Creating desired potentials by embedding small inhomogeneities

Journal Article · Tue Dec 15 04:00:00 EST 2009 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3267887· OSTI ID:21294531

Ramm, A G ^[1]

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1. Department of Mathematics, Kansas State University, Manhattan, Kansas 66506-2602 (United States)

The governing equation is [{nabla}{sup 2}+k{sup 2}-q(x)]u=0 in R{sup 3}. It is shown that any desired potential q(x), vanishing outside a bounded domain D, bounded in D, Riemann integrable, can be obtained if one embeds into D many small scatterers q{sub m}(x), vanishing outside balls B{sub m}:={l_brace}x:|x-x{sub m}|<a{r_brace}, such that q{sub m}=A{sub m} in B{sub m}, q{sub m}=0 outside B{sub m}, 1{<=}m{<=}M, M=M(a). It is proven that if the number of small scatterers in any subdomain {delta} is defined as N({delta}):={sigma}{sub x{sub m}}{sub isanelementof{delta}}1 and is given by the formula N({delta})=|V(a)|{sup -1}{delta}n(x)dx[1+o(1)] as a{yields}0, where V(a)=4{pi}a{sup 3}/3, then the limit of the function u{sub M}(x), lim{sub a{yields}}{sub 0} u{sub M}=u{sub e}(x), does exist and solves the equation [{nabla}{sup 2}+k{sup 2}-q(x)]u=0 in R{sup 3}, where q(x)=n(x)A(x), A(x{sub m})=A{sub m}, and u{sub M}(x) is a solution to the equation [{nabla}{sup 2}+k{sup 2}-p(x)]u=0, where p(x):=p{sub M}(x) is some piecewise-constant potential. The total number M of small inhomogeneities is equal to N(D) and is of the order O(a{sup -3}) as a{yields}0. A similar result is derived in the one-dimensional case.

• OSTI ID: 21294531
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 12 Vol. 50; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
FUNCTIONS
INTEGRAL CALCULUS
MATHEMATICAL SOLUTIONS
ONE-DIMENSIONAL CALCULATIONS
S MATRIX
SCHROEDINGER EQUATION