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We investigate operator growth in quantum systems with two-dimensionalSchr\"odinger group symmetry by studying the Krylov complexity. While feasiblefor semisimple Lie algebras, cases such as the Schr\"odinger algebra which ischaracterized by a semi-direct sum structure are complicated. We propose tocompute Krylov complexity for this algebra in a natural orthonormal basis,which produces a pentadiagonal structure of the time evolution operator,contrasting the usual tridiagonal Lanczos algorithm outcome. The resultingcomplexity behaves as expected. We advocate that this approach can provideinsights to other non-semisimple algebras.

Krylov complexity in a natural basis for the Schr\"odinger algebra