Analytic solutions of the quantum two-state problem in terms of the double-, bi- and tri-confluent Heun functions

TL;DR

This paper introduces new classes of exactly solvable two-state quantum models using advanced Heun functions, expanding the scope of solvable models beyond traditional hypergeometric functions.

Contribution

It generalizes known solvable quantum two-state models to four-parametric classes involving confluent Heun functions, enabling analysis of complex level-crossing and resonance processes.

Findings

Derived five classes solvable with double confluent Heun functions

Identified five classes solvable with biconfluent Heun functions

Presented an example for each of the three confluent Heun equations

Abstract

We derive five classes of quantum time-dependent two-state models solvable in terms of the double confluent Heun functions, five other classes solvable in terms of the biconfluent Heun functions, and a class solvable in terms of the triconfluent Heun functions. These classes generalize all the known families of two- or three-parametric models solvable in terms of the confluent hypergeometric functions to more general four-parametric classes involving three-parametric detuning modulation functions. The particular models derived describe different non-linear (parabolic, cubic, sinh, cosh, etc.) level-sweeping or level-glancing processes, double- or triple-level-crossing processes, as well as periodically repeated resonance-glancing or resonance-crossing processes. We show that more classes can be derived using the equations obeyed by certain functions involving the derivatives of the confluent Heun functions. We present an example of such a class for each of the three discussed confluent Heun equations.