The sine kernel, two corresponding operator identities, and random matrices

TL;DR

This paper explores the sine kernel integral operator, deriving operator identities and differential systems, and analyzes their asymptotics using operator methods and random matrix theory.

Contribution

It introduces new operator identities and differential systems associated with the sine kernel, and studies their asymptotic behavior using combined analytical approaches.

Findings

Derived two operator identities for the sine kernel operator.

Established asymptotics of the resolvent and Hamiltonians.

Connected operator identities with random matrix theory results.

Abstract

In the present paper, we consider the integral operator, which acts in Hilbert space and has sine kernel. This operator generates two operator identities and two corresponding canonical differential systems. We find the asymptotics of the corresponding resolvent and Hamiltonians. We use both the method of operator identities and the theory of random matrices.