Polynomial changes of the initial state in Krylov complexity are solved exactly via Christoffel transforms of the spectral measure, yielding finite-band amplitude transfer and projected-kernel complexity formulas without re-running Lanczos.
Toda-Darboux maps and vertex operators
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abstract
The purpose of this paper is to study Toda-Darboux transforms, i.e., Darboux transforms for operators L(t) flowing according to the Toda lattice. Each element of the null-space $L(t)-z$ specifies a factorization for all t and thus a Toda-Darboux transform on $L(t)$. The Toda-Darboux map induces a transformation on the tau-vectors, given by a certain vertex operator, and on eigenfunctions, given by a Wronskian. .
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Polynomial Initial-State Jumps and Christoffel Transforms in Krylov Complexity
Polynomial changes of the initial state in Krylov complexity are solved exactly via Christoffel transforms of the spectral measure, yielding finite-band amplitude transfer and projected-kernel complexity formulas without re-running Lanczos.