Krylov complexity and orthogonal polynomials

TL;DR

This paper explores Krylov complexity, connecting it to orthogonal polynomials, and provides a pedagogical overview with analytical examples involving classical and special orthogonal polynomials.

Contribution

It offers a comprehensive introduction to the mathematical framework of Krylov complexity and demonstrates its application through explicit examples with various orthogonal polynomials.

Findings

Krylov complexity can be described using orthogonal polynomials.

Analytical examples include classical, Hahn class, and Tricomi-Carlitz polynomials.

The paper clarifies the mathematical structure underlying operator growth.

Abstract

Krylov complexity measures operator growth with respect to a basis, which is adapted to the Heisenberg time evolution. The construction of that basis relies on the Lanczos algorithm, also known as the recursion method. The mathematics of Krylov complexity can be described in terms of orthogonal polynomials. We provide a pedagogical introduction to the subject and work out analytically a number of examples involving the classical orthogonal polynomials, polynomials of the Hahn class, and the Tricomi-Carlitz polynomials.