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Completely integrable systems and groups generated by reflections.

E Gutkin¹, B Sutherland

¹Department of Mathematics, University of Utah, Salt Lake City, Utah 84112.

Proceedings of the National Academy of Sciences of the United States of America

|December 1, 1979

Summary

We present a new class of quantum systems with delta-function potentials. These systems are fully integrable, allowing for explicit solutions and new identities for reflection groups.

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Area of Science:

Quantum mechanics
Mathematical physics

Background:

Quantum Hamiltonian systems are fundamental to understanding particle behavior.
Systems with delta-function potentials model specific physical interactions.
Groups generated by reflections have applications in various mathematical and physical fields.

Purpose of the Study:

To introduce and analyze a novel class of quantum Hamiltonian systems.
To explore the connection between these systems and groups generated by reflections.
To demonstrate the complete integrability of these systems.

Main Methods:

Development of a theoretical framework for quantum systems with delta-function potentials.
Application of techniques for analyzing systems related to groups generated by reflections.

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Explicit integration of the Hamiltonian systems.

Main Results:

The introduced quantum Hamiltonian systems are shown to be completely integrable.
An explicit integration method for these systems is provided.
New identities for groups generated by reflections are derived using this technique.

Conclusions:

The study establishes a new integrable model in quantum mechanics.
The findings offer a method to derive mathematical identities from physical systems.
This work generalizes existing models of particles on a line.