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THU: Thursday Contributed Sessions

THU 10: Foundational / Mathematical Aspects – Methods and Approximations

THU 10.7: Talk

Thursday, September 11, 2025, 15:45–16:00, ZHG103

The perils of finite dimensional approximations — •Felix Fischer — FAU Erlangen, Staudtstr. 7 91058 Erlangen

When numerically simulating the unitary time evolution of an infinite-dimensional quantum system, one is usually led to treat the Hamiltonian H as an "infinite-dimensional matrix" by expressing it in some orthonormal basis of the Hilbert space, and then truncate it to some finite dimensions. However, the solutions of the Schrödinger equations generated by the truncated Hamiltonians need not converge, in general, to the solution of the Schrödinger equation corresponding to the actual Hamiltonian. In some cases, the approximate solutions do not converge to any valid state at all, whilst in others they converge to the dynamics generated by a "wrong" Hamiltonian different from the initial one. In this talk, I present multiple necessary and sufficient conditions for the convergence of finite dimensional approximations to the correct dynamics. Multiple examples from quantum chemistry and quantum optics illustrate the convergence issues which can appear in practice. Using our abstract results, I discuss why these issues arise and showcase how to ensure convergence to the correct dynamics we aim to simulate.

Keywords: Galerkin approximations; squeezing; Finite dimensional approximations; particle in a box; quantum chemistry