Inverse spectral analysis for singular differential operators with matrix coefficients

• Main Authors: Nour el Houda Mahmoud, Imen Yaich
• Format: Article
• Language: English
• Published: Texas State University 2006-02-01
• Series: Electronic Journal of Differential Equations
• Subjects: Inverse problem; Fourier-Bessel transform; spectral measure; Hilbert-Schmidt operator; Fredholm's equation.
• Online Access: http://ejde.math.txstate.edu/Volumes/2006/16/abstr.html

Let $L_alpha$ be the Bessel operator with matrix coefficients defined on $(0,infty)$ by $$ L_alpha U(t) = U''(t)+ {I/4-alpha^2over t^2}U(t), $$ 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.