On inverse problems for left-definite discrete Sturm-Liouville equations

Academic Article

Abstract

  • We establish an expansion theorem and investigate inverse spectral and inverse scattering problems for the discrete Sturm-Liouville problem -u″(n-1)+q(n)u(n) =?w(n)u(n) where q is nonnegative and w may change sign. If w is positive, the l2 -space weighted by w is a Hilbert space and it is customary to use that space for setting the problem. In the present situation the right-hand-side of the equation does not give rise to a positive-definite quadratic form and we use instead the left-hand side to define such a form and hence a Hilbert space (such problems are called left-definite). The difference equation gives rise to a linear relation which, upon proper restrictions, generates a self-adjoint operator. For this operator we define a Fourier transform and investigate the relationship between two operators with the same transform (the inverse spectral problem). If q - q0 and w - 1 are summable one may define the scattering process and we solve the inverse scattering problem. For coefficients decaying sufficiently fast to q0 and 1, respectively, the concept of a resonance is introduced as a generalization of the notion of an eigenvalue and the set of iso-resonant operators, i.e., operators having the same eigenvalues and resonances, is described.
  • Authors

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    Digital Object Identifier (doi)

    Author List

  • Alahmad R; Weikard R
  • Start Page

  • 35
  • End Page

  • 70
  • Volume

  • 7
  • Issue

  • 1