Spectral function for relatively Hilbert-Schmidt perturbations (Q2732240)

The author considers the pair of unbounded selfadjoint operators \(H_0=-\Delta, H=-\Delta_g+V\) acting on \(\mathbb Z^2(\mathbb R^d), d\geq 2,\) where \(\Delta\) is the usual Laplacian \(\Delta_g\) is the Laplace-Beltrami operator associated to the metric \(g(x)=(g^{jk}(x))_{1\leq j,k\leq d}\) and \(V\) is a potential such that the coefficients of \(H-H_0\) decay as \(| x|^{-\rho}\) at infinity, \(\rho>\frac{d}{2}\). Under a non-trapping assumption he gives the existence of a high energy asymptotic expansion and deduces a Levinson formula in the special case \(d=3\) under a weaker condition than the usual one \(\rho>3\).