A Floer Theoretic Approach to Energy Eigenstates on one Dimensional Configuration Spaces

TL;DR

This paper applies Floer homology from symplectic topology to analyze quantum energy eigenstates for particles on a ring and in a box, extending Rabinowitz Floer homology to non-autonomous Hamiltonians.

Contribution

It extends Rabinowitz Floer homology to non-autonomous Hamiltonians and uses it to prove existence of energy eigenstates in quantum systems with complex potentials.

Findings

Proved compactness of moduli space of J-holomorphic curves for extended Floer homology.

Established existence of energy eigenstates for a wide range of potentials.

Linked quantum problems to symplectic topology methods.

Abstract

In this article we consider two classical problems in Quantum Mechanics, namely the 'particle on a ring' and the 'particle in a box' from the viewpoint of symplectic topology. Interpreting the solutions of the corresponding time independent Schr\"odinger equation as orbits in a suitably chosen time dependent Hamiltonian system allows us to investigate them using Floer theory. More precisely we extend the definition of Rabinowitz Floer homology to non-autonomous Hamiltonians on $\mathbb{R}^{2n}$ with its standard symplectic structure and show that compactness of the moduli space of J-holomorphic curves still holds. With this homology we are then able to prove existence results for energy $E$ eigenstates on the 'ring' or in the 'box' for a big range of exterior potentials.