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arXiv:2406.06325

n Distinguishable Particles on the Real Line interacting via Two Body Delta Potentials

Moscato, Antonio

Published Jun 10, 2024• 0 citations
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Abstract

This paper studies a system of $n \in \mathbb{N}: \, n \geq 2$ non-relativistic, spinless quantum particles moving on the real line and interacting via a two-body delta potential. The Hamiltonian of such a system is proved to be affiliated to the resolvent algebra of the case, $\mathcal{R}\left( \mathbb{R}^{2n},σ\right)$; it is further shown the existence of a $\text{C}^{\ast}-$dynamical system and of a subalgebra $π_S\left( \mathfrak{S}_0 \right)^{-1} \subset \mathcal{R}\left( \mathbb{R}^{2n},σ\right)$, stable under time evolution, where $π_S$ is the Schr{ö}dinger representation of the algebra.

DOI
10.48550/arxiv.2406.06325

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This paper shows that the resolvent algebra R(R2,σ)\mathcal{R}\left( \mathbb{R}^2,σ\right)R(R2,σ) can accommodate dynamics induced by self-adjoint Hamiltonians on L2(R)L^2\left( \mathbb{R} \right)L2(R) describing a single non-relativistic spinless particle undergoing one up to countably many different fixed point interactions located on the real line.
Abstract We study the dynamics of non-relativistic fermions in Rd\mathbb {R}^dRd R d interacting through a pair potential. Employing methods developed by Buchholz in the framework of resolvent algebras, we consider an extension of the CAR algebra where the dynamics acts as a group of ∗*∗ ∗ -automorphisms, which are continuous in time in all sectors for fixed particle numbers. In addition, we construct a C∗C^*C∗ C ∗ -dynamical system by identifying a suitable dense subalgebra. Finally, we briefly discuss how this framework could be used to construct KMS states in the future.
This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.
In this paper we will report on a one-dimensional, non-separable quantum many-particle system.It consists of two (distinguishable) particles moving on the half-line R + being subjected to two different kinds of two-particle interactions: singular many-particle interactions localized at the origin and a binding-potential leading to a molecular-like state.We will formulate the model precisely, obtaining a well-defined self-adjoint operator (the Hamiltonian for our system) and elaborate on its spectral properties.In addition, we will present possible directions for future research.
Time evolution of relativistic particle systems can be accomplished by means of a Schrödinger wave equation, provided the Hamiltonian of the N particle system satisfies some commutator equations involving the other generators of a suitable representation of the Poincaré group on the initial data space of the Schrödinger equation. This set of operator equations is solved in the N free particle case. Further, the structure of N interacting particle Hamiltonians is worked out to include second-order relativistic corrections, i.e., (1/c2)-terms. It is shown that Breit and Barker–O’Connell Hamiltonians (nonzero spin particles) and the Bažański Hamiltonian (zero-spin particles) are particular cases of this more general Hamiltonian.
There is no relativistic Hamiltonian for many particles systems except for free particles and this has been accepted since the 1960s from the work of Currie, Jordan and Sudarshan, Cannon and Jordan, and Leutwyler. This is the problem we will address in this paper by coupling relativistic particles with a scalar field. The Hamiltonian we derive is explicitly relativistic and in the non-relativistic limit gives a Hamiltonian with two body interaction. However, we find that the Poincare algebra is still not satisfied because of the inherent time delayed interaction in relativistic particle systems.