A new approach to inverse spectral theory. I. Fundamental formalism.

We present a new approach (distinct from Gel'fand-Levitan) to the theorem o f Borg-Marchenko that the m-function (equivalently, spectral measure) for a finite interval or half-line Schrodinger operator determines the potential . Our approach is an analog of the continued fraction approach for the mome nt problem. We prove there is a representation for the m-function m(-kappa( 2)) = -kappa - integral(0)(b) A(alpha)e(-2 alpha kappa) d alpha + O(e(-(2b- epsilon)kappa)). A on [0,a] is a function of q on [0, a] and vice-versa. A key role is played by a differential equation that A Obeys after allowing x -dependence: delta A/delta x = delta A/delta alpha + integral(0)(alpha) A(beta, x)A(alph a - beta, x) d beta. Among our new results are necessary and sufficient conditions on the m-func tions for potentials q(1) and q(2) for q(1) to equal q(2) on [0, a].