Solutions of the two-level problem in terms of biconfluent Heun functions (Q2766162)

In this paper a quantum mechanical two-level model \(U^*, \delta^*\) NEWLINE\[NEWLINE i\delta^*_z-\frac{U^*_z}{U^*}=2\frac{\varphi_z}{\varphi}+f,\quad U^{**}=\frac{\varphi_{zz}}{\varphi}+f\frac{\varphi_z}{\varphi}+g NEWLINE\]NEWLINE are investigated, where \(f\) and \(g\) are the coefficients of the biconfluent Heun equation. For this model five-four parametric classes are derived, permitting reduction of the initial problem to the biconfluent Heun equation. Three of these classes are generalizations of the well known classes of Landau-Zenger, Nikitin and Crothers. The authors show that two other classes describe super and sublinear and essentially nonlinear level crossings, as well as processes with three crossing points. For the essentially nonlinear cubic-crossing model, \(\delta_t\sim\delta_2t^3\), the general solution of the two-level problem as series of confluent hypergeometric functions is constructed.