# Exact Classical and Quantum Solutions for a Covariant Oscillator Near the Black Hole Horizon in Stueckelberg-Horwitz-Piron Theory

## TL;DR

This paper derives exact classical and quantum solutions for a covariant harmonic oscillator near a black hole horizon within the Stueckelberg-Horwitz-Piron framework, revealing a mixed energy spectrum and orthonormal eigenfunctions.

## Contribution

It provides the first exact solutions for the classical and quantum dynamics of a covariant oscillator in black hole spacetime using the SHP theory.

## Key findings

- Existence of an orthonormal basis of eigenfunctions.
- The energy spectrum is mixed with a conserved momentum quantum number.
- Ground state energy obtained via variational method.

## Abstract

We found exact solutions for canonical classical and quantum dynamics for general relativity in Horwitz general covarience theory. These solutions can be obtained by solving the generalized geodesic equation and Schr\"{o}dinger-Stueckelberg -Horwitz-Piron (SHP) wave equation for a simple harmonic oscilator in the background of a two dimensional dilaton black hole spacetime metric. We proved the existence of an orthonormal basis of eigenfunctions for generalized wave equation. This basis functions form an orthogonanl and normalized (orthonormal) basis for an appropriate Hilbert space. The energy spectrum has a mixed spectrum with one conserved momentum $p$ according to a quantum number $n$. To find the ground state energy we used a variational method with appropriate boundary conditions. A set of mode decomposed wave functions and calculated for the Stueckelberg-Schrodinger equation on a general five dimensional blackhole spacetime in Hamilton gauge.