The integrating factor technique is widely used to solve numerically (in particular) the Schr\"odinger equation in the context of spectral methods. Here, we present an improvement of this method exploiting the freedom provided by the gauge condition of the potential. Optimal gauge conditions are derived considering the equation and the temporal numerical resolution with an adaptive embedded scheme of arbitrary order. We illustrate this approach with the nonlinear Schr\"odinger (NLS) and with the Schr\"odinger-Newton (SN) equations. We show that this optimization increases significantly the overall computational speed, sometimes by a factor five or more. This gain is crucial for long time simulations.

翻译：积分因子技术广泛应用于谱方法框架下对薛定谔方程进行数值求解。本文利用势能规范条件所提供的自由度，提出该方法的改进方案。通过考虑方程本身及采用任意阶自适应嵌入式格式的时间数值分辨率，推导出最优规范条件。我们以非线性薛定谔（NLS）方程和薛定谔-牛顿（SN）方程为例验证该方法。结果表明，该优化能显著提升整体计算速度，有时可提高五倍甚至更多。这一增益对于长时间模拟至关重要。