# Modification of Schr\"odinger-Newton equation due to braneworld models with minimal length

## TL;DR

This paper investigates how braneworld models and generalized uncertainty principles modify the energy spectrum of a gravitational quantum well, suggesting a potential bridge between semiclassical and full quantum gravity theories.

## Contribution

It introduces a combined correction to the Schr"odinger-Newton equation from braneworld and minimal length effects, aligning theoretical predictions closer to experimental results.

## Key findings

- Energy spectrum corrections are closer to GRANIT experiment values.
- Combined effects suggest a possible intermediate quantum gravity theory.
- Proposes a schematic experimental setup for laboratory testing.

## Abstract

We study the correction of the energy spectrum of a gravitational quantum well due to the combined effect of the braneworld model with infinite extra dimensions and generalized uncertainty principle. The correction terms arise from a natural deformation of a semiclassical theory of quantum gravity governed by the Schr\"odinger-Newton equation based on a minimal length framework. The two fold correction in the energy yields new values of the spectrum, which are closer to the values obtained in the GRANIT experiment. This raises the possibility that the combined theory of the semiclassical quantum gravity and the generalized uncertainty principle may provide an intermediate theory between the semiclassical and the full theory of quantum gravity. We also prepare a schematic experimental set-up which may guide to the understanding of the phenomena in the laboratory.