Browsing by Author "van der Mee, Cornelis"

Now showing 1 - 10 of 10
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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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    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 operators for the matrix Zakharov-Shabat system
    Klaus, Martin; van der Mee, Cornelis (AIP Publishing, 2010-05)
    In this article, we prove the similarity (and, in the focusing case, the J-unitary equivalence) of the free Hamiltonian and the restriction of the full Hamiltonian to the maximal invariant subspace on which its spectrum is real for the matrix Zakharov-Shabat system under suitable conditions on the potentials. This restriction of the full Hamiltonian is shown to be a scalar-type spectral operator whose resolution of the identity is evaluated. In the focusing case, the restricted full Hamiltonian is an absolutely continuous, J-self-adjoint non-J-definitizable, operator allowing a spectral theorem without singular critical points. To illustrate the results, two examples are provided. (C) 2010 American Institute of Physics. [doi:10.1063/1.3377048]
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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].