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A new approach to inverse spectral theory. I: Fundamental formalism. (English) Zbl 0945.34013

The inverse spectral problem of recovering the potential \(q(x)\) in the Schrödinger operator \(-(d^2/dx^2)\allowbreak+q\) on \((0,b)\), with Dirichlet, Neumann or mixed-type boundary condition at \(x=b\) if \(b\) is finite, from the Weyl \(m\)-function is solved by writing the \(m\)-function in the form \[ m(-\kappa^2,x)=-\kappa-\int_0^b A(\alpha,x)e^{-2\alpha\kappa} d\alpha +O(e^{-(2b-\varepsilon)\kappa}),\qquad\kappa\to+\infty, \] solving the integrodifferential equation \[ {{\partial A(\alpha,x)}\over{\partial x}}= {{\partial A(\alpha,x)}\over{\partial\alpha}} +\int_0^\alpha A(\beta,x)A(\alpha-\beta,x) d\beta, \] and putting \(q(x)=A(0^+,x)\). Smoothness properties of \(q\) are related to those of \(A\). Known asymptotic results for the Weyl \(m\)-function and Borg-type uniqueness results for the potential are rederived.

MSC:

34B20 Weyl theory and its generalizations for ordinary differential equations
34A55 Inverse problems involving ordinary differential equations
47E05 General theory of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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