Geometry and analysis on isospectral sets. I: Riemannian geometry, asymptotic case (Q1911155)

The author considers the Dirichlet equation \[ -{y''} + q(x)y = \lambda y(x), \qquad y(0) = y(1) = 0, \quad q \in L^{2} \text{ over }[0,1], \] \(\lambda\) some complex number and \(x \in [0,1]\). To solve the inverse Dirichlet problem of the above equation, let \(M[q]\) be the iso-spectral set (\(M[q] := \{ p \in L^{2}: \mu (p) = \mu (q)\}\)), where \(\mu (p)\) is the spectral set of \(p\). In the present setting, a Riemannian manifold structure is given to \(M[e]\) in the asymptotic case. Metrics, connection, curvature and parallel transport are examined. A sufficient condition for a curve to be geodesic is given using the Banach fixed point theorem (Theorem 6).