Phase Space Approach to Solving the Time-independent Schr"odinger Equation

We propose a method for solving the time independent Schr"odinger equation based on the von Neumann (vN) lattice of phase space Gaussians. By incorporating periodic boundary conditions into the vN lattice [F. Dimler et al., New J. Phys. 11, 105052 (2009)] we solve a longstanding problem of convergence of the vN method. This opens the door to tailoring quantum calculations to the underlying classical phase space structure while retaining the accuracy of the Fourier grid basis. The method has the potential to provide enormous numerical savings as the dimensionality increases. In the classical limit the method reaches the remarkable efficiency of 1 basis function per 1 eigenstate. We illustrate the method for a challenging two-dimensional potential where the FGH method breaks down.

💡 Research Summary

The paper introduces a novel numerical scheme for solving the time‑independent Schrödinger equation (TISE) that combines the advantages of the von Neumann (vN) phase‑space lattice with the robustness of a Fourier‑grid representation. Traditional Fourier‑grid Hamiltonian (FGH) methods employ a globally orthogonal basis of plane‑wave‑like functions; they are highly accurate but suffer from exponential growth of the basis size as the dimensionality of the problem increases. The vN approach, by contrast, uses a set of Gaussian wave packets placed on a regular grid in phase space (position × momentum). While this offers a natural connection to the underlying classical dynamics, the original vN basis lacks proper boundary conditions, leading to severe non‑orthogonality and poor convergence when applied to bounded quantum systems.

To overcome this long‑standing obstacle, the authors adopt the periodic‑boundary‑condition (PBC) construction proposed by Dimler et al. (2009). They embed the vN lattice inside a finite phase‑space box of length L and impose periodicity on both the position and momentum coordinates. Each basis function is defined as

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