Resonances in the two-center Coulomb systems (Q2822030)

The Coulomb system is the oldest since 90 years ago Erwin Schrödinger derived his famous equation bearing his name. The present paper extends this study on two-center Coulomb systems in two dimensions which have their own history that goes back to the pioneering works of Euler, Jacobi, and Pauli, and mainly comes from molecular physics. A little in this area is known on the regularity of the solutions of the corresponding Schrödinger equation, especially on the problem of quantum resonances which are the main concern of this review. Its layout is the following: The two-center problem in both, classical and quantum formulations, is treated in Section 2. Section 3 describes the spectrum of the operator obtained from the angular differential equation and the properties of its analytical continuation. In the next Section, the authors deal with the spectrum of the operator obtained from the radial differential equation and with the analytic continuation of its resolvent, and summarize their main results in Theorems 4.5 and 4.14, and their corollaries. Section 5 explains how the resolvent of the two-center system relates to the angular and radial operators. In Section 6, the authors apply the aforesaid approach and define the resonances for the two-center problem, and analyze some their properties. This approximation is used in Section 7 to compute the resonances. This numerical approach strongly supports the relation between the resonances and the classical closed hyperbolic trajectories. Section 8 concludes with some additional comments that link the results for the two-center problem to the three-dimensional one and the \(n\)-center problem. Appendix A outlines the modification of the generalized Prüfer transformation in the semiclassical limit in order to obtain more precise estimates.