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Krylov complexity and orthogonal polynomials. (English) Zbl 1514.81117

Summary: 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.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81P68 Quantum computation
68Q12 Quantum algorithms and complexity in the theory of computing
81P55 Special bases (entangled, mutual unbiased, etc.)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
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