Three-body problem in quantum mechanics: hyperspherical elliptic coordinates and harmonic basis sets

Area of Science:

• Quantum mechanics
• Theoretical chemistry
• Mathematical physics

Background:

• Elliptic coordinates in hyperspherical formalism were previously introduced for three-body problems.
• Applications have emerged in areas like chemical reaction theory.

Purpose of the Study:

• To explore the role of hyperspherical elliptic coordinates in bridging symmetric and asymmetric parametrizations.
• To focus on the properties and applications of hyperspherical elliptic harmonics.

Main Methods:

• Utilizing the hyperspherical formalism with elliptic coordinates.
• Defining and analyzing hyperspherical elliptic harmonics, which are products of associated Lame polynomials.
• Developing an expansion of these harmonics in terms of standard hyperspherical harmonics.

Main Results:

• Demonstrated the utility of hyperspherical elliptic coordinates for smooth transitions between parametrizations.
• Introduced hyperspherical elliptic harmonics involving associated Lame polynomials.
• Provided a method to expand these new harmonics using standard ones.

Conclusions:

• Hyperspherical elliptic harmonics offer a valuable new tool for quantum mechanical three-body problems.
• The expansion method facilitates future applications in scattering and bound-state calculations.
• This work advances the theoretical framework for analyzing complex few-body systems.