Asymptotics of the resonance number of the Schrödinger operator having linear and matricial potential
The author defines the resonances of the operator \(P= -\Delta\times I_r+ V(x)\) with \(r\times r\) Hermitian matrix potential via the complex dilation as eigenvalues of the operator \[ P_\theta= e^{-2i\theta} \Delta+ e^{i\theta} V(x),\quad \theta\in(0,\pi/3]. \] Assuming that the eigenvalues of the matrix \(V(x)\) do not vanish and are linearly growing of infinity, the author develops quasi-classical techniques for relevant operators \(P^h_\theta= -h^2e^{-2i\theta}\Delta+ e^{i\theta}V(x)\) and estimates the counting function for them as \[ N^h(r)= \{|\lambda(h)|\leq r\}\leq ch^{-h},\quad r\leq c, \] and then derives from the similarity of the operators \(P^h_\theta\) and \(\lambda P_\theta^{h\lambda^{-3/2}}\) that \(N^h\leq Cr^{3n/2}\).
scientific article; zbMATH DE number 1053511