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

Barry Simon

Review from Zentralblatt MATH:

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 $${{

tial A(\alpha,x)}\over{

tial x}}= {{

tial A(\alpha,x)}\over{

tial\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.

Reviewed by Cornelis van der Mee

Keywords: inverse spectral problem; Weyl $m$-function

Classification (MSC2000): 34B20 34A55 47E05 34L40

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