Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 16, pp. 1-19.
Title: Inverse spectral analysis for singular
differential operators with matrix coefficients
Authors: Nour el Houda Mahmoud (Faculte des Sciences de Tunis, Tunisia)
Imen Yaich (Faculte des Sciences de Tunis, Tunisia)
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.