On eigenstates for some $sl_2$ related Hamiltonian

TL;DR

This paper analyzes the eigenstates of a specific $sl_2$-related Hamiltonian, proving it has a continuous spectrum and providing a detailed construction of its eigenstates.

Contribution

It introduces a new analysis of a self-conjugated $sl_2$ Hamiltonian and explicitly constructs its eigenstates, demonstrating the presence of a continuous spectrum.

Findings

The Hamiltonian has a continuous spectrum.

Explicit construction of eigenstates is provided.

The Hamiltonian is self-conjugated and related to $sl_2$ operators.

Abstract

In this paper we consider the stationary Schroedinger equation for a self-conjugated Hamiltonian $H = \frac{e+f}{i}$, where $e$ and $f$ is an anti-unitary pair of the canonical Cartan "creating" and "annihilation" operators for the classical Lie algebra $sl_2$ taken in the representation with "the lowest weight equals to $1$". In this paper we prove that this operator has the continuous spectrum. Construction of eigenstates for $H$ is given in details.