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

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 Schrödinger 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.

• The paper establishes a Floer theoretic framework that reinterprets energy eigenstates as periodic Hamiltonian trajectories.
• It extends Rabinowitz Floer homology to non-autonomous Hamiltonians, ensuring compactness and control through careful analytic bounds.
• The work rigorously proves the existence of energy eigenstates for ring and box configurations under broad conditions on external potentials.

Floer Theoretic Methods for Quantum Energy Eigenstates on One-Dimensional Configuration Spaces

Introduction and Motivation

The paper "A Floer Theoretic Approach to Energy Eigenstates on one Dimensional Configuration Spaces" (2603.29201) systematically develops a novel connection between the spectral theory of the quantum mechanical Schrödinger operator on low-dimensional systems and the machinery of symplectic topology, notably (time-dependent) Rabinowitz Floer homology (RFH). The author focuses on two paradigmatic quantum systems: the "particle on a ring" and the "particle in a box," each subjected to general external potentials. The main technical innovation is the extension of Rabinowitz Floer homology to non-autonomous Hamiltonians and the construction of moduli spaces of $J$-holomorphic curves in this context, which, in turn, enables rigorous existence results for energy eigenstates.

Reformulation of Quantum Eigenstates as Periodic Orbits

The standard approach to the spectral analysis of quantum systems involves direct operator-theoretic or PDE methods for the Schrödinger equation. Instead, this work interprets energy eigenstates $\psi$ solving $\hat{H}\psi = E\psi$ as position components of periodic orbits in an associated time-dependent Hamiltonian system on phase space. For configuration spaces given by $S^1$ (the ring) or an interval (the box), the quantum Hamiltonian is mapped to an infinite-dimensional geometric setting where classical symplectic tools become applicable.

Specifically, for a Hamiltonian operator

$$\hat{H} = -\frac{1}{R^2}\frac{d^2}{d\varphi^2} + V,$$

on $L^2(S^1, \mathbb{C})$ and a target energy $E$, the author constructs a Hamiltonian function on $\mathbb{R}^4$,

$$H(t, q, p) = \frac{\|p\|^2}{2} + \left(E - V\left(\frac{t}{R}\right)\right)\frac{\|q\|^2}{2} - c,$$

where $c > 0$ is a parameter, and $\psi$0 is extended periodically or restricted according to the system. Periodic orbits of this Hamiltonian correspond to energy eigenstates of the quantum system; this correspondence is made explicit and exploited throughout.

Extension and Construction of Rabinowitz Floer Homology for Non-Autonomous Hamiltonians

The canonical version of RFH necessitates autonomous Hamiltonians to ensure the existence of a well-defined energy hypersurface of contact type. In the present setting, the one-dimensional quantum systems with time-dependent potentials demand working with non-autonomous Hamiltonians. The paper provides a detailed construction ensuring compactness and regularity required for Floer-theoretic homology, building on the analytic framework pioneered in [cieliebak2009a], and extending it by employing average energy constraints and bounds on moduli spaces even in the absence of a global contact hypersurface.

A careful use of the action functional

$\psi$1

yields critical point equations for periodic (or Lagrangian boundary) orbits with the correct energy scaling. The analysis secures $\psi$2 bounds on the gradient flow lines (ensured via maximum principles) and uniform control of the Lagrange multiplier, crucial for obtaining compactness of the moduli space.

A key technical result is that in the case where the Hessian of $\psi$3 is everywhere positive-definite and diagonally dominant (as for time-dependent harmonic oscillators and their generalizations), the Conley-Zehnder index strictly increases with the period, ensuring transversality and control over the algebraic structure of the Floer chain complex.

Main Theorems: Existence of Energy Eigenstates for General Potentials

Two central theorems are established. Let $\psi$4 be a radially invariant potential, and let $\psi$5.

Ring Case: There exists a radius $\psi$6 such that on the circle $\psi$7, the spectral problem

$\psi$8

has a nontrivial solution, with $\psi$9 the restriction of $\hat{H}\psi = E\psi$0 to $\hat{H}\psi = E\psi$1.

Box Case: There exists a length $\hat{H}\psi = E\psi$2 such that on any line segment of that length, the eigenvalue problem

$\hat{H}\psi = E\psi$3

exhibits an energy-$\hat{H}\psi = E\psi$4 eigenstate, with $\hat{H}\psi = E\psi$5 restricted to the segment.

These are proved via contradiction using the vanishing of the (extended) RFH for the systems, which yields the existence (by Morse inequalities and index arguments) of a Hamiltonian trajectory with prescribed period (or boundary conditions) corresponding to an eigenstate.

Notably, these results hold for a broad class of external potentials and for all energies $\hat{H}\psi = E\psi$6 above the maximum of the potential: this claim is nontrivial and highlights the power of the topological method compared to spectral-theoretic techniques, especially in non-integrable or spatially non-homogeneous cases.

Lagrangian Boundary Conditions and Chord Existence

The treatment of the "particle in a box" uses the Lagrangian version of RFH, where boundary conditions are imposed on a Lagrangian subspace representing the endpoints of the interval. Analyses of Hamiltonian chords beginning and ending on this subspace, together with anti-symplectic symmetry considerations, allow reduction to the periodic orbit framework and establish index-doubling properties, robustly guaranteeing the existence of the required eigenstates.

Implications and Perspectives

The approach taken in this work elevates the connection between symplectic topology and quantum mechanics from analogy to rigorous machinery for existence theorems. Its implications are multifaceted:

• Spectral Theory: The ability to guarantee the existence of eigenstates for arbitrary high energies and broad classes of spatially arbitrary potentials using topological counts rather than analytic estimates opens a new perspective for studying the spectrum of Schrödinger operators on constrained domains.
• Symplectic Topology: The extension of RFH to non-autonomous Hamiltonians without contact-type hypersurfaces could be relevant well beyond quantum systems, for instance in time-dependent celestial mechanics or the study of forced Hamiltonian dynamics.
• Mathematical Physics: The method bridges geometric and analytic quantum approaches and has potential for generalization to higher-dimensional or more complex configuration spaces.
• Floer Theory: The methodology for moduli space compactness, action estimates, and index theory in the non-autonomous setting enriches the foundational toolbox for Hamiltonian Floer theory.

Conclusion

This paper establishes a geometrically motivated, Floer-theoretic framework for establishing the existence of energy eigenstates for quantum particles constrained to one-dimensional manifolds in external fields, realized for both ring and box topologies. Through analytic extension and algebraic analysis of Rabinowitz Floer homology to non-autonomous systems, the author delivers nontrivial existence results for energy eigenstates for a wide range of external potentials. The work solidifies the connection between symplectic topology and quantum spectral theory and provides a robust set of analytic and topological tools for future exploration in quantum Hamiltonian analysis and symplectic geometry (2603.29201).