Electron. J. Diff. Eqns., Vol. 2006(2006), No. 16, pp. 1-19.

Inverse spectral analysis for singular differential operators with matrix coefficients

Nour el Houda Mahmoud, Imen Yaich

Abstract:
Let $L_\alpha$ be the Bessel operator with matrix coefficients defined on $(0,\infty)$ by
$$ L_\alpha U(t) = U''(t)+ {I/4-\alpha^2\over t^2}U(t), $$
where $\alpha$ is a fixed diagonal matrix. The aim of this study, is to determine, on the positive half axis, a singular second-order differential operator of $L_\alpha+Q$ kind and its various properties from only its spectral characteristics. Here $Q$ is a matrix-valued function. Under suitable circumstances, the solution is constructed by means of the spectral function, with the help of the Gelfund-Levitan process. The hypothesis on the spectral function are inspired on the results of some direct problems. Also the resolution of Fredholm's equations and properties of Fourier-Bessel transforms are used here.

Submitted October 14, 2005. Published February 2, 2006.
Math Subject Classifications: 45Q05, 45B05, 45F15, 34A55, 35P99.
Key Words: Inverse problem; Fourier-Bessel transform; spectral measure; Hilbert-Schmidt operator; Fredholm's equation.

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Nour el Houda Mahmoud
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire, 1060 Tunis, Tunisia
email: houda.mahmoud@fst.rnu.tn
Imen Yaïch
Département de Mathématiques
Faculté des Sciences de Tunis
Campus Universitaire, 1060 Tunis, Tunisia
email: imen.maalej@fst.rnu.tn

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