Abstract

Linear systems of differential equations with an invariant in the form of a positive definite quadratic form in a real Hilbert space are considered. It is assumed that the system has a simple spectrum and the eigenvectors form a complete orthonormal system. Under these assumptions, the linear system can be represented in the form of the Schrödinger equation by introducing a suitable complex structure. As an example, we present such a representation for the Maxwell equations without currents. In view of these observations, the dynamics defined by some linear partial differential equations can be treated in terms of the basic principles and methods of quantum mechanics.

中文翻译：

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具有二次不变量作为Schrödinger方程的微分方程的线性系统

摘要

考虑在实希尔伯特空间中具有定正二次形式形式的不变量的微分方程组的线性系统。假设系统具有简单的频谱，特征向量形成完整的正交系统。在这些假设下，可以通过引入合适的复杂结构以Schrödinger方程的形式表示线性系统。作为一个例子，我们给出了没有电流的麦克斯韦方程的这种表示。鉴于这些观察，可以根据量子力学的基本原理和方法来处理由某些线性偏微分方程定义的动力学。