arXiv subjects

Kevin Ruck

Publications and source records attributed to Kevin Ruck.

4 recordsLinked to original sources

arXiv2026

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

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.

math.SG↗

arXiv2025

Spatial Stark-Zeeman Systems and Their Regularizations

In this article, we study spatial Stark-Zeeman systems which describe the dynamics of a charged particle moving in three-dimensional space under the influence of a Coulomb potential, a magnetic field, and an electric field, possibly time-dependent. Such systems are modeled by Hamiltonian flows on the cotangent bundle of an open subset of $\mathbb{R}^3, $ equipped with a twisted symplectic structure. The presence of the Coulomb singularity leads to the study of collision orbits, and hence understanding the regularization of these orbits is essential for global dynamical properties. We investigate regularization techniques for spatial Stark-Zeeman systems, both in time-independent and time-dependent cases. In particular, in the time-dependent case, following a new regularization method developed by Barutello, Ortega, and Verzini, we formulate the corresponding regularized variational principles and carefully analyze the effects of magnetic and electric terms under the Kustaanheimo-Stiefel transformation. The resulting regularized action functional yields a variational characterization of collision orbits and facilitates further analysis of periodic solutions. Our results provide a general scheme for regularizing spatial Stark-Zeeman systems, opening the door for further applications in symplectic geometry, Floer theory, and celestial mechanics.

math.DS↗

arXiv2025

Consecutive collision orbits in the restricted three-body problem above the first critical energy value

In this paper, we study the planar circular restricted three-body problem for energy levels slightly above the first critical value. We first observe that the energy hypersurfaces in the Birkhoff regularization corresponding to these energy levels are of contact type. Then, using a version of Rabinowitz Floer homology, we establish the existence of either a periodic symmetric collision orbit or infinitely many symmetric consecutive collision orbits. Furthermore, by an analytic continuation argument, for generic mass ratios and energy levels, we prove that there is no periodic symmetric collision orbit with odd number of collisions. This in turn implies the existence of at least two symmetric consecutive collision orbits.

math.SG↗

arXiv2022

Tate Homology and Powered Flybys

In this paper we show that in the planar circular restricted three body problem there are either infinitely many symmetric consecutive collision orbits or at least one periodic symmetric consecutive collision orbit for all energies below the first critical energy value. Using Levi-Civita regularization allows us to distinguish two different kinds of symmetric consecutive collision orbits and prove the above claim for both of them separately, one corresponding to a solar eclipse and the other to a lunar eclipse. By interpreting the orbits as Hamiltonian chords between two different Lagrangian submanifolds we can use a perturbed version of $G$-equivariant Lagrangian Rabinowitz Floer homology to prove the existence of this kind of consecutive collision orbit. To calculate this homology we show that under certain conditions the $G$-equivariant Lagrangian Rabinowitz Floer homology is equal to the Tate homology of $G$.

math.SG↗