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SUMMARY:Successes and Failures of Numerical Solutions to the 1-Dimentional
Marchenko Integral Equation for Quantum Inverse Scattering
DTSTART;VALUE=DATE-TIME:20110713T091500Z
DTEND;VALUE=DATE-TIME:20110713T093000Z
DTSTAMP;VALUE=DATE-TIME:20240620T093419Z
UID:indico-contribution-4911@events.saip.org.za
DESCRIPTION:Speakers: Serle Shuman (UNISA)\nThe origin of numerical errors
\, which in certain cases lead\nto complete failure of the direct method f
or solving the 1-dimensional Marchenko\nintegral equation\, are investigat
ed. Bargmann and block potentials\, for which exact analytical\nexpression
s exist\, are used to compare the accuracy of numerical results. In\nparti
cular\, numerical results from the Nystrom method are compared to those of
two\nnovel collocation methods. The results indicate that\, for roughly t
he same\ncomputational effort\, the three method may be ranked in order of
increasing accuracy: \nNystrom method\, equidistant collocation method an
d non-equidistant collocation method. \nIn all three cases the origin of t
he failure\, which occurs for barrier-like potentials of relatively \nlarg
e width and height\, may be attributed to the numerical instability introd
uced in both the Fourier \ninversion and the linearization steps. Subseque
ntly these errors produce perturbations on \nthe matrix elements of the in
version matrix\, a matrix which is found to become increasingly \nill-cond
itioned as the height and width of the potential increases. At the current
limits of\nsuccessful inversion\, we attribute the "early" onset of inver
sion failures\, to the\naforementioned matrix element perturbations. Howev
er at still higher potential height\nand width\, the inherent ill-conditio
ning of the inversion matrix alone\, accounts for\nthe inversion failures.
We therefore conclude that there is an inherent upper-bound on the\nappli
cation of the direct inversion method for inverting relatively high and wi
de\npotentials.\n\nhttps://events.saip.org.za/event/7/contributions/4911/
LOCATION: Acro5
URL:https://events.saip.org.za/event/7/contributions/4911/
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