Inverse spectral problem

zetafunction

could someone give me some info about inverse spectral proble ??

my doubt is the following, let us suppose we know the spectrum or more precisely the value of the function/distribution

[tex] Z(u)= \sum _{n} Cos( u \lambda _n) [/tex] is known for u >0 , so i can calculate Z(u) exactly for every positive u

my next assumption is that this set of eigenvalues (lambda) belongs to the self-adjoint operator

[tex] -D^2y(x) + W(x)y(x) [/tex]

my problem is that i do not know what W(x) is , is ther using the inverse spectral theory a method to obtain a differential , integral or other kind of equation for W(x) ??

another possible question , let us suppose that [tex] -D^2y(x) + W(x)y(x) [/tex] is a 1-D Scrodinger equation associated to a scattering problem, then how could i recover W(x) ??? , if possible give me analytic (approximate) solutions please, thank you very much.