A Poincar\'e determinant on the torus

TL;DR

This paper introduces a Poincaré determinant for operators on the torus and uses it to prove the existence of nontrivial solutions for certain elliptic equations via Hill's method.

Contribution

It presents a new Poincaré determinant concept for torus operators and applies it to establish solutions for elliptic equations.

Findings

Defined a Poincaré determinant for torus operators

Proved existence of solutions for elliptic equations on the torus

Utilized Hill's method for the analysis

Abstract

In this work we introduce a Poincar\'e determinant type for operators on the torus $\To^n$. As an application we establish the existence of nontrivial solutions for elliptic equations of the form $(-\Delta)^{\frac{\nu}{2}}u+Qu=0$ on $\To^n$ by using the Hill's method.