Department of Mathematics

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Browsing Department of Mathematics by Author "Aktosun, T."

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    Asymptotics of the scattering coefficients for a generalized Schrödinger equation
    Aktosun, T.; Klaus, Martin (AIP Publishing, 1999-08)
    The generalized Schrodinger equation d(2)psi/dx(2) + F(k)psi=[ikP(x) + Q(x)]psi is considered, where P and Q are integrable potentials with finite first moments and F satisfies certain conditions. The behavior of the scattering coefficients near zeros of F is analyzed. It is shown that in the so-called exceptional case, the values of the scattering coefficients at a zero of F may be affected by P(x). The location of the k-values in the complex plane where the exceptional case can occur is studied. Some examples are provided to illustrate the theory. (C) 1999 American Institute of Physics. [S0022-2488(99)03007-8].
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    Inverse scattering in 1-D nonhomogeneous media and recovery of the wave speed
    Aktosun, T.; Klaus, Martin; van der Mee, Cornelis (AIP Publishing, 1992-04)
    The inverse scattering problem for the 1-D Schrodinger equation d2-psi/dx2 + k2-psi = k2P(x)psi + Q(x)psi is studied. This equation is equivalent to the 1-D wave equation with speed 1/ square-root 1 - P(x) in a nonhomogeneous medium where Q(x) acts as a restoring force. When Q(x) is integrable with a finite first moment, P(x) < 1 and bounded below and satisfies two integrability conditions, P(x) is recovered uniquely when the scattering data and Q(x) are known. Some explicitly solved examples are provided.
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    Inverse wave scattering with discontinuous wave speed
    Aktosun, T.; Klaus, Martin; van der Mee, Cornelis (AIP Publishing, 1995-06)
    The inverse scattering problem on the line is studied for the generalized Schrödinger equation (d 2ψ/dx 2)+k 2 H(x)2ψ=Q(x)ψ, where H(x) is a positive, piecewise continuous function with positive limits H ± as x → ±∞. This equation, in the frequency domain, describes the wave propagation in a nonhomogeneous medium, where Q(x) is the restoring force and 1/H(x) is the variable wave speed changing abruptly at various interfaces. A related Riemann–Hilbert problem is formulated, and the associated singular integral equation is obtained and proved to be uniquely solvable. The solution of this integral equation leads to the recovery of H(x) in terms of the scattering data consisting of Q(x), a reflection coefficient, either of H ±, and the bound state energies and norming constants. Some explicitly solved examples are provided.
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    On the number of bound states for the one-dimensional Schrödinger equation
    Aktosun, T.; Klaus, Martin; van der Mee, Cornelis (AIP Publishing, 1998-09)
    The number of bound states of the one-dimensional Schrodinger equation is analyzed in terms of the number of bound states corresponding to ''fragments'' of the potential. When the potential is integrable and has a finite first moment, the sharp inequalities 1 -p + Sigma(j=1)(p) N(j)less than or equal to N less than or equal to Sigma(j=1)(p) N-j are proved, where p is the number of fragments, N is the total number of bound states, and N-j is the number of bound states for the jth fragment. When p=2 the question of whether N=N-1 +N-2 or N=N-1+N-2-1 is investigated in detail. An illustrative example is also provided. (C) 1998 American Institute of Physics.
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    On the Riemann–Hilbert problem for the one dimensional Schrödinger equation
    Aktosun, T.; Klaus, Martin; van der Mee, Cornelis (AIP Publishing, 1993-07)
    A matrix Riemann-Hilbert problem associated with the one-dimensional Schrodinger equation is considered, and the existence and uniqueness of its solutions are studied. The solution of this Riemann-Hilbert problem yields the solution of the inverse scattering problem for a larger class of potentials than the usual Faddeev class. Some examples of explicit solutions of the Riemann-Hilbert problem are given, and the connection with ambiguities in the inverse scattering problem is established.
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    Scattering and inverse scattering in one-dimensional nonhomogeneous media
    Aktosun, T.; Klaus, Martin; van der Mee, Cornelis (AIP Publishing, 1992-05)
    The wave propagation in a one-dimensional nonhomogeneous medium is considered, where the wave speed and the restoring force depend on location. In the frequency domain this is equivalent to the Schrodinger equation d2-psi/dx2 + k2-psi = k2P(x)psi + Q(x)psi with an added potential proportional to energy. The scattering and bound-state solutions of this equation are studied and the properties of the scattering matrix are obtained; the inverse scattering problem of recovering the restoring force when the wave speed and the scattering data are known are also solved.
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    Small-energy asymptotics of the scattering matrix for the matrix Schrodinger equation on the line
    Aktosun, T.; Klaus, Martin; van der Mee, Cornelis (AIP Publishing, 2001-10)
    The one-dimensional matrix Schrodinger equation is considered when the matrix potential is self-adjoint with entries that are integrable and have finite first moments. The small-energy asymptotics of the scattering coefficients are derived, and the continuity of the scattering coefficients at zero energy is established. When the entries of the potential have also finite second moments, some more detailed asymptotic expansions are presented. (C) 2001 American Institute of Physics.
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    Wave scattering in one dimension with absorption
    Aktosun, T.; Klaus, Martin; van der Mee, Cornelis (AIP Publishing, 1998-04)
    Wave scattering is analyzed in a one-dimensional nonconservative medium governed by the generalized Schrodinger equation d(2) psi/dx(2)+k(2) psi=[ikP(x)+Q(x)]psi, where P(x) and Q(x) are real, integrable potentials with finite first moments. Various properties of the scattering solutions are obtained. The corresponding scattering matrix is analyzed, and its small-k and large-k asymptotics are established. The bound states, which correspond to the poles of the transmission coefficient in the upper-half complex plane, are studied in detail. When the medium is not purely absorptive, i.e., unless P(x)less than or equal to 0, it is shown that there may be bound states at complex energies, degenerate bound states, and singularities of the transmission coefficient imbedded in the continuous spectrum. Some explicit examples are provided illustrating the theory. (C) 1998 American Institute of Physics. [S0022-2488(98)01503-5].