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Direct computation of the spectral function

Author: Amin Boumenir
Journal: Proc. Amer. Math. Soc. 123 (1995), 3431-3436
MSC: Primary 34B24; Secondary 34L05
MathSciNet review: 1283541
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Abstract: We would like to find an explicit formula for the spectral function of the following Sturm-Liouville problem:

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {Lf \equiv - \frac{{{d^2}}}{{d{x... ...\quad x \geq 0,} \hfill \\ {f'(0) - mf(0) = 0.} \hfill \\ \end{array} } \right.$

A simple operational calculus argument will help us obtain an explicit formula for the transmutation kernel. The expression of the spectral function is then obtained through the nonlinear integral equation found in the Gelfand-Levitan theory.

References [Enhancements On Off] (What's this?)

  • [1] A. Boumenir, Comparison theorem for self-adjoint operators, Proc. Amer. Math. Soc. 111 (1991), 161-175. MR 1021896 (91i:47035)
  • [2] R. W. Carroll, Transmutations, scattering theory and special functions, North-Holland, Math. Stud., vol. 69, North-Holland, Amsterdam and New York, 1982. MR 677111 (84b:35114)
  • [3] I. M. Gelfand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. Ser. 2 1 (1951), 239-253. MR 0073805 (17:489c)

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