Asymptotic analysis for Schrödinger Hamiltonians via Birkhoff-Gustavson normal form (Q2864764)

The paper deals with the computation of the Birkhoff-Gustavson normal form (the Hamiltonian version of Poincaré-Dulac normal forms) and its application to the calculation of quantum mechanical spectra for the Schroedinger operator.NEWLINENEWLINEThe theory of normal forms is well developed, and the computation of the Birkhoff-Gustavson normal form (BGNF) is -- at least for problems arising from motion in a smooth potential -- rather standard. However, at the time of considering quantum mechanical versions of the approach, one is faced with a substantial problem: the very essence of the BGNF lies in performing a suitable change of coordinates, so that quantizing a BGNF calls into play the issue of how to quantize a classical problem with coordinates (by this one means coordinates in phase space, i.e., space coordinates and momenta) are not standard; see, e.g. [\textit{M. K. Ali}, J. Math. Phys. 26, 2565--2572 (1985)] and [\textit{B. Eckhardt}, J. Phys. A 19, 2961--2972 (1986; Zbl 0622.70015)].NEWLINENEWLINEDespite these problems, the BGNF provides very accurate information in computing quantum spectra; see, e.g. [\textit{R. T. Swimm} and \textit{J. B. Delos}, J. Chem. Phys. 71, 1706--1717 (1979); \textit{D. Robert} and \textit{M. Joyeux}, J. Chem. Phys. 119, 8761--8772 (2003); \textit{M. Joyeux}, J. Chem. Phys. 109, 337--408 (1998)].NEWLINENEWLINEIn the present paper, the authors exhibit explicit computations of the BGNF in some simple resonant situations like those which can be met in physical situations, e.g., studying small molecules. In particular, they analyze the 1:1, 1:2 and 1:3 resonances (without considering concrete physical applications). The authors also provide a computer code (source files being available through their website) which can be used to implement a quantum BGNF computation following their approach.NEWLINENEWLINEIt should be noted that, in contrast to the most usual approach (where quantization is applied after obtaining the BGNF), here the normalization and quantization step are performed sequentially at each order in perturbation.