Spectral analysis of a two body problem with zero-range perturbation (Q935064)

The authors consider a quantum system in \(\mathbb R^d\) (\(d=1,2,3\)) composed of a test particle and a harmonic oscillator interacting through a zero-range force concentrated on a hyperplane. The Hamiltonian is formally written as \(H_\alpha^\omega =H_0^\omega +\alpha \delta (\overarrow{x}- \overarrow{y})\), \(H_0^\omega =-\frac{1}2\Delta_{\overarrow{x}} -\frac{1}2\Delta_{\overarrow{y}}+\frac{\omega^2y^2}2 -\frac{\omega d}2\). An accurate introduction of \(H_\alpha^\omega\) as a selfadjoint operator on \(L_2(\mathbb R^{2d})\) is achieved by the method of quadratic forms. The authors give (separately for each dimension \(d\)) a detailed description of point spectum, its asymptotic behavior with respect to the parameters \(\omega\) and \(\alpha\). Some results regarding the positive spectrum and the scattering properties of the Hamiltonian are also given.