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Functional inversion for potentials in quantum mechanics
Richard L. Hall
abstract: Let E = F(v) be the ground-state eigenvalue of the Schroedinger Hamiltonian H
= -Delta + vf(x), where the potential shape f(x) is symmetric and monotone
increasing for x > 0, and the coupling parameter v is positive.
If the 'kinetic potential' bar{f}(s) associated with f(x) is defined by the
transformation: bar{f}(s) = F'(v), s = F(v)-vF'(v),then f can be reconstructed
from F by the sequence: f^{[n+1]} = bar{f} o bar{f}^{[n]^{-1}} o f^{[n]}.
Convergence is proved for special classes of potential shape; for other test
cases it is demonstrated numerically. The seed potential shape f^{[0]} need not
be 'close' to the limit f.
- oai_identifier:
- oai:arXiv.org:quant-ph/9912032
- categories:
- quant-ph math-ph math.MP
- comments:
- 14 pages, 2 figures
- doi:
- 10.1016/S0375-9601(99)00872-5
- arxiv_id:
- quant-ph/9912032
- journal_ref:
- Phys. Lett. A265, 28-34 (2000)
- report_no:
- CUQM-74
- created:
- 1999-12-07
Full article ▸
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