Quantum Tunneling and Complex Dynamics in the Suris's Integrable Map.

Area of Science:

• Quantum mechanics
• Mathematical physics
• Dynamical systems

Background:

• Quantum tunneling is a phenomenon where particles pass through potential barriers.
• Integrable maps offer a simplified model for studying quantum dynamics.
• Previous studies focused on tunneling splitting, with less attention to wave function tails.

Purpose of the Study:

• To investigate quantum tunneling in a two-dimensional integrable map.
• To compare tunneling behavior with associated one-dimensional Hamiltonian systems.
• To explore the origins of differences in wave function tunneling tails.

Main Methods:

• Analyzing orbits confined to curves defined by a one-dimensional Hamiltonian.
• Comparing tunneling splitting in the integrable map and Hamiltonian system.
• Superposing eigenfunctions to form doublets and examining wave function tails.
• Observing classical dynamics in the complex plane to identify potential influences.

Main Results:

• Tunneling splitting in the integrable map and Hamiltonian system are qualitatively similar.
• Significant differences are observed in the tunneling tails of wave functions.
• Classical dynamics in the complex plane reveal the role of branch points.
• Branch points in the potential function are linked to non-trivial tunneling tail behavior.

Conclusions:

• Quantum tunneling in integrable maps shares similarities with Hamiltonian systems regarding splitting.
• Wave function tunneling tails exhibit distinct behaviors not fully explained by real-plane dynamics.
• Complex plane dynamics, particularly branch points, are crucial for understanding subtle quantum tunneling effects.
• A comprehensive understanding of quantum tunneling requires considering dynamics beyond the real plane.