A Second Regularized Trace Formula for a Fourth Order Differential Operator

Abstract

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.

Second regularized trace of a differential operator of Sturm–Liouville type

“…Trace formulas for differential operators with operator coefficients are included in the works of Khalilova [27], Adıgüzelov [28], Adıgüzelov et al [29], Gül [30–33], Bayramov et al [34], Badalova [35], Karayel and Sezer [36], and Gül and Ceyhan [37]. The second regular trace formula of the Sturm–Liouville operator with inverse periodic boundary conditions and scalar coefficient is given in the study Akgun et al [38].…”

“…The calculation of the second regularized trace formula of the Sturm–Liouville operator with operator coefficient introduced in Adıgüzelov et al [29] where the second regularized trace formula of the Sturm–Liouville operator with the bounded operator coefficient given by Dirichlet boundary conditions is given. In Gül and Ceyhan [37], second regularized trace formulas were found for a fourth‐order differential operator with a bounded operator coefficient given with symmetric boundary conditions, and in Gül [32], second regularized trace formula are found for a high‐order differential operator with unbounded operator coefficient given with symmetric boundary conditions. In addition, in Gül and Gill [39, 40], the first regularized trace formulas for differential operators with bounded and unbounded operator coefficients, respectively, are calculated on a separable Banach space.…”