Department of Mathematics

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Browsing Department of Mathematics by Author "Aktosun, Tuncay"

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    Factorization of scattering matrices due to partitioning of potentials in one-dimensional Schrödinger-type equations
    Aktosun, Tuncay; Klaus, Martin; van der Mee, Cornelis (AIP Publishing, 1996-12)
    The one-dimensional Schrodinger equation and two of its generalizations are considered, as they arise in quantum mechanics, wave propagation in a nonhomogeneous medium, and wave propagation in a nonconservative medium where energy may be absorbed or generated. Generically, the zero-energy transmission coefficient vanishes when the potential is nontrivial, but in the exceptional case this coefficient is nonzero, resulting in tunneling through the potential. It is shown that any nontrivial exceptional potential can always be fragmented into two generic pieces. Furthermore, any nontrivial potential, generic or exceptional, can be fragmented into generic pieces in infinitely many ways. The results remain valid when Dirac delta functions are included in the potential and other coefficients are added to the Schrodinger equation. For such Schrodinger equations, factorization formulas are obtained that relate the scattering matrices of the fragments to the scattering matrix of the full problem. (C) 1996 American Institute of Physics.
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    Integral equation methods for the inverse problem with discontinuous wave speed
    Aktosun, Tuncay; Klaus, Martin; van der Mee, Cornelis (AIP Publishing, 1996-07)
    The recovery of the coefficient H(x) in the one-dimensional generalized Schrodinger equation d(2) psi dx(2)+k(2)H(x)(2) psi=Q(x)psi, where H(x) is a positive, piecewise continuous function with positive limits H-+/- as x-->+(+/-infinity), is studied. The large-k asymptotics of the wave functions and the scattering coefficients are analyzed. A factorization formula is given expressing the total scattering matrix as a product of simpler scattering matrices. Using this factorization an algorithm is presented to obtain the discontinuities in H(x) and H'(x)/H(x) in terms of the large-k asymptotics of the reflection coefficient. When there are no bound states, it is shown that H(x) is recovered from an appropriate set of scattering data by using the solution of a singular integral equation, and the unique solvability of this integral equation is established. An equivalent Marchenko integral equation is derived and is shown to be uniquely solvable; the unique recovery of H(x) from the solution of this Marchenko equation is presented. Some explicit examples are given, illustrating the recovery of H(x) from the solution of the singular integral equation and from that of the Marchenko equation. (C) 1996 American Institute of Physics.
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    Small-energy analysis for the self-adjoint matrix Schrodinger operator on the half line
    Aktosun, Tuncay; Klaus, Martin; Weder, Ricardo (AIP Publishing, 2011-10)
    The matrix Schrodinger equation with a self-adjoint matrix potential is considered on the half line with the most general self-adjoint boundary condition at the origin. When the matrix potential is integrable and has a first moment, it is shown that the corresponding scattering matrix is continuous at zero energy. An explicit formula is provided for the scattering matrix at zero energy. The small-energy asymptotics are established also for the related Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3640029]