A Second Regularized Trace Formula for a Fourth Order Differential Operator

Erdal Gül,Aylan Ceyhan

doi:10.3390/sym13040629

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.

Highlights

We introduced and computed a new second regularized trace formula for a fourthorder differential operator with a bounded operator coefficient defined on a separable
The regularized trace can be computed on a separable Banach space, which is a continuous dense embedding in a separable Hilbert space [23]
The trace formulae of these operators are used in many branches of mathematics, mathematical physics and quantum mechanics

Summary

Introduction

The regularized trace formula of a scalar differential operator was first introduced by I. Several works on the regularized traces of scalar differential operators appeared (see [4,5,6,7,8,9]). A second regularized trace formula was obtained in [19] for the Sturm–Liouville operator with the antiperiodic boundary conditions. Let σ1 (H) denote the space of trace-class operators from H to H [21]. Denote by trA the sum of the eigenvalues of a trace-class operator A [22] and the notation “·” stands for the product. Belonging to the interval [m4 − 1/2, m4 + 1/2], and C is a constant depending on Q(t) This formula is said to be a second regularized trace formula of the operator L

Some Relations about Eigenvalues and Resolvents

The Second Regularized Trace Formula

Conclusions