We will obtain a formula for the trace of the Sturm–Liouville operator with operator coefficient.

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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.

In this paper, we obtain a second regularized trace formula on L2([0, π]; H) for a higher order self‐adjoint differential operator with unbounded operator‐valued coefficient, where H is a separable Hilbert space.