The inverse spectral problem for the generalized second-order operator

Given a non-decreasing function Gamma ( lambda ) with growth Gamma ( lambda )*c lambda ^1-1( alpha +1/) as lambda to infinity were alpha >0, and a non-negative function omega (x)>or=0, we find a function h(x) so that Gamma ( lambda ) is the spectral function associated with the generalized second-order differential operator L(f)(x) identical to -1 d²/ omega (x)dx²f(x)+h(x)f(x) x>or=0. The results obtained generalize the well known Gelfand-Levitan result, corresponding to omega (x) identical to 1, i.e. alpha identical to 1.