Stokes graphs of the Rabi problem with real parameters (Q6545363)

The authors consider the Rabi model of a reaction of atoms to the harmonic electric field with a frequency close to the natural one of the atoms. This leads to considering a second order differential equation\N\[\Nd^2y/dz^2+Q(\Delta ,E,g,z)y=0,\N\]\Nwhere \(Q(\Delta ,E,g,z)=-(1/4)(z^4+a_3z^3+a_2z^2+a_1z+a_0)z^{-2}(z+4g^2)^{-2}\) is a rational function with coefficients \(a_k=a_k(\Delta ,E,g)\) depending on the parameters of the Rabi problem.\N\NThe authors give a complete classification of generic topological types of domain configurations and Stokes graphs of the quadratic differential \(Q_0(z)dz^2=-((z^4+c_3z^3+c_2z^2+c_1z+c_0)/(z^2-1)^2)dz^2\) with \(c_k\in \mathbb{R}\), assuming that the zeros are distinct from \(\pm 1\). They identify the set of coefficients \((c_3,c_2,c_1,c_0)\in \mathbb{R}^4\), with the particular choices of physical parameters \(\Delta\), \(g\) and \(E\) describing the Rabi model.\N\NThe asymptotic structure of Stokes graphs and domain configurations of quadratic differentials, when the Rabi parameters tend to infinity, is also discussed.