“…The problem of reconstructing potentials from bound states has been studied numerically in Ref. [6]. In [7], Rudyak and Zakhariev have constructed potentials from E-and ℓ-dependent data, however assuming that aE + bℓ(ℓ + 1) is a constant.…”

Section: Introductionmentioning

confidence: 99%

“…They conjectured that the knowledge of the ground-state energies E (0) , for all non-negative integers , allows one to recover the potential in a unique way. The problem of reconstructing potentials from bound states has been studied numerically in [6]. In [7], Rudyak and Zakhariev have constructed potentials from E-and -dependent data, however assuming that aE + b ( + 1) is a constant.…”

Section: Introductionmentioning

confidence: 99%

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Construction of potentials using mixed scattering data

et al. 2008

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The long-standing problem of constructing a potential from mixed scattering data is discussed. We first consider the fixed-ℓ inverse scattering problem. We show that the zeros of the regular solution of the Schrödinger equation, r n (E) which are monotonic functions of the energy, determine a unique potential when the domain of energy is such that the r n (E)'s range from zero to infinity. The latter method is applied to the domain {E ≥ E 0 , ℓ = ℓ 0 } ∪ {E = E 0 , ℓ ≥ ℓ 0 } for which the zeros of the regular solution are monotonic in both parts of the domain and still range from zero to infinity. Our analysis suggests that a unique potential can be obtained from the mixed scattering data {δ(ℓ 0 , k), k ≥ k 0 } ∪ {δ(ℓ, k 0 ), ℓ ≥ ℓ 0 } provided that certain integrability conditions required for the fixed ℓ-problem, are fulfilled. The uniqueness is demonstrated using the JWKB approximation. *

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The Heisenberg Model of the He Atom Revisited

et al. 2011

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The present work is devoted to a study of the Heisenberg model of the He atom. We consider the subset of the singlet states, in which one of the electron is kept on its 1s orbital. This spectrum is treated as a pseudo two-body problem. Use is made of a method designed to solve the inverse problem from bound states to determine the local equivalent potential seen by the second electron. The problem is simplified by the use of parametric expressions instead of a full numerical determination of the local equivalent potential. It keeps the numerical effort to a reasonable level. By construction, the local equivalent potential should fit the experimental eigenvalues, which is well achieved with the proposed simplified expressions. It shows the validity of the Heisenberg model for the excited states. It yields also a value for the mean square radius of the He atom. This one is in good qualitative agreement with the value derived from large basis calculations of the He ground state

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The inverse problem in the case of bound states

Cited by 12 publications

References 15 publications

Energy-dependent potentials and the problem of the equivalent local potential

Energy-dependent potentials and the problem of the equivalent local potential

Construction of potentials using mixed scattering data

The Heisenberg Model of the He Atom Revisited

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