Kernel-based integral derivative operators with applications to quantum systems

Abstract We develop the analytical and numerical consequences of a class of multidimensional Lanczos-type integral operators that realize derivatives via compactly supported kernel integrations. The discrete operator induced by this construction is a finite-range convolution whose Fourier symbol reproduces − θ 2 at low frequencies while remaining uniformly bounded at high frequencies. This boundedness acts as an inherent low-pass filter, in sharp contrast with the O ( h − 2 ) growth of the standard finite-difference Laplacian. Two applications illustrate the impact of this spectral structure. For the disordered Schrödinger equation, the kernel-based Laplacian allows significantly larger time steps with greatly reduced norm drift. For a momentum-dephasing Lindblad model–where high-frequency noise is especially harmful–the operator preserves positivity across all tested scales, whereas the finite-difference discretization becomes unstable. These results show that multidimensional Lanczos-type kernels yield Laplacian discretizations that are consistent, compactly supported, and spectrally stable, providing a robust alternative for simulations of both closed and open quantum systems.