The quantum dynamics of a complex system can be efficiently described in Krylov space, the minimal subspace in which the dynamics unfolds. We apply the Krylov subspace method to Hamiltonian deformations, which provides a systematic way to construct solvable models from known instances. In doing so, we relate the evolution of deformed and undeformed theories and investigate their complexity. For a certain class of deformations, the resulting Krylov subspace remains unchanged, and time evolution takes place in a reorganized basis. The tridiagonal form of the generator in the Krylov space is maintained, and we obtain generalized Toda equations as a function of the deformation parameters. The imaginary-time-like evolutions can be described by real-time unitary ones. As possible applications, we discuss coherent Gibbs states for thermodynamic systems, for which we analyze the survival probability, spread complexity, Krylov entropy, and associated time-averaged quantities. We further discuss the statistical properties of random matrices and supersymmetric systems for quadratic deformations.

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• Integrable systems
• Ising model
• Random matrix theory