Direct computation of the spectral function
HTML articles powered by AMS MathViewer
- by Amin Boumenir PDF
- Proc. Amer. Math. Soc. 123 (1995), 3431-3436 Request permission
Abstract:
We would like to find an explicit formula for the spectral function of the following Sturm-Liouville problem: \[ \left \{ {\begin {array}{*{20}{c}} {Lf \equiv - \frac {{{d^2}}}{{d{x^2}}}f(x) + q(x)f(x),\quad x \geq 0,} \hfill \\ {f’(0) - mf(0) = 0.} \hfill \\ \end {array} } \right .\] 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
- Amin Boumenir, A comparison theorem for selfadjoint operators, Proc. Amer. Math. Soc. 111 (1991), no. 1, 161–175. MR 1021896, DOI 10.1090/S0002-9939-1991-1021896-3
- Robert Wayne Carroll, Transmutation, scattering theory and special functions, Notas de Matemática [Mathematical Notes], vol. 87, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 677111
- I. M. Gel′fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. (2) 1 (1955), 253–304. MR 0073805, DOI 10.1090/trans2/001/11
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 3431-3436
- MSC: Primary 34B24; Secondary 34L05
- DOI: https://doi.org/10.1090/S0002-9939-1995-1283541-5
- MathSciNet review: 1283541