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

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.

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Cited by in corpus (16)

• Exact solution of the Schrödinger equation for the inverse square root potential $V_0/\sqrt{x}$
• Solutions of the bi-confluent Heun equation in terms of the Hermite functions
• Schrödinger potentials solvable in terms of the general Heun functions
• Discretization of Natanzon potentials
• The Lambert-W step-potential - an exactly solvable confluent hypergeometric potential
• A conditionally exactly solvable generalization of the inverse square root potential
• The third exactly solvable hypergeometric quantum-mechanical potential
• Exact solutions of the sextic oscillator from the bi-confluent Heun equation
• A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions
• Four five-parametric and five four-parametric independent confluent Heun potentials for the stationary Klein-Gordon equation
• Analytical results for a parity-time symmetric two-level system under synchronous combined modulations
• Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory
• Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions
• A periodic level-crossing two-state model of a general Heun class
• Analytical Results for the Dynamics of Parabolic Level-Crossing Model
• Inverse square root level-crossing quantum two-state model