Inverse spectral analysis for singular differential operators with matrix coefficients
Mahmoud, Nour el Houda
Texas State University-San Marcos, Department of Mathematics
Let Lα be the Bessel operator with matrix coefficients defined on (0, ∞) by LαU(t) = U″ (t) + I/4 - α2 / t2 U(t), where α 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 Lα + 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.
Inverse problem, Fourier-Bessel transform, Spectral measure, Hilbert-Schmidt operator, Fredholm's equation
Mahoud, N. H., & Yaïch, I. (2006). Inverse spectral analysis for singular differential operators with matrix coefficients. <i>Electronic Journal of Differential Equations, 2006</i>(16), pp. 1-19.