In applications, many states given for a system can be expressed by orthonormal elements, called “state elements,” taken in a separable Hilbert space (called “state space”). The exact nature of the Hilbert space depends on the system; for example, the state space for position and momentum states is the space of square-integrable functions. The symmetries of a quantum system can be represented by a class of unitary operators that act in the Hilbert space. The operators called ladder operators have the effect of lowering or raising the energy of the state. In this paper, we study the spectral properties of a self-adjoint, fourth-order differential operator with a bounded operator coefficient and establish a second regularized trace formula for this operator.

中文翻译：

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四阶微分算子的第二正则化跟踪公式

在应用中，给定系统的许多状态可以用在可分离的希尔伯特空间（称为“状态空间”）中的正交元素（称为“状态元素”）表示。希尔伯特空间的确切性质取决于系统。例如，位置和动量状态的状态空间是平方可积函数的空间。量子系统的对称性可以由作用在希尔伯特空间中的一类operators算子来表示。称为梯形运算符的运算符具有降低或提高状态能量的作用。在本文中，我们研究了具有界算子系数的自伴伴四阶微分算子的光谱性质，并为此算子建立了第二个正则化的跟踪公式。