# Spectral inequality for Dirac right triangles

## TL;DR

This paper investigates the spectral properties of the Dirac operator on right triangles with infinite-mass boundary conditions, proposing a conjecture about the minimization of the lowest positive eigenvalue and proving it under certain geometric assumptions.

## Contribution

The paper formulates a conjecture about eigenvalue minimization for Dirac operators on right triangles and provides a proof under specific geometric conditions, extending previous work on Dirac rectangles.

## Key findings

- Conjecture that the isosceles right triangle minimizes the lowest positive eigenvalue.
- Proof of the conjecture under additional geometric hypotheses.
- Extension of methods from Dirac rectangles to right triangles.

## Abstract

We consider the Dirac operator on right triangles, subject to infinite-mass boundary conditions. We conjecture that the lowest positive eigenvalue is minimised by the isosceles right triangle both under the area or perimeter constraints. We prove this conjecture under extra geometric hypotheses relying on a recent approach of Ph. Briet and D. Krej{\v{c}}i{\v{r}}{\'i}k for Dirac rectangles [2].